My Session on Mathematical Optimisation Fundamentals and Industry applications for the Academic Knowledge Refresher Program organised by Kerala Technology University and College of Engineering Trivandrum, Department of Interdisciplinary Studies.

1. MATHEMATICAL OPTIMISATION
AN INTRODUCTION TO THE INDUSTRIAL APPICATIONS OF

2. KNOWN AS
NON-LINEAR PROGRAMMING
OR NUMERICAL OPTIMISATION
OVERVIEW

3. INTRODUCTION
MATHEMATICAL OPTIMISATION IN THE REAL WORLD
• Manufacturing
• Production
• Inventory Control
• Transportation
• Scheduling
• Networks
• Finance
• Control Engineering
• Marketing
• Policy Modelling

4. A NARRATIVE
MATHEMATICAL OPTIMISATION IN THE REAL WORLD
• Optimising time in the production
cycle of an industry, optimising tax in
a tax return, optimising length in a
tour are mathematical optimisation
problems we encounter in our daily
life.
• A solution that satisfying all the
constraints is called a feasible
solution
• The set of all solutions, satisfying all
the constraints is called the feasible
region

5. A SNAPSHOT
HISTORY
• 1940s
• Linear Programming
• 1950s
• Quadratic Programming
• 1960s
• Geometric Programming
• 1990s
• Semidefinite Programming
• Quadratically Constrained Quadratic Programming
• Robust Optimizaton
• Sum of Squares Programming

6. A QUICK SURVEY
NEW APPLICATIONS SINCE 1990
• Linear matrix inequality techniques in control
• Circuit design via geometric programming
• Support vector machine learning via
quadratic programming
• Semidefinite programming relaxations in
combinatorial optimisation
• L1- norm optimisation for sparse signal
reconstruction
• Applications in quantum information theory,
computer vision, image processing, finance

7. MATHEMATICAL OPTIMISATION
MANY OTHER KINDS …
• Linear Network Optimization
• Specialization of LP to problems with
graph structure ( shortest path,
transportation, trans-shipment, etc.)
• Integer Programming
• Decision variables are allowed only
integer values
• Combinatorial optimisation
• Decision variables have nice
combinatorial structures ( e.g. trees,
permutations, matchings )

8. DEEPER INSIGHTS
MATHEMATICAL OPTIMISATION : SPECIAL CASES
• Optimal solution need not be unique
• One of the special case is when
variables have symmetry.
• In this case, some kind of
permutation can be applied to get
multiple optimal solution

9. MATHEMATICAL OPTIMISATION
ESSENTIAL NOTIONS
• Design Variables
• Objective Functions
• Inequality constraint functions
• Equality constraint functions
• Optimum Vector

10. WHAT IS
MATHEMATICAL
OPTIMISATION
FORMALLY, MATHEMATICAL
OPTIMISATION IS THE PROCESS OF
FORMULATION AND SOLUTION OF
A CONSTRAINED OPTIMISATION PROBLEM
OF THE GENERAL MATHEMATICAL FORM:
MINIMIZE F(X), X = [X1, X2, X3, …. ] T E R N
SUBJECT TO THE CONSTRAINTS
G(X) < 0 , J = 1, 2, … , M
H(X) = 0 , J = 1, 2, …, R
WHERE F(X), G(X), AND H(X) ARE SCALAR
FUNCTIONS OF THE REAL COLUMN VECTOR
X.

11. MATHEMATICAL OPTIMISATION
OBJECTIVE AND CONSTRAINT FUNCTIONS
• The values of the functions f(x), g(x),
and h(x) at any point x = [x1, x2,
…xn] may in practice be obtained in
different ways
• From analytically known formulae
• as the outcome of some
computational process, g(x) = a(x)
- amax
• From measurement taken from a
physical process, e.g. h1(x) = T(x)
- T0

12. INTRODUCTION
MATHEMATICAL OPTIMISATION : PROBLEM TYPES
• Limited or Unlimited
• One variable or Many variables
• Discrete Variable or Continuous
Variables
• Static Problems or Dynamic
Problems
• Deterministic or Stochastic Problems
• Linear Equations or Nonlinear
Equations

13. MATHEMATICAL OPTIMISATION
SENSITIVITY AND ROBUSTNESS
• Every model we write is only a coarse
description of reality
• Conclusions about the model may
more may not correspond to actual
behaviour
• Validity of models often informal,
implicit
• These considerations sometimes
incorporated through sensitivity
analysis

14. MATHEMATICAL OPTIMISATION
SENSITIVITY AND ROBUSTNESS
• More recently, better techniques to
explicit account of difference between
real world and model ( robust control,
robust optimisation )
• Constraints and / or objective
known only approximately
• Implemented solution different
from the computed one

15. MODEL CLASSIFICATIONS
MATHEMATICAL OPTIMISATION
• Unconstrained Optimisation
• Linear Optimisation
• Linear Constrained Optimisation :
• If the constraint functions are linear / affine
• Conic Linear Optimisation :
• If both the objective and the constraint functions
are linear / affine but variables in a convex cone
• Quadratically constrained Quadratic Optimisation :
• If both the objective and constraint functions
are quadratic

16. MATHEMATICAL OPTIMISATION
GLOBAL OPTIMISATION
• Consider unconstrained optimisation
• Typically extremely difficult, many
local minima
• Many questions can be posed in
these terms ( protein folding )
• Very flexible formulation
• But hard to do anything substantial
with it
• Complexity theoretic obstacles

17. BASIC OVERVIEW
CONVEX OPTIMISATION
• Convex optimisation is a generation
of linear programming where the
constraints and object functions are
convex.
• Many subclasses of convex
optimisation like semidefinite
programming and least square
problem are also widely used and
have important applications in
various fields.
• This case guarantees that we are
able to find global minimum.

