This document outlines a talk on the use of 1 in mathematical programming and summarizes several topics to be covered, including degeneracy and the simplex method, nonlinear optimization, equality constraints, inequality constraints, and data envelopment analysis. It provides details on linear programming and the simplex method, including preliminary results, basic feasible solutions, the single iteration process, and the lexicographic rule for selecting leaving variables. The document contains mathematical notation and definitions to explain these concepts.

1. The use of ① in Mathematical Programming
R. De Leone
School of Science and Tecnology
Universit `a di Camerino
June 2013
NUMTA2013 1 / 46

2. Outline of the talk
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Degeneracy and the Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment Analysis
NUMTA2013 2 / 46

3. Degeneracy and the Simplex
Method
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
NUMTA2013 3 / 46

4. Linear Programming and the Simplex Method
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
min
x
cT x
subject to Ax = b
x ≥ 0
The simplex method proposed by George Dantzig in 1947
• start at a corner point (a Basic Feasible Solution, BFS)
• verify if the current point is optimal
• if not, moves along an edge to a new corner point
until the optimal corner point is identified or it discovers that the
problem has no solution.
NUMTA2013 4 / 46

5. Preliminary results and notations
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Let
X = {x ∈ IRn : Ax = b, x ≥ 0}
where A ∈ IRm×n, b ∈ IRm, m ≤ n.
A point ¯x ∈ X is a Basic Feasible Solution (BFS) iff the columns of
A corresponding to positive components of ¯x are linearly
independent.
NUMTA2013 5 / 46

6. Preliminary results and notations
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Let
X = {x ∈ IRn : Ax = b, x ≥ 0}
where A ∈ IRm×n, b ∈ IRm, m ≤ n.
A point ¯x ∈ X is a Basic Feasible Solution (BFS) iff the columns of
A corresponding to positive components of ¯x are linearly
independent.
Let ¯x be a BFS and define ¯I = I(¯x) := {j : ¯xj > 0} then
rank(A.¯I ) = |¯I|. Note: |¯I| ≤ m
NUMTA2013 5 / 46

7. Preliminary results and notations
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Let
X = {x ∈ IRn : Ax = b, x ≥ 0}
where A ∈ IRm×n, b ∈ IRm, m ≤ n.
A point ¯x ∈ X is a Basic Feasible Solution (BFS) iff the columns of
A corresponding to positive components of ¯x are linearly
independent.
Let ¯x be a BFS and define ¯I = I(¯x) := {j : ¯xj > 0} then
rank(A.¯I ) = |¯I|. Note: |¯I| ≤ m
BFS ≡ Vertex ≡ Extreme Point
Vertex Point, Extreme Points and Basic Fea-sible
Solution Point coincide
NUMTA2013 5 / 46

8. BFS and associated basis
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Assume now
rank(A) = m ≤ n
A base B is a subset of m linearly independent columns of A.
B ⊆ {1, . . . , n} , det(A.B)6= 0
N = {1, . . . , n} − B
Let ¯x be a BFS. .
NUMTA2013 6 / 46

9. BFS and associated basis
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Assume now
rank(A) = m ≤ n
A base B is a subset of m linearly independent columns of A.
B ⊆ {1, . . . , n} , det(A.B)6= 0
N = {1, . . . , n} − B
Let ¯x be a BFS. .
non-degenerate BFS
If |{j : ¯xj > 0}| = m the BFS is said to be non–
degenerate and there is only a single base B :=
{j : ¯xj > 0} associated to ¯x
NUMTA2013 6 / 46

10. BFS and associated basis
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Assume now
rank(A) = m ≤ n
A base B is a subset of m linearly independent columns of A.
B ⊆ {1, . . . , n} , det(A.B)6= 0
N = {1, . . . , n} − B
Let ¯x be a BFS. .
degenerate BFS
If |{j : ¯xj > 0}| < m the BFS is said to be degener-ate
and there are more than one base B1,B2, . . . ,Bl
associated to ¯x with {j : ¯xj > 0} ⊆ Bi
NUMTA2013 6 / 46

11. BFS and associated basis
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Assume now
rank(A) = m ≤ n
A base B is a subset of m linearly independent columns of A.
B ⊆ {1, . . . , n} , det(A.B)6= 0
N = {1, . . . , n} − B
Let ¯x be a BFS. Let B a base associated to ¯x.
Then
¯xN = 0, ¯xB = A−1
.B b ≥ 0
NUMTA2013 6 / 46

12. BFS and associated basis
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Assume now
rank(A) = m ≤ n
A base B is a subset of m linearly independent columns of A.
B ⊆ {1, . . . , n} , det(A.B)6= 0
N = {1, . . . , n} − B
Let ¯x be a BFS. Let B a base associated to ¯x.
Any feasible point x in X can be expressed in term of the base B as
follows:
xB = A−1
.B b + A−1
.B A.NxN
with xN ≥ 0 (and xB ≥ 0)
NUMTA2013 6 / 46

13. Single iteration of the simplex method
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Step 0 Let B ⊆ {1, . . . , n} be the current base and let x ∈ X the
current BFS
xB = A−1
.B b ≥ 0, xN = 0, |B| = m.
Assume
B = {j1, j2, . . . , jm}
and
N = {1, . . . , n} − B = {jm+1, . . . , jn} .
NUMTA2013 7 / 46

14. Single iteration of the simplex method
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Step 1 Compute
π = A−T
.B cB
and the reduced cost vector
¯cjk = cjk − A.jk
T π, k = m + 1, . . . , n.
Step 2 If
¯cjk ≥ 0, ∀k = m + 1, . . . , n
the currect point is an optimal BFS and the algorithm stops.
Instead if ¯cN6≥ 0 choose jr with r ∈ {m + 1, . . . , n}) with
¯cjr < 0. This is the variable candidate to enter the base.
NUMTA2013 7 / 46

15. Single iteration of the simplex method
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Step 3 Compute
A¯.jr = A−1
.B A.jr
Step 4 If
A¯.jr ≤ 0
the problem is unbounded below and the algorithm stops.
Otherwise, compute
¯ρ = min
i=1,...m
¯A
ijr
>0
(
A−1
i
.B b
A¯ijr
)
and let s ∈ {1, . . . ,m} such that
(A−1
.B b)s
A¯sjr
= ¯ρ
js is the leaving variable.
NUMTA2013 7 / 46

