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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.