Engineering Optimization: Theory and Practice, Fourth Edition

36 Introduction to Optimization

The most widely circulated journals that publish papers related to engineering opti-

mization areEngineering Optimization, ASME Journal of Mechanical Design, AIAA

Journal, ASCE Journal of Structural Engineering, Computers and Structures, Interna-

tional Journal for Numerical Methods in Engineering, Structural Optimization, Journal

of Optimization Theory and Applications, Computers and Operations Research, Oper-

ations Research,Management Science, Evolutionary Computation, IEEE Transactions

on Evolutionary Computation, European Journal of Operations Research, IEEE Trans-

actions on Systems, Man and Cybernetics, andJournal of Heuristics. Many of these

journals are cited in the chapter references.

1.8 Solution of Optimization Problems Using MATLAB

The solution of most practical optimization problems requires the use of computers.

Several commercial software systems are available to solve optimization problems that

arise in different engineering areas. MATLAB is a popular software that is used for

the solution of a variety of scientific and engineering problems.†MATLAB has several

toolboxes each developed for the solution of problems from a specific scientific area.

The specific toolbox of interest for solving optimization and related problems is called

theoptimization toolbox. It contains a library of programs or m-files, which can be

used for the solution of minimization, equations, least squares curve fitting, and related

problems. The basic information necessary for using the various programs can be found

in the user’s guide for the optimization toolbox [1.124]. The programs or m-files, also

called functions, available in the minimization section of the optimization toolbox are

given in Table 1.2. The use of the programs listed in Table 1.2 is demonstrated at the end

of different chapters of the book. Basically, the solution procedure involves three steps

after formulating the optimization problem in the format required by the MATLAB

program (or function) to be used. In most cases, this involves stating the objective

function for minimization and the constraints in “≤” form with zero or constant value

on the righthand side of the inequalities. After this, step 1 involves writing an m-file

for the objective function. Step 2 involves writing an m-file for the constraints. Step 3

involves setting the various parameters at proper values depending on the characteristics

of the problem and the desired output and creating an appropriate file to invoke the

desired MATLAB program (and coupling the m-files created to define the objective and

constraints functions of the problem). As an example, the use of the program,fmincon,

for the solution of a constrained nonlinear programming problem is demonstrated in

Example 1.11.

Example 1.11 Find the solution of the following nonlinear optimization problem

(same as the problem in Example 1.1) using the MATLAB functionfmincon:

Minimizef (x 1 , x 2 ) = 9. 82 x 1 x 2 + 2 x 1

subject to

g 1 (x 1 , x 2 )=

2500

π x 1 x 2

− 005 ≤ 0

†The basic concepts and procedures of MATLAB are summarized inAppendix C.