16. Single iteration of the simplex method
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Step 5 Define
¯xjk = 0, k = m + 1, . . . , n, k6= r
¯xjr = ¯ρ
¯xB(ρ) = A−1
.B b − ρ¯A¯.jr .
and
¯B
= B − {js} ∪ {jr} = {j1, j2, . . . , js−1, jr, js+1, . . . , jm}
NUMTA2013 7 / 46

17. Lexicographic Rule
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
At each iteration of the simplex method we choose the leaving
variable using the lexicographic rule
NUMTA2013 8 / 46

18. Lexicographic Rule
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Let B0 be the initial base and N0 = {1, . . . , n} − B0.
We can always assume, after columns reordering, that A has the
form
A =
A.Bo
...
A.No
NUMTA2013 8 / 46

19. Lexicographic Rule
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Let
¯ρ = min
i:A¯ijr0
(A.−1
B b)i
A¯ijr
if such minimum value is reached in only one index this is the leaving
variable.
OTHERWISE
NUMTA2013 8 / 46

20. Lexicographic Rule
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Among the indices i for which
min
i:A¯ijr0
(A.−1
B b)i
A¯ijr
= ¯ρ
we choose the index for which
min
i:A¯ijr0
(A.−1
B A.Bo)i1
A¯ijr
If the minimum is reached by only one index this is the leaving
variable.
OTHERWISE
NUMTA2013 8 / 46

21. Lexicographic Rule
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Among the indices reaching the minimum value, choose the index for
which
min
i:A¯ijr0
(A.−1
B A.Bo)i2
A¯ijr
Proceed in the same way.
This procedure will terminate providing a single index: the rows of
the matrix (A.−1
B A.Bo) are linearly independent.
NUMTA2013 8 / 46

22. Lexicographic rule and RHS perturbation
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
The procedure outlined in the previous slides is equivalent to perturb
each component of the RHS vector b by a very small quantity.
If this perturbation is small enough, the new linear programming
problem is nondegerate and the simplex method produces exactly
the same pivot sequence as the lexicographic pivot rule
However, is very difficult to determine how small this perturbation
must be. More often a symbolic perturbation is used (with higher
computational costs)
NUMTA2013 9 / 46

23. Lexicographic rule and RHS perturbation and ①
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Replace bi witheb
i with
bi +
X
j∈Bo
Aij①−j .
NUMTA2013 10 / 46

24. Lexicographic rule and RHS perturbation and ①
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Replace bi witheb
i with
bi +
X
j∈Bo
Aij①−j .
Let
e =


①−1
①−2
...
①−m


and
eb
= A.−1
B (b + A.Boe) = A.−1
B b + A.−1
B A.Boe.
NUMTA2013 10 / 46

25. Lexicographic rule and RHS perturbation and ①
• Outline of the talk
Degeneracy and the
Simplex Method
• Linear Programming
and the Simplex Method
• Preliminary results
and notations
• BFS and associated
basis
• Single iteration of the
simplex method
• Lexicographic Rule
• Lexicographic rule
and RHS perturbation
• Lexicographic rule
and RHS perturbation
and ①
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
Replace bi witheb
i with
bi +
X
j∈Bo
Aij①−j .
Thereforeeb= (A.−1
B b)i +
Xm
k=1
(A.−1
B A.Bo)ik①−k
and
min
i:A¯ijr0
(A.−1
B b)i +
Xm
k=1
(A.−1
B A.Bo)ik①−k
A¯ijr
=
min
i:A¯ijr0
(A.−1
B b)i
A¯ijr
+
(A.−1
B A.Bo)i1
A¯ijr
①−1+. . .+
(A.−1
B A.Bo)im
A¯ijr
①−m
NUMTA2013 10 / 46

26. Nonlinear Optimization
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
NUMTA2013 11 / 46

27. Equality Constraint
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
NUMTA2013 12 / 46

28. The Equality Constraint Nonlinear Problem
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
min
x
f(x)
subject to h(x) = 0
where f : IRn → IR and h : IRn → IRk
L(x, π) := f(x) +
Xk
j=1
πjhj(x) = f(x) + πT h(x)
NUMTA2013 13 / 46

29. First Order Optimality Conditions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Let
F(x∗) :=
n
d ∈ Rn : ∇hi(x∗)T d = 0
o
d ∈ TX(x∗) ⇐⇒ ∃{xl}l feasible points with {xl}l → x∗ and
{tl}l real positive number with {tl}l → 0 such that
liml
xl − x∗
tl
= d
NUMTA2013 14 / 46

30. First Order Optimality Conditions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Constraints Qualifications
The set of tangent directions TX(x∗) coincides with the
set of “linearized” feasible directions F(x∗).
Regularity Conditions
NUMTA2013 14 / 46

31. First Order Optimality Conditions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Let x∗ ∈ IRn and assume that rank(∇hi(x∗)) = k (i.e., the
columns are linearly independent)
If x∗ is a local minimizer then
NUMTA2013 14 / 46

32. First Order Optimality Conditions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Let x∗ ∈ IRn and assume that rank(∇hi(x∗)) = k (i.e., the
columns are linearly independent)
If x∗ is a local minimizer then
there exists π∗ ∈ IRk such that
∇xL(x∗, π∗) = ∇f(x∗) +
Xk
j=1
∇hj(x∗)π∗j = 0
∇L(x∗, π∗) = h(x∗) = 0
NUMTA2013 14 / 46

33. First Order Optimality Conditions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Let x∗ ∈ IRn and assume that rank(∇hi(x∗)) = k (i.e., the
columns are linearly independent)
If x∗ is a local minimizer then
there exists π∗ ∈ IRk such that
∇xL(x∗, π∗) = ∇f(x∗) + ∇h(x∗)T π∗ = 0
∇L(x∗, π∗) = h(x∗) = 0
NUMTA2013 14 / 46

34. First Order Optimality Conditions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Let x∗ ∈ IRn and assume that rank(∇hi(x∗)) = k (i.e., the
columns are linearly independent)
If x∗ is a local minimizer then
there exists π∗ ∈ IRk such that
∇xL(x∗, π∗) = ∇f(x∗) + ∇h(x∗)T π∗ = 0
∇L(x∗, π∗) = h(x∗) = 0
KKT (Karush–Kuhn–Tucker) Conditions
NUMTA2013 14 / 46

35. Penalty and barrier functions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
A penalty function P : IRn → IR satisfies the following condition
P(x)
= 0 if x belongs to the feasible region
0 otherwise
NUMTA2013 15 / 46

