CH5150: Optimization Techniques I

This project explores solving optimization problems for chemical engineering applications using deterministic and evolutionary algorithms.

Problem 1 : Determining rate constants

The system involves parallel reactions happening in an isothermal batch reactor. The rate equations need to be solved to determine the rate constants k1 and k2. I lost the data for this problem but the data contained time, concentration of components CA, CB, and CC.

$\ce{A ->[k1] B}$

$\ce{A ->[k2] C}$

Rate equations:

$\frac{dC_{A}}{dt} = -(k_{1}C_{A} + k_{2}C_{A})$

$\frac{dC_{B}}{dt} = k_{1}C_{A}$

$\frac{dC_{C}}{dt} = k_{2}C_{A}$

This problem utilizes the following python libraries:

import numpy # to do math
import pandas # to import data
import scipy # for scipy.integrate and scipy.optimize
import timeit # for comparing runtimes

Algorithms Used

• Nelder-Mead Method
• Powell Method
• Conjugate Gradient Method (CG)
• BFGS Method (Broyden-Fletcher-Goldfarb-Shanno)
• COBYLA Method (Constrained Optimization BY Linear Approximations)
• SLSQP Method (Sequential Least SQuares Programming)

Results

Method	Convergence	k1_opt	k2_opt	Error	Execution Time
Nelder-Mead	Converged	1.9998748692788715	3.9997615838621496	1.962460e-05	0.149742
Powell	Converged	1.9999018222782379	3.999807785445786	1.954439e-05	0.304135
CG	Did not converge	1.9999017781607618	3.9998076688535855	1.954439e-05	0.846164
BFGS	Did not converge	1.9999018263820614	3.9998077521324666	1.954439e-05	0.508273
TNC	Did not converge	1.9857536272294474	3.9316821455453663	3.748240e-03	0.775833
COBYLA	Converged	1.9994226343894554	3.9990020736425738	3.684053e-05	0.258513
SLSQP	Converged	1.9998997663393545	3.999809061609218	1.954768e-05	0.132066

Problem 2 : Himmelblau function optimization

Minimize the Himmelblau function:

$\min_{x, y} (x_{1}^2 + x_{2} - 11)^2 + (x_{1} + x_{2}^2 - 7)^2$

Subject to the constraint:

$x_{1}^2 + x_{2}^2 \geq 25$

With bounds on ( x_1 ) and ( x_2 ):

$-5 \leq x_{1}, x_{2} \leq 5$

Algorithms

This problem utilizes the following python libraries:

import numpy # to do math
import scipy # for deterministic algorithms

!pip install deap
import deap # for genetic algorithms

Three deterministic optimization algorithms are implemented:

• Gradient Descent (using BFGS method)
• Nelder-Mead Method
• CG Method

Three variants of the genetic algorithms are explored

• eaSimple
• eaMuPlusLambda
• eaMuCommaLambda

Results

Method	x1_opt	x2_opt	Minimum Value
Gradient Descent	-3.779310265	-3.283186	1.4142791614207676e-15
Nelder-Mead	-3.77929277	-3.2832138	6.563218571685403e-08
CG	-3.77931029	-3.28318601	5.833980405874394e-14
eaSimple	-3.779310253414729	-3.2831859914210866	7.416028657579182e-19
eaMuPlusLambda	-3.779310253489113	-3.2831859912216483	1.1076340901563927e-18
eaMuCommaLambda	-3.779310253377747	-3.283185991286169	7.888609052210118e-31

Genetic algorithms are far superior compared to deterministic methods and produce highly optmized solutions.