1. Hands on with Optimization solvers in PYTHON
Presentation on
Dr. Kishalay Mitra
Professor, Department of Chemical Engineering
Indian Institute of Technology Hyderabad
kishalay@che.iith.ac.in

2. Indian Institute of Technology Hyderabad
Department of Chemical Engineering
Type Formulation Solver
Scalar minimization min
𝑥
𝑓(𝑥)
such that lb<x<ub (x is a scalar)
fminbound,
minimize_scalar
Unconstrained minimization min
𝑥
𝑓(𝑥) fmin, fmin_powell
Linear programming min
𝑥
𝑓𝑇
𝑥
such that 𝐴. 𝑥 ≤ 𝑏, 𝐴𝑒𝑞. 𝑥 = 𝑏𝑒𝑞, 𝑙𝑏 ≤
𝑥 ≤ 𝑢𝑏
linprog
Mixed integer linear
programming
min
𝑥
𝑓𝑇
𝑥
such that 𝐴. 𝑥 ≤ 𝑏, 𝐴𝑒𝑞. 𝑥 = 𝑏𝑒𝑞, 𝑙𝑏 ≤
𝑥 ≤ 𝑢𝑏
x is integer valued
milp
Constrained minimization min
𝑥
𝑓(𝑥)
such that 𝑐 𝑥 ≤ 0, 𝑐𝑒𝑞 𝑥 = 0,
𝐴. 𝑥 ≤ 𝑏, 𝐴𝑒𝑞. 𝑥 = 𝑏𝑒𝑞, 𝑙𝑏 ≤ 𝑥 ≤ 𝑢𝑏
minimize
Root finding methods min
𝑥
𝑓(𝑥)
such that lb<x<ub
root_scalar, root
Minimization problems : scipy.optimize

3. Department of Chemical Engineering Indian Institute of Technology Hyderabad
Type Formulation Solver
Linear least squares
min
𝑥
1
2
𝑐. 𝑥 − 𝑑 2
m equations, n variables
lsq_linear
Non negative linear least squares
min
𝑥
1
2
𝑐. 𝑥 − 𝑑 2
Such that x≥0
nnls
Nonlinear least squares
min
𝑥
𝐹 𝑥 = min
𝑥
𝑖
𝐹𝑖
2
(𝑥)
such that 𝑙𝑏 ≤ 𝑥 ≤ 𝑢𝑏
least_squares
Nonlinear curve fitting min
𝑥
𝐹 𝑥, 𝑥𝑑𝑎𝑡𝑎 − 𝑦𝑑𝑎𝑡𝑎 2
such that 𝑙𝑏 ≤ 𝑥 ≤ 𝑢𝑏
curve_fit
Least squares (curve fitting) problems: scipy.optimize
Single objective optimization min
𝑥
𝑓(𝑥) GA
Multi objective optimization min
𝑥
𝑓1(𝑥)
m𝑎𝑥
𝑥
𝑓2(𝑥)
NSGA2
Optimization problems: pymoo

4. Optimizer
myfun
Decision
Variables
Objective
function
Unconstrained nonlinear function minimization
Result = optimizer (myfun, x0)
Indian Institute of Technology Hyderabad
Department of Chemical Engineering

5. Case study 1 min
𝑥
𝑓 𝑥 = 12𝑥5
− 45𝑥4
+ 40𝑥3
+ 5
Optimizer
myfun
Decision
Variables
Objective
function
Result = optimizer (myfun, x0)
Unconstrained unbounded nonlinear function minimization
Indian Institute of Technology Hyderabad
Department of Chemical Engineering

6. def f = myfun(x):
f = 12*x^5 - 45*x^4 + 40*x^3 + 5
Case study 1 min
𝑥
𝑓 𝑥 = 12𝑥5
− 45𝑥4
+ 40𝑥3
+ 5
 Result= fmin (myfun, x0)
 Result = fmin_powell (myfun, x0)
Indian Institute of Technology Hyderabad
Department of Chemical Engineering
Result = optimizer (myfun, x0)
Unconstrained unbounded nonlinear function minimization

7. Case study 2 min
𝑥
𝑓 𝑥 = 12𝑥5
− 45𝑥4
+ 40𝑥3
+ 5
Optimizer
myfun
Decision
Variables
Objective
function
Result = optimizer (myfun, 𝑥1, 𝑥2)
such that x ∈ (0.5, 2.5)
Indian Institute of Technology Hyderabad
Department of Chemical Engineering
Unconstrained bounded nonlinear function minimization

8. def f = myfun(x):
f = 12*x^5 - 45*x^4 + 40*x^3 + 5
Case study 2 min
𝑥
𝑓 𝑥 = 12𝑥5
− 45𝑥4
+ 40𝑥3
+ 5
Result = optimizer (myfun, 𝑥1, 𝑥2)
such that x ∈ (0.5, 2.5)
 Result = fminbound (myfun, 𝑥1, 𝑥2)
Indian Institute of Technology Hyderabad
Department of Chemical Engineering
Unconstrained bounded nonlinear function minimization