36. Penalty and barrier functions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
A penalty function P : IRn → IR satisfies the following condition
P(x)
= 0 if x belongs to the feasible region
0 otherwise
Examples of penalty functions
P(x) =
Xk
j=1
|hj(x)|
P(x) =
Xk
j=1
h2j
(x)
NUMTA2013 15 / 46

37. Exactness of a Penalty Function
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
The optimal solution of the constrained problem
min
x
f(x)
subject to h(x) = 0
can be obtained by solving the following unconstrained minimization
problem
min f(x) +
1
σ
P(x)
for sufficiently small but fixed σ 0.
NUMTA2013 16 / 46

38. Exactness of a Penalty Function
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
The optimal solution of the constrained problem
min
x
f(x)
subject to h(x) = 0
can be obtained by solving the following unconstrained minimization
problem
min f(x) +
1
σ
P(x)
for sufficiently small but fixed σ 0.
P(x) =
Xk
j=1
|hj(x)|
NUMTA2013 16 / 46

39. Exactness of a Penalty Function
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
The optimal solution of the constrained problem
min
x
f(x)
subject to h(x) = 0
can be obtained by solving the following unconstrained minimization
problem
min f(x) +
1
σ
P(x)
for sufficiently small but fixed σ 0.
P(x) =
Xk
j=1
|hj(x)|
Non–smooth function!
NUMTA2013 16 / 46

40. Sequential Penalty method
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
min
x −x1 − x2
subject to x11
+ x22
− 1 = 0
The unique solution is x∗ =
1/√2
1/√2
and π∗ = 1/√2.
NUMTA2013 17 / 46

41. Sequential Penalty method
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
min
x −x1 − x2
subject to x11
+ x22
− 1 = 0
The unique solution is x∗ =
1/√2
1/√2
and π∗ = 1/√2.
In fact
∇f(x) =
−1
−1
, ∇h1(x)
2x1
2x2
∇f(x∗) + π∗∇h1(x∗) =
−1
−1
+
1
√2
2/√2
2/√2
= 0
NUMTA2013 17 / 46

42. Sequential Penalty method
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
min
x −x1 − x2
subject to x11
+ x22
− 1 = 0
The unique solution is x∗ =
1/√2
1/√2
and π∗ = 1/√2.
For no finite values of σ the solution of
min−x1 − x2 +
1
2σ
x11
+ x22
− 1
2
is also the solution of the constrained problem.
NUMTA2013 17 / 46

43. Sequential Penalty method
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Let {σl} ↓ 0 and P(x) =
Xk
j=1
h2j
(x)
Step 0 Set l = 0
Step 1 Let x(σl) be an optimal solution of the unconstrained
differentiable problem
min f(x) +
1
σl
P(x)
Step 2 Set l = l + 1 and return to Step 1
NUMTA2013 18 / 46

44. Convergence Results
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Assume that
• f(x) be bounded below in the (nonempty) feasible region,
• {σl} be a monotonic non-increasing sequence such that
{σlk} ↓ 0,
• for each l there exists a global minimum x(σl) of
f(x) + 1
l
P(x) =: φ(x, σl).
NUMTA2013 19 / 46

45. Convergence Results
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Assume that
• f(x) be bounded below in the (nonempty) feasible region,
• {σl} be a monotonic non-increasing sequence such that
{σlk} ↓ 0,
• for each l there exists a global minimum x(σl) of
f(x) + 1
l
P(x) =: φ(x, σl).
Then
n
φ(x(σl); σl)
o
is monotonically non–decreasing
n
P(x(σl))
o
is monotonically non–increasing
f (x (σl)) is monotonically non–decreasing
NUMTA2013 19 / 46

46. Convergence Results
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Assume that
• f(x) be bounded below in the (nonempty) feasible region,
• {σl} be a monotonic non-increasing sequence such that
{σlk} ↓ 0,
• for each l there exists a global minimum x(σl) of
f(x) + 1
l
P(x) =: φ(x, σl).
Moreover, liml h (x (σl)) = 0 and each limit point x∗ of the
sequence
n
(x (σl))
o
l
solves the nonlinear constrained problem.
NUMTA2013 19 / 46

47. Convergence Results
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Assume that
• f(x) be bounded below in the (nonempty) feasible region,
• {σl} be a monotonic non-increasing sequence such that
{σlk} ↓ 0,
• for each l there exists a global minimum x(σl) of
f(x) + 1
l
P(x) =: φ(x, σl).
Moreover, liml h (x (σl)) = 0 and each limit point x∗ of the
sequence
n
(x (σl))
o
l
solves the nonlinear constrained problem.
If x∗ is regular limit point of
n
(x (σl))
o
l
then (x∗, π∗) satisfy
KKT–conditions where 1
l
h (x (σl)) → π∗.
NUMTA2013 19 / 46

48. Introducing ①
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Let
P(x) =
Xk
j=1
h2j
(x)
Solve
min f(x) + ①P(x)φ (x,①)
NUMTA2013 20 / 46

49. Assumptions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
In the sequel we assume that
x = x0 + ①−1x1 + ①−2x2 + . . .
with xi ∈ IRn
NUMTA2013 21 / 46

50. Assumptions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
In the sequel we assume that
f(x) = f(x0) + ①−1f(1)(x) + ①−2f(2)(x) + . . .
h(x) = h(x0) + ①−1h(1)(x) + ①−2h(2)(x) + . . .
where f(i) : IRn → IR, h(i) : IRn → IRk are all finite–value functions.
NUMTA2013 21 / 46

51. Assumptions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
In the sequel we assume that
∇f(x) = ∇f(x0) + ①−1F(1)(x) + ①−2F(2)(x) + . . .
∇h(x) = ∇h(x0) + ①−1H(1)(x) + ①−2H(2)(x) + . . .
where F(i) : IRn → IRn, H(i) : IRn → IRk×n are all finite–value
functions.
NUMTA2013 21 / 46

52. Assumptions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Previous conditions are satisfied (for example) by functions that are
product of polynomial functions in a single variable, i.e.,
p(x) = p1(x1)p2(x2) · · · pn(xn)
where pi(xi) is a polynomial function.
NUMTA2013 21 / 46

53. Convergence Results for Equality Constrained Problems
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
min
x
f(x)
subject to h(x) = 0
(1)
NUMTA2013 22 / 46

54. Convergence Results for Equality Constrained Problems
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
min
x
f(x)
subject to h(x) = 0
(1)
min
x
f(x) +
1
2
①kh(x)k2 (2)
NUMTA2013 22 / 46