18. MATHEMATICAL OPTIMISATION
CONVEX OPTIMISATION
• Objective function is convex
• Feasible set is convex
• Many advantages
• Modelling Flexibility
• Tractability and Scalability
• Sensitivity Analysis relatively simple
• Can naturally incorporate robust
considerations

19. FUNDAMENTAL CONCEPTS
LINEAR PROGRAMMING
• Linear programming is one of the well studied classes of
optimisation problem.
• A linear program is one which has linear objectives and
constraint functions.
• Examples:
• Max flow: Given a graph, start and end node, capacities on
every edge, find out the maximum flow possible through
edges.
• Simplex, Ellipsoid, Interior Point Method are some of the
well known algorithms for solving linear programming
problems
• Simplex method was one of the first methods to solve these
programs. But all the initial versions took exponential time
• The first polynomial time algorithm was ellipsoid algorithm.
Few years later interior point method was developed and
shown to be in polynomial time.

20. MATHEMATICAL OPTIMISATION
OTHER CONVEX OPTIMISATION METHODS
• Least Squares Problems
• Used for data fitting
• Semidefinite programming
• Quadratic Programming

21. MATHEMATICAL OPTIMISATION
SEMIDEFINITE PROGRAMMING
• A broad generalisation of LP to
symmetric matrices
• The intersection of an affine subspace
L and the cone of positive semidefinite
matrices
• Originated in control theory and
combinatorial optimisation
• Convex finite dimensional optimisation
• Essentially solvable in polynomial time

22. COMPLEXITY MATTERS
NONLINEAR OPTIMISATION
• The general optimisation problem is
intractable
• Even simple looking optimisation
problems can be very hard
• Examples
• Quadratic optimisation problem
with many constraints
• Minimising a multivariate
polynomial

23. EMBEDDED OPTIMISATION
PRACTICAL EXAMPLES OF USING OPTIMISATION
• In the last decades, the size of
computers and their components
decreased and therefore it became
beneficial to optimise the device
sizing in electric circuits
• For this problem the objective
function is a power consumption
• The variables are widths and lengths
of the device and the constraints are
manufacturing limits, timing
requirements, and maximum area.

24. PORTFOLIO OPTIMISATION
PRACTICAL EXAMPLES OF USING MATHEMATICAL
OPTIMISATION
• When investing in assets there is a big
risk of loosing invested money and so
it would be a competitive advantage to
posses a control system, that would
find the most risk free way for your
investment
• For this purpose, the objective function
could be overall risk or return variance,
the variables amounts invested in
different assets, and the budget,
maximum and minimum investment
per asset and minimum return would
be our constraints.

25. DATA FITTING
PRACTICAL EXAMPLES OF USING MATHEMATICAL
OPTIMISATION
• In this case, we are looking for the
best fitting model for our observed
data.
• The objective function can represent
the measure of misfit or prediction
error.
• The variables model parameters and
constraints can feature prior
information and parameter limits.

26. MODEL PREDICTIVE CONTROL
PRACTICAL EXAMPLES OF USING MATHEMATICAL
OPTIMISATION
• This is a concept of advanced control
systems such as heating control of
intelligent buildings or control of
chemical processes
• This method works with complex
dynamic behaviour of the system and
is aiming mainly to minimise the
performance criterion in the future
that would possibly be subject to
constraints
• In MPC there are dependent and
independent variables

27. QUANTUM COMPUTING AND MATHEMATICAL OPTIMISATION
MATHEMATICAL OPTIMISATION

28. LOREM IPSUM
QUANTUM INFORMATION AND MATHEMATICAL OPTIMISATION
• Convexity naturally arises in many
places in Quantum Information
Theory as the possible preparations,
processes and measurements for
Quantum systems are convex sets
• Quantum Error Correction
• Quantum Entanglement Estimation
• Quantum Tomography

29. LOREM IPSUM
CONVEX OPTIMISATION FOR QUANTUM ENTANGLEMENT
• Using off the shelf semidefinite
programming solvers
• Measurement of relative entropy of
entanglement
• Measurement of Rains bond, known upper
bound on the distillable entanglement
• Identifying the lower bound on the
quantum conditional mutual information
interns of the relative entropy
• At least one subsystem of a multipart state
is a quit

30. MATHEMATICAL OPTIMISATION IN DIGITAL TECHNOLOGY
INFINITYLABS EXPERIMENTS

31. MALWARE
MODELLING
RBM, DBN

32. BIOMETRIC
IDENTITY
GABBOR FILTER

33. IDEA
ANALYTICS
SVM

34. KNOWLEDGE
GRAPH
CNN, RCNN
GRADIENT ALGORITHM

35. REINFORCEMENT
LEARNING
TREE LSTM

36. NEURAL
DATA
STREAM
HASH GRAPH

37. OBJECT
DETECTION
FROM
STREAMING
MEDIA
RCNN

38. QUANTUM
BLOOM
FILTER
GROVER ALGORITHM