9. def f = myfun(x):
f = 12*x^5 - 45*x^4 + 40*x^3 + 5
f = -f;
Case study 3 m𝑎𝑥
𝑥
𝑓 𝑥 = 12𝑥5
− 45𝑥4
+ 40𝑥3
+ 5
such that x ∈ (0.5, 2.5)
Indian Institute of Technology Hyderabad
Department of Chemical Engineering
Unconstrained bounded nonlinear function maximization
Result = optimizer (myfun, 𝑥1, 𝑥2)
 Result = fminbound (myfun, 𝑥1, 𝑥2)

10. Case study 4
Root finder
myfun
Variable
Function
value
Result = rootfinder (myfun, x0)
Root finding
min
𝑥,𝑦
𝑓 𝑥, 𝑦 = 3𝑥2
− 6
𝑥0 = 0.2
Indian Institute of Technology Hyderabad
Department of Chemical Engineering
 Result = root_scalar(fun, x0, fprime=fprime)

11. • lsq_linear, nnls, least_squares, curve_fit are the operators used for
solving an overdetermined system of linear equations
• Result = optimize.curve_fit(myfun, 𝑥𝑑𝑎𝑡𝑎, 𝑦𝑑𝑎𝑡𝑎, p0)
lsqcurvefit
myfun
p0 – decision
variables/
Parameters and
𝑥𝑑𝑎𝑡𝑎
𝑦𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑒𝑑
𝑦𝑑𝑎𝑡𝑎
Indian Institute of Technology Hyderabad
Department of Chemical Engineering
Linear & Nonlinear regression and curve fitting

12. Result= optimize.minimize(c=f, x0, constraints, method,
bounds=[lb,ub])
Optimizer
Model
Decision
Variables
Objective and
Constraints
Constrained nonlinear function minimization
Indian Institute of Technology Hyderabad
Department of Chemical Engineering

13. Result= optimize.minimize(c=f, x0, constraints, method,
bounds=[lb,ub])
Constrained nonlinear function minimization
Indian Institute of Technology Hyderabad
Department of Chemical Engineering

14. Case study 5
Indian Institute of Technology Hyderabad
Department of Chemical Engineering
Constrained nonlinear function minimization

15.    
5
x
,
x
5
25
x
x
.
t
.
s
7
x
x
11
x
x
2
1
2
2
2
1
2
2
1
2
2
2
1
x
,
x
2
2
1
Min










Constrained nonlinear function minimization
Case study 6.
Indian Institute of Technology Hyderabad
Department of Chemical Engineering

16. Indian Institute of Technology Hyderabad
Department of Chemical Engineering
𝒎𝒊𝒏 𝒇𝟏 𝒙, 𝒚 = 𝟒𝒙𝟐 + 𝟒𝒚𝟐
𝒎𝒊𝒏 𝒇𝟐 𝒙, 𝒚 = 𝒙 − 𝟓 𝟐
+ 𝒚 − 𝟓 𝟐
𝒈𝟏 𝒙, 𝒚 = 𝒙 − 𝟓 𝟐 + 𝒚 𝟐 ≤ 𝟐𝟓
𝒈𝟐 𝒙, 𝒚 = 𝒙 − 𝟖 𝟐 + 𝒚 + 𝟑 𝟐 ≤ 𝟕. 𝟕
𝟎 ≤ 𝒙 ≤ 𝟓 𝟎 ≤ 𝒚 ≤ 𝟑
Constrained nonlinear Multi objective optimization
Case study 7.

17. Unconstrained nonlinear function minimization
Linear & Nonlinear Regression and Curve Fitting
Case study 8. ODE Solving
First order series reactions happening in an isothermal
batch reactor
0
)
0
(
c
),
t
(
b
k
dt
dc
0
)
0
(
b
),
t
(
b
k
)
t
(
a
k
dt
db
1
)
0
(
a
),
t
(
a
k
dt
da
2
2
1
1








A B C
k1 k2
Solve using solve_ivp
& then perform parameter
estimation using minimize
Indian Institute of Technology Hyderabad
Department of Chemical Engineering

18. Unconstrained nonlinear function minimization
Linear & Nonlinear Regression and Curve Fitting
First order series reactions happening in an isothermal
batch reactor
0
)
0
(
,
0
)
0
(
,
1
)
0
(
)
(
)
(
)
(
)
(
2
2
1
1








c
b
a
t
b
k
dt
dc
t
b
k
t
a
k
dt
db
t
a
k
dt
da
A B C
k1 k2
def my_model(t,y,k):
f =[]
f[0] = -k[0]*y[0]
f[2] = k[0]*y[0]-k[1]*y[1]
f[3] = k[1]*y[1]
return f
Indian Institute of Technology Hyderabad
Department of Chemical Engineering

19. def fun:
ic = [1,0,0]
tspan = [0 0.05 0.1 0.15 0.2 0.3 0.4 0.45 0.56
0.67 0.7 0.75 0.78 0.8 0.9 0.92 1]
k = [6.3,4.23]
sol = solve_ivp(my_model(t,y,k),tspan, ic)
Case study 10. ODE Solving
Indian Institute of Technology Hyderabad
Department of Chemical Engineering