55. Convergence Results for Equality Constrained Problems
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
min
x
f(x)
subject to h(x) = 0
(1)
min
x
f(x) +
1
2
①kh(x)k2 (2)
Let
x∗ = x∗0 + ①−1x∗1 + ①−2x∗2 + . . .
be a stationary point for (2) and assume that the LICQ condition
holds true at x∗0.
NUMTA2013 22 / 46

56. Convergence Results for Equality Constrained Problems
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
min
x
f(x)
subject to h(x) = 0
(1)
min
x
f(x) +
1
2
①kh(x)k2 (2)
Let
x∗ = x∗0 + ①−1x∗1 + ①−2x∗2 + . . .
be a stationary point for (2) and assume that the LICQ condition
holds true at x∗0.
Then, the pair
x∗0, π∗ = h(1)(x∗)
is a KKT point of (1).
NUMTA2013 22 / 46

57. Proof
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Since x∗ = x∗0 +①−1x∗1 +①−2x∗2 +. . . is a stationary point we
have
∇f(x∗) + ①
Xk
j=1
∇hj(x∗)hj(x∗) = 0
NUMTA2013 23 / 46

58. Proof
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Since x∗ = x∗0 +①−1x∗1 +①−2x∗2 +. . . is a stationary point we
have
∇f(x∗) + ①
Xk
j=1
∇hj(x∗)hj(x∗) = 0
∇f(x0) + ①−1F(1)(x) + ①−2F(2)(x) + . . .+
+①
Xk
j=1
∇hj(x0) + ①−1H(1)
j (x) + ①−2H(2)
j (x) + . . .
hj(x0) + ①−1h(1)
j (x) + ①−2h(2)
j (x) + . . .
= 0
NUMTA2013 23 / 46

59. Proof
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Xk
①
j=1
!
∇hj(x∗0)hj(x∗0)+
∇f(x∗0) +
+
Xk
j=1
∇hj(x∗0)h(1)
j (x∗) +
Xk
j=1
H(1)
j (x∗)hj(x∗0)
!
+①1
. . .
!
+ ①2
. . .
!
+ ....
NUMTA2013 23 / 46

60. Proof
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Xk
①
j=1
∇hj(x∗0)hj(x∗0)
!
Assuming LICQ we obtain
h(x∗0) = 0
NUMTA2013 23 / 46

61. Proof
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
∇f(x∗0) +
Xk
j=1
∇hj(x∗0)h(1)
j (x∗) +
Xk
j=1
H(1)
j (x∗)hj(x∗0) = 0
⇓
∇f(x∗0) +
Xk
j=1
∇hj(x∗0)h(1)
j (x∗) = 0
NUMTA2013 23 / 46

62. Example 1
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
min
x
1
2x21
+ 1
6x22
subject to x1 + x2 = 1
The pair (x∗, π∗) with x∗ =


1
4
3
4

, π∗ = −1
4 is a KKT point.
NUMTA2013 24 / 46

63. Example 1
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
min
x
1
2x21
+ 1
6x22
subject to x1 + x2 = 1
The pair (x∗, π∗) with x∗ =


1
4
3
4

, π∗ = −1
4 is a KKT point.
f(x) + ①P(x) =
1
2
x21
+
1
6
x22
−
1
2
①(1 − x1 − x2)2
NUMTA2013 24 / 46

64. Example 1
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
f(x) + ①P(x) =
1
2
x21
+
1
6
x22
−
1
2
①(1 − x1 − x2)2
First Order Optimality Condition
x1 − ①(1 − x1 − x2) = 0
1
3x2 − ①(1 − x1 − x2) = 0
NUMTA2013 24 / 46

65. Example 1
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
f(x) + ①P(x) =
1
2
x21
+
1
6
x22
−
1
2
①(1 − x1 − x2)2
x∗1
=
1①
1 + 4①
, x∗2
=
3①
1 + 4①
NUMTA2013 24 / 46

66. Example 1
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
f(x) + ①P(x) =
1
2
x21
+
1
6
x22
−
1
2
①(1 − x1 − x2)2
x∗1
=
1①
1 + 4①
, x∗2
=
3①
1 + 4①
x∗1
=
1
4 − ①−1(
1
16 −
1
64
①−1 . . .)
x∗2=
3
4 − ①−1(
3
16 −
3
64
①−1 . . .)
NUMTA2013 24 / 46

67. Example 1
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
f(x) + ①P(x) =
1
2
x21
+
1
6
x22
−
1
2
①(1 − x1 − x2)2
x∗1
=
1①
1 + 4①
, x∗2
=
3①
1 + 4①
−①(1 − x
2) = −①
1 − x
1 −
1
4
+ ①−1 1
16 −
1
64
①−2 . . .
−
3
4
+ ①−1 3
16 −
3
64
①−2 . . .
= −①
①−1 1
16
+ ①−1 3
16 − ①−2 4
64
. . .
= −
1
4
+
4
64
①−1 . . .
and h(1)(x∗) = −1
4 = π∗
NUMTA2013 24 / 46

68. Example 2
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
min x1 + x2
subject to x21
+ x22
− 2 = 0
L(x, π) = x1 + x2 + π
x21
+ x22
− 2
The optimal solution is x∗ =
−1
−1
and
the pair
x∗, π∗ = 1
2
satisfies the KKT conditions.
NUMTA2013 25 / 46

69. Example 2
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
φ (x,①) = x1 + x2 +
①
2
x21
+ x22
− 2
2
First–Order Optimality Conditions


x1 + 2①x1
x21
+ x22
− 2
2 = 0
x2 + 2①x2
x21
+ x22
− 2
2 = 0
The solution is given by


8 + ①−2C
x1 = −1 − ①−1 1
x2 = −1 − ①−1 1
8 + ①−2C
NUMTA2013 25 / 46

70. Example 2
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
−1−
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
In fact
2①x1 = −2① −
1
4
+ 2①−1C
x21
+ x22
− 2 = 1 +
1
64
①−2 + ①−4C2 1
4
①−1 − 2①−2 −
1
4
①−3C +
1 +
1
64
①−2 + ①−4C2 1
4
①−1 − 2①−2 −
1
4
①−3C
=
1
2
①−1 +
1
32 − 4C
①−2 +
−
1
2
C
①−3 +
−2C2
①−4
−1 −
h
2①x1
ih
x21
+ x22
− 2
i
=
h
−2①−
1
4
+2①−1C
ih 1
2
①−1+
1
32 − 4C
①−2+
−
1
2
C
①−3+
−2C2
①−4
i
=
−1 + 2
1
2
+ (. . .)①−1 + (. . .)①−2 + . . .
NUMTA2013 25 / 46

71. Example 2
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
−1−
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
and
2①x2 = −2① −
1
4
+ 2①−1C
x21
+ x22
− 2 = 1 +
1
64
①−2 + ①−4C2 1
4
①−1 − 2①−2 −
1
4
①−3C +
1 +
1
64
①−2 + ①−4C2 1
4
①−1 − 2①−2 −
1
4
①−3C
=
1
2
①−1 +
1
32 − 4C
①−2 +
−
1
2
C
①−3 +
−2C2
①−4
−1 −
h
2①x2
ih
x21
+ x22
− 2
i
=
h
−2①−
1
4
+2①−1C
ih 1
2
①−1+
1
32 − 4C
①−2+
−
1
2
C
①−3+
−2C2
①−4
i
=
−1 + 2
1
2
+ (. . .)①−1 + (. . .)①−2 + . . .
NUMTA2013 25 / 46

72. Example 2
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
• The Equality
Constraint Nonlinear
Problem
• First Order Optimality
Conditions
• Penalty and barrier
functions
• Exactness of a
Penalty Function
• Sequential Penalty
method
• Sequential Penalty
method
• Convergence Results
• Introducing ①
• Assumptions
• Convergence Results
for Equality Constrained
Problems
• Proof
• Example 1
• Example 2
Inequality Constraint
Data Envelopment
Analysis
Finally,
h
x21
①
+x22
−2
i
= ①
ih1
2
①−1+
1
32 − 4C
①−2+
−
1
2
C
①−3+
NUMTA2013 25 / 46
−2
1
2
+ (. . .)①−1 + (. . .)①−2 + . . .

73. Inequality Constraint
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
NUMTA2013 26 / 46

74. Inequality Constraints
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min
x
f(x)
subject to g(x) ≤ 0
h(x) = 0
where f : IRn → IR, g : IRn → IRm h : IRn → IRk.
L(x, π, μ) := f(x) +
Xm
i=1
μigi(x) +
Xk
j=1
πjhj(x)
= f(x) + μT g(x) + πT h(x)
NUMTA2013 27 / 46

75. First Order Optimality Conditions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
Let x∗ ∈ IRn with
n
∇gi(x∗), i : gi(x∗) = 0,∇hj(x∗), j = 1, . . . , k
o
linearly independent
NUMTA2013 28 / 46

76. First Order Optimality Conditions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
Let x∗ ∈ IRn with
n
∇gi(x∗), i : gi(x∗) = 0,∇hj(x∗), j = 1, . . . , k
o
linearly independent
If x∗ is a local minimizer then
NUMTA2013 28 / 46

77. First Order Optimality Conditions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
Let x∗ ∈ IRn with
n
∇gi(x∗), i : gi(x∗) = 0,∇hj(x∗), j = 1, . . . , k
o
linearly independent
If x∗ is a local minimizer then
there exists μ∗ ∈ IRm+
, π∗ ∈ IRk such that
∇xL(x∗, μ∗, π∗) = ∇f(x∗) +
Xk
j=1
∇hj(x∗)π∗j = 0
∇μL(x∗, μ∗, π∗) = g(x∗) ≤ 0
∇L(x∗, μ∗, π∗) = h(x∗) = 0
μ∗ ≥ 0
μ∗T∇L(x∗, μ∗, π∗) = 0
NUMTA2013 28 / 46

78. Additional Assumptions
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
In addition to the conditions stated before we similarly require that
the following conditions hold:
g(x) = g(x0) + ①−1g(1)(x) + ①−2g(2)(x) + . . .
∇g(x) = ∇g(x0) + ①−1G(1)(x) + ①−2G(2)(x) + . . .
where g(i) : IRn → IRm and G(i) : IRn →→ IRm×n are all
finite–value functions
NUMTA2013 29 / 46

79. Modified LICQ condition
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
Let x0 ∈ IRn. The Modified LICQ (MLICQ) condition is said to hold
true at x0 if the vectors
n
∇gi(x0), i : gi(x0) ≥ 0,∇hj(x0), j = 1, . . . , k
o
are linearly independent.
NUMTA2013 30 / 46

80. Convergence Results for Inequality Constrained Problems
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min
x
f(x)
subject to g(x) ≤ 0
h(x) = 0
min
x
f(x) +
①
2 kmax{0, gi(x)}k2 +
①
2 kh(x)k2
x∗ = x∗0 + ①−1x∗1 + ①−2x∗2 + . . .
⇓ (MLICQ)
NUMTA2013 31 / 46

81. Convergence Results for Inequality Constrained Problems
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min
x
f(x)
subject to g(x) ≤ 0
h(x) = 0
min
x
f(x) +
①
2 kmax{0, gi(x)}k2 +
①
2 kh(x)k2
x∗ = x∗0 + ①−1x∗1 + ①−2x∗2 + . . .
⇓ (MLICQ)
KKT–point
x0, μ = g(1)(x), π = h(1)(x)
NUMTA2013 31 / 46

82. Proof
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
∇f(x) + ①
mX
i=1
∇gi(x)max {0, gi(x)} + ①
Xp
j=1
∇hj (x)hj (x) = 0.
−−∇f(x0) + ①1F(1)(x) + ①2F(2)(x) + . . .+
mX
+①
i=1
∇gi(x0) + ①−1G
(1)
i (x) + ①−2G
(2)
i (x) + . . .
max
n
0, gi(x0) + ①−1g
(1)
i (x) + ①−2g
o#
(2)
i (x) + . . .
+
+①
Xp
j=1
∇hj (x0) + ①−1H
(1)
j (x) + ①−2H
(2)
j (x) + . . .
hj (x0) + ①−1h
(1)
j (x) + ①−2h
#
(2)
j (x) + . . .
= 0
NUMTA2013 32 / 46

83. Proof, cont.
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
(1)
i (x) + ①−2g
gi(x0) 0 ⇒ max {0, gi(x)} = gi(x0) + ①−1g
(2)
i (x) + . . .
gi(x0) 0 ⇒ max {0, gi(x)} = 0
gi(x0) = 0 ⇒ max {0, gi(x)} = ①−1 max
n
0, g
(1)
i (x) + ①−1g
o
(2)
i (x) + . . .
NUMTA2013 33 / 46

