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17
Solution of Optimization Problems Using MATLAB

17.1 Introduction

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.1 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 the optimization 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]. The programs or m‐files, also called functions, available in the minimization section of the optimization toolbox are given in Table 17.1. The use of the programs listed in Table 17.1 is demonstrated in this chapter. 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 right hand 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).

images — Table 17.1 MATLAB Programs or Functions for Solving Optimization Problems.

Type of optimization problem	Standard form for solution by MATLAB	Name of MATLAB program or function to solve the problem
Function of one variable or scalar minimization	Find x to minimize f(x) with x ₁ < x < x ₂	`fminbnd`
Unconstrained minimization of function of several variables	Find x to minimize f(x)	`fminunc` or `fminsearch`
Linear programming problem	Find x to minimize f ^T x subject to [A]x ≤ b, [A _eq]x = b _eq, l ≤ x ≤ u	`linprog`
Quadratic programming problem	Find x to minimize x ^T[H]x + f ^T x subject to [A]x ≤ b, [A _eq]x = b _eq, l ≤ x ≤ u	`quadprog`
Minimization of function of several variables subject to constraints	Find x to minimize f(x) subject to c(x) ≤ 0, c _eq = 0 [A]x ≤ b, [A _eq]x = b _eq, l ≤ x ≤ u	`fmincon`
Goal attainment problem	Find x and γ to minimize γ such that F(x) – w γ ≤ goal, c(x) ≤ 0, c _eq = 0 [A]x ≤ b, [A _eq]x = b _eq, l ≤ x ≤ u	`fgoalattain`
Minimax problem	such that c(x) ≤ 0, c _eq = 0 [A]x ≤ b, [A _eq]x = b _eq, l ≤ x ≤ u	`fminimax`
Binary integer programming problem	Find x to minimize f ^T x subject to [A]x ≤ b, [A _eq]x = b _eq, each component of x is binary	`bintprog`

The examples presented in this chapter illustrate the application of MATLAB to solve the problems/methods presented in the previous chapters.

17.2 Solution of General Nonlinear Programming Problems

Example 17.1

Find the solution of the following nonlinear optimization problem (same as the problem in Example 1.1) using the MATLAB function fmincon:

SOLUTION

Step 1: Write an M‐file probofminobj.m for the objective function.
```
function f= probofminobj (x)
f= 9.82*x(1)*x(2)+2*x(1);
```

Step 2: Write an M‐file conprobformin.m for the constraints.

function [c, ceq] = conprobformin(x)
% Nonlinear inequality constraints
c = [2500/(pi*x(1)*x(2))-500;2500/(pi*x(1)*x(2))-(pi^2*(x(1)^2+x(2)^2))/0.5882;-x(1)+2;x(1)-14;-x(2)+0.2;
x(2)-0.8];
% Nonlinear equality constraints
ceq = [];

Step 3: Invoke constrained optimization program (write this in new MATLAB file).

clc
clear all
warning off
x0 = [7 0.4]; % Starting guess
fprintf ('The values of function value and constraints
at starting point
');
f=probofminobj (x0)
[c, ceq] = conprobformin (x0)
options = optimset ('LargeScale', 'off');
[x, fval]=fmincon (@probofminobj, x0, [], [], [], [], [],
[], @conprobformin, options)
fprintf('The values of constraints at optimum solution
');
[c, ceq] = conprobformin(x) % Check the constraint values at x

This produces the solution or output as follows:

The values of function value and constraints at starting point f=
 41.4960
c =
-215.7947
-540.6668
-5.0000
-7.0000
-0.2000
-0.4000
ceq =
 []
Optimization terminated: first-order optimality measure
less than options. TolFun and maximum constraint violation is less than options.TolCon.
Active inequalities (to within options.TolCon = 1e-006):
lower upper ineqlin ineqnonlin
     1
     2
x=
 5.4510 0.2920
fval =
 26.5310
The values of constraints at optimum solution
c=
 -0.0000
 -0.0000
 -3.4510
 -8.5490
 -0.0920
 -0.5080
ceq =
 []

17.3 Solution of Linear Programming Problems

The solution of linear programming problems, using the simplex method, can be found as illustrated by the following example.

Example 17.2

Find the solution of the following linear programming problem using MATLAB (simplex method):

subject to

SOLUTION

Step 1 Express the objective function in the form f (x) = f ^T x and identify the vectors x and f as

Express the constraints in the form Ax