84. Proof, cont.
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
−−∇f(x0) + ①1F(1)(x) + ①2F(2)(x) + . . .+
mX
+①
i=1
gi(x0)0
∇gi(x0) + ①−1G
(1)
i (x) + ①−2G
(2)
i (x) + . . .
gi(x0) + ①−1g
(1)
i (x) + ①−2g
#
(2)
i (x) + . . .
+
+①
mX
i=1
gi(x0)=0
∇gi(x0) + ①−1G
(1)
i (x) + ①−2G
(2)
i (x) + . . .
①−1 max
n
0, g
(1)
i (x) + ①−1g
o#
(2)
i (x) + . . .
+
+①
Xp
j=1
∇hj (x0) + ①−1H
(1)
j (x) + ①−2H
(2)
j (x) + . . .
hj (x0) + ①−1h
(1)
j (x) + ①−2h
(2)
j (x) + . . .
#
= 0
NUMTA2013 33 / 46

85. Proof, cont.
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
Xm
i=1
gi(x0)≥0
∇gi(x∗0)gi(x∗0) +
Xp
j=1
∇hj(x∗0)hj(x∗0) = 0
NUMTA2013 33 / 46

86. Proof, cont.
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
Xm
i=1
gi(x0)≥0
∇gi(x∗0)gi(x∗0) +
Xp
j=1
∇hj(x∗0)hj(x∗0) = 0
and hence from MLICQ
g(x∗0) ≤ 0 and h(x∗0) = 0
NUMTA2013 33 / 46

87. Example 3
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min
x∈IR
x
subject to x ≥ 1
The solution is ¯x = 1 with associated multiplier μ∗ = 1.
NUMTA2013 34 / 46

88. Example 3
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min
x∈IR
x
subject to x ≥ 1
The solution is ¯x = 1 with associated multiplier μ∗ = 1.
min
x∈IR
x +
①
2
max {0, 1 − x}2
NUMTA2013 34 / 46

89. Example 3
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min
x∈IR
x
subject to x ≥ 1
The solution is ¯x = 1 with associated multiplier μ∗ = 1.
min
x∈IR
x +
①
2
max {0, 1 − x}2
1 − ①max {0, 1 − x} = 0
For x 1 the only solution is
x∗ =
① − 1
①
= 1 − ①−1
NUMTA2013 34 / 46

90. Example 3
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min
x∈IR
x
subject to x ≥ 1
The solution is ¯x = 1 with associated multiplier μ∗ = 1.
min
x∈IR
x +
①
2
max {0, 1 − x}2
1 − ①max {0, 1 − x} = 0
For x 1 the only solution is
x∗ =
① − 1
①
= 1 − ①−1
Therefore x∗0 = 1.
Moreover, g(x∗) = 1 −
1 − ①−1
= ①−1 and μ∗ = 1
NUMTA2013 34 / 46

91. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min x1 + x2
subject to x21
+ x22
− 2 ≤ 0
−x2 ≤ 0
NUMTA2013 35 / 46

92. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min x1 + x2
subject to x21
+ x22
− 2 ≤ 0
−x2 ≤ 0
L(x, π) = x1 + x2 + μ1
x21
+ x22
− 2
− μ2x2
The solution is x∗ =
−√2
0
and (x+, μ∗) with
μ∗ =
1/2√2
0
satisfies KKT conditions.
∇f(x) =
1
1
, ∇g1(x) =
2x1
2x2
, ∇g2(x) =
0
−1
1
1
+
1
2√2
−2√2
0
+ 1
0
−1
= 0
NUMTA2013 35 / 46

93. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min x1 + x2
subject to x21
+ x22
− 2 ≤ 0
−x2 ≤ 0
φ(x,①) = x1+x2+
①
2
max
0, x21
+ x22
− 2
2
+
①
2
max {0,−x2}2
NUMTA2013 35 / 46

94. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
φ(x,①) = x1+x2+
①
2
max
0, x21
+ x22
− 2
2
+
①
2
max {0,−x2}2
NUMTA2013 35 / 46

95. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
φ(x,①) = x1+x2+
①
2
max
0, x21
+ x22
− 2
2
+
①
2
max {0,−x2}2


1 + 2x1①max
0, x21
+ x22
− 2
= 0
1 + 2x2①max
0, x21
+ x22
− 2
− ①max {0,−x2} = 0
NUMTA2013 35 / 46

96. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
φ(x,①) = x1+x2+
①
2
max
0, x21
+ x22
− 2
2
+
①
2
max {0,−x2}2


1 + 2x1①max
0, x21
+ x22
− 2
= 0
1 + 2x2①max
0, x21
+ x22
− 2
− ①max {0,−x2} = 0


x∗1
= −√2 + A①−1 + B①−2 + . . .
x∗2
= 0 + C①−1 + D①−2 + . . .
NUMTA2013 35 / 46

97. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
2x∗1
① = −2√2① + 2A + 2B①−1 + · · ·
(x∗1
)2 + (x∗2
)2 − 2 =
h
−
√2+A①−1+B①−2+· · ·
i2
+
h
C①−1+D①−2+· · ·
i2
−2 =
2+A2①−2+B2①−4−2√2A①−1−2√2B①−2+2AB①−3+· · ·+
C2①−2 + D2①−4 + 2CD①−3 + h
· · · − 2 =
−2√2A①−1 +
A2 2√− 2B + C2
i
①−2 + · · ·
NUMTA2013 35 / 46

98. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
1 + 2x∗1
①
(x∗1
)2 + (x∗2
)2 − 2
=
1 +
h
−2√2① + 2A + 2B①−1 + · · ·
i
−2√2A①−1 +
h
A2 − 2√2B + C2
i
①−2 + · · ·
#
=
1 +
−2√2
−2√2
A +
h
· · ·
i
①−1 +
h
· · ·
i
①−2 + · · ·
A = −
1
8
NUMTA2013 35 / 46

99. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
2x∗2
① = 2C + 2D①−1 + · · ·
1 + 2x∗2
①
(x∗1
)2 + (x∗2
)2 − 2
− x∗2
① =
1 +
h
2C + 2D①−1 + · · ·
i
−2√2A①−1+
h
A2 − 2√2B + C2
i
①−2+· · ·
i
−
h
−C①−1−D①−2+· · ·
#
① =
1 + C +
h
· · ·
i
①−1 +
h
· · ·
i
①−2 + · · ·
C = −1
NUMTA2013 35 / 46

100. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
x∗1
= −√2 1
− 8
①−1 + B①−2 + · · ·
x∗2
= 0 − ①−1 + D①−2 + · · ·
NUMTA2013 35 / 46

101. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
x∗1
= −√2 1
− 8
①−1 + B①−2 + · · ·
x∗2
= 0 − ①−1 + D①−2 + · · ·
①h1(x∗) = ①((x∗1
)2 + (x∗2
)2 − 2) =
+2√2
①
1
8
①−1 +
1
64 − 2√2B + C2
①−2 + · · ·
#
μ∗1= 2√2
1
8
=
1
2√2
NUMTA2013 35 / 46

102. Example 4
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
x∗1
= −√2 1
− 8
①−1 + B①−2 + · · ·
x∗2
= 0 − ①−1 + D①−2 + · · ·
①h2(x∗) = ①(−x∗2
)
①
−①−1 − D①−2 + · · ·
#
μ∗2
= 1
NUMTA2013 35 / 46

103. Example 5
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min 1
2
3
x1 − 2
2
+ 1
2
1
x2 − 2
4
subject to x1 + x2 − 1 ≤ 0
x1 − x2 − 1 ≤ 0
−x1 + x2 − 1 ≤ 0
−x1 − x2 − 1 ≤ 0
The solution is x∗ =
1
0
and (x+, μ∗) with μ∗ =


3/8
1/8
0
0


satisfies KKT conditions.
NUMTA2013 36 / 46

104. Example 5
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
(x,①) =
1
2
x1 −
3
2
2
+
1
2
x2 −
1
2
4
+
①
2
(
max{0, x1 + x2 − 1}2+
max{0, x1 − x2 − 1}2 + max{0,−x1 + x2 − 1}2 + max{0,−x1 − x2 − 1}2
)
NUMTA2013 36 / 46

105. Example 5
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
(x,①) =
1
2
x1 −
3
2
2
+
1
2
x2 −
1
2
4
+
①
2
(
max{0, x1 + x2 − 1}2+
max{0, x1 − x2 − 1}2 + max{0,−x1 + x2 − 1}2 + max{0,−x1 − x2 − 1}2
)


x1 − 3
2
max{0, x1 + x2 − 1}+
+ ①
max{0, x1 − x2 − 1} − max{0,−x1 + x2 − 1} − max{0,−x1 − x2 − 1}
#
= 0
2
x2 − 1
2
3
+ ①
max{0, x1 + x2 − 1}−
max{0, x1 − x2 − 1} + max{0,−x1 + x2 − 1} − max{0,−x1 − x2 − 1}
#
= 0
NUMTA2013 36 / 46

106. Example 5
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis


x∗1
= 1 + 1
4
①−1 + · · ·
x∗2
= 1
8
①−1 + · · ·
2 − 1 = 1 + 1
x
1 + x
4
①−1 + 1
8
−①1 + 3
· · · = 8
①−1 + · · · 0
2 − 1 = 1 + 1
x
1 − x
4
−①1 1
− 8
−①1 + 1
· · · = 8
①−1 + · · · 0
−x
2 − 1 = −1 − 1
1 + x
4
①−1 + 1
8
①−1 + · · · 0
2 − 1 = −1 − 1
1 − x
−x
4
−①1 1
− 8
①−1 + · · · 0
NUMTA2013 36 / 46

107. Example 5
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis


x∗1
= 1 + 1
4
①−1 + · · ·
x∗2
= 1
8
①−1 + · · ·
2 − 1 = 1 + 1
x
1 + x
4
①−1 + 1
8
−①1 + 3
· · · = 8
①−1 + · · · 0
2 − 1 = 1 + 1
x
1 − x
4
−①1 1
− 8
−①1 + 1
· · · = 8
①−1 + · · · 0
−x
2 − 1 = −1 − 1
1 + x
4
①−1 + 1
8
①−1 + · · · 0
2 − 1 = −1 − 1
1 − x
−x
4
−①1 1
− 8
①−1 + · · · 0
x∗1
−
3
2
+ ①
h3
8
①−1 +
1
8
①−1 + · · ·
i
=
1+
1
4
①−1+· · ·−
3
2
h3
8
+①
①−1+
1
8
①−1+· · ·
i
= 0+
h
· · ·
i
①−1+· · ·
NUMTA2013 36 / 46

108. Example 5
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis


x∗1
= 1 + 1
4
①−1 + · · ·
x∗2
= 1
8
①−1 + · · ·
2 − 1 = 1 + 1
x
1 + x
4
①−1 + 1
8
−①1 + 3
· · · = 8
①−1 + · · · 0
2 − 1 = 1 + 1
x
1 − x
4
−①1 1
− 8
−①1 + 1
· · · = 8
①−1 + · · · 0
−x
2 − 1 = −1 − 1
1 + x
4
①−1 + 1
8
①−1 + · · · 0
2 − 1 = −1 − 1
1 − x
−x
4
−①1 1
− 8
①−1 + · · · 0
2
x
2 −
1
2
3
h 3
8
+ ①
①−1 +
1
8
①−1 + · · ·
i
=
2
h 1
8
①−1 + · · · −
1
2
i
−
3
2
h 3
8
+ ①
①−1 +
1
8
①−1 + · · ·
i
=
2
−
1
2
3
+
h
· · ·
i
①−1+· · ·−
1
2
i
−
3
2
h 3
8
+①
①−1+
1
8
①−1+· · ·
i
= 0+
h
· · ·
i
①−1
NUMTA2013 36 / 46

109. Example 5
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis


x∗1
= 1 + 1
4
①−1 + · · ·
x∗2
= 1
8
①−1 + · · ·
2 − 1 = 1 + 1
x
1 + x
4
①−1 + 1
8
−①1 + 3
· · · = 8
①−1 + · · · 0
2 − 1 = 1 + 1
x
1 − x
4
−①1 1
− 8
−①1 + 1
· · · = 8
①−1 + · · · 0
−x
2 − 1 = −1 − 1
1 + x
4
①−1 + 1
8
①−1 + · · · 0
2 − 1 = −1 − 1
1 − x
−x
4
−①1 1
− 8
①−1 + · · · 0
x∗1
+ x∗2
− 1 =
3
8
①−1 + · · · =⇒ μ∗1
=
3
8
x∗1
− x∗2
− 1 =
1
8
①−1 + · · · =⇒ μ∗1
=
1
8
NUMTA2013 36 / 46

110. Example 6
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
min x1 + x2
subject to
x21
+ x22
− 2
2 = 0
L(x, π) = x1 + x2 + π
x21
+ x22
− 2
2
The optimal solution is x∗ =
−1
−1
and
the pair
x∗, π∗ = 1
2
satisfies the KKT conditions.
NUMTA2013 37 / 46

111. Example 6
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis
φ (x,①) = x1 + x2 +
①
2
x21
+ x22
− 2
4
First–Order Optimality Conditions

 x1 + 4①x1
x21
+ x22
− 2
3 = 0
x2 + 4①x2
x21
+ x22
− 2
3 = 0
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112. Example 6
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis


x1 = −1 1
− 8
①−1 + ①−2C
x2 = −1 1
− 8
①−1 + ①−2C
1 + 4①x∗1
h
x21
+ x22
− 2
i3
=
1 +
h
−4① −
1
2
① + 2①−1C
i
1
2
①−1 +
h
· · ·
i
①−2 + · · ·
#3
=
1 +
h
−4① −
1
2
① + 2①−1C
i
①−3
h
· · ·
i
6= 0
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113. Example 6
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
• Inequality Constraints
• First Order Optimality
Conditions
• Additional
Assumptions
• Modified LICQ
condition
• Convergence Results
for Inequality
Constrained Problems
• Proof
• Proof, cont.
• Example 3
• Example 4
• Example 5
• Example 6
Data Envelopment
Analysis


x1 = A + B①−1 + C①−2
x2 = D + E①−1 + F①−2
1 + 4①x∗1
h
x21
+ x22
− 2
i3
=
1 +
h
4A① + 4B + 4C①−1
i
R +
h
· · ·
i
①−1 + · · · + · · ·
#3
=
where R = A2 + B2 − 2. If R = 0 there is still a term multiplying
①. If R = 0, a term ①−3 can be factored out. The only possibility to
eliminate the term multiplying ① is A = 0. Spurious solution!
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114. Data Envelopment Analysis
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
• Problem Data
• First DEA model
• CCR primal model
• CCR dual model
• Assurance interval for
ǫ
• CCR dual model
• Conclusions (?)
•
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115. Problem Data
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
• Problem Data
• First DEA model
• CCR primal model
• CCR dual model
• Assurance interval for
ǫ
• CCR dual model
• Conclusions (?)
•
n Decision Making Unit (DMUs)
j = 1, . . . , n
(
Inputj = {xj
i , i = 1, . . . ,m}
Outputj = {yj
r, r = 1, . . . , s}
Effk(π, σ) =
Xs
r=1
σryk
r
Xm
i=1
πixk
i
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116. First DEA model
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
• Problem Data
• First DEA model
• CCR primal model
• CCR dual model
• Assurance interval for
ǫ
• CCR dual model
• Conclusions (?)
•
DMUk
max
,
σT yk
πT xk
subject to
σT yj
πT xj ≤ 1 j = 1, . . . , n
π ≥ 0, σ ≥ 0
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117. First DEA model
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
• Problem Data
• First DEA model
• CCR primal model
• CCR dual model
• Assurance interval for
ǫ
• CCR dual model
• Conclusions (?)
•
DMUk
max
,
σT yk
πT xk
subject to
σT yj
πT xj ≤ 1 j = 1, . . . , n
π ≥ 0, σ ≥ 0
Input Oriented: reduce input as much as possible while keeping at
least the present level of outputs
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118. First DEA model
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
• Problem Data
• First DEA model
• CCR primal model
• CCR dual model
• Assurance interval for
ǫ
• CCR dual model
• Conclusions (?)
•
DMUk
max
,
σT yk
πT xk
subject to
σT yj
πT xj ≤ 1 j = 1, . . . , n
π ≥ 0, σ ≥ 0
Output Oriented: increase output level as much as possible under at
most the present level of input consumption
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119. CCR primal model
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
• Problem Data
• First DEA model
• CCR primal model
• CCR dual model
• Assurance interval for
ǫ
• CCR dual model
• Conclusions (?)
•
Constant Return to Scale
Input Oriented
max
u,v
vT yk
subject to −uT xj + vT yj ≤ 0
j = 1, . . . , n
uT xk = 1
π ≥ ǫ, σ ≥ ǫ
where ǫ is an non-Archimedean infinitesimal.
Charnes, Cooper, Rhodes (1978)
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120. CCR dual model
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
• Problem Data
• First DEA model
• CCR primal model
• CCR dual model
• Assurance interval for
ǫ
• CCR dual model
• Conclusions (?)
•
Constant Return to Scale
Input Oriented
min
,,s,s−
θ − ǫ
eT s∗ + eT s−
subject to
Xn
j=1
xjλj + s∗ = θxk
Xn
j=1
yjλj − s− = yk
λ ≥ 0, s∗ ≥ 0, s− ≥ 0
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121. Assurance interval for ǫ
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
• Problem Data
• First DEA model
• CCR primal model
• CCR dual model
• Assurance interval for
ǫ
• CCR dual model
• Conclusions (?)
•
Acceptability intervals for ǫ can be obtained by solving n linear
programs (n is the number of DMUs).
B. Daneshian, G. R. Jahanshahloo et al, Mathematical and
Computational Applications, 2005
A polynomial-time algorithm for finding in DEA models has been
proposed by Gholam R. Amin and Mehdi Toloo (Computers
Operations Research,2004)
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122. CCR dual model
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
• Problem Data
• First DEA model
• CCR primal model
• CCR dual model
• Assurance interval for
ǫ
• CCR dual model
• Conclusions (?)
•
Constant Return to Scale
Input Oriented
min
,,s,s−
θ − ①−1
eT s∗ + eT s−
subject to
Xn
j=1
xjλj + s∗ = θxk
Xn
j=1
yjλj − s− = yk
λ ≥ 0, s∗ ≥ 0, s− ≥ 0
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123. Conclusions (?)
• Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
• Problem Data
• First DEA model
• CCR primal model
• CCR dual model
• Assurance interval for
ǫ
• CCR dual model
• Conclusions (?)
•
• The use of ① is extremely beneficial in various aspects in Linear
and Nonlinear Optimization
• Difficult problems in NLP can be approached in a simpler way
using ①
• A new convergence theory for standard algorithms (gradient,
Newton’s, Quasi-Newton) needs to be developed in theis new
framework
NUMTA2013 45 / 46

124. • Outline of the talk
Degeneracy and the
Simplex Method
Nonlinear Optimization
Equality Constraint
Inequality Constraint
Data Envelopment
Analysis
• Problem Data
• First DEA model
• CCR primal model
• CCR dual model
• Assurance interval for
ǫ
• CCR dual model
• Conclusions (?)
•
Thanks for your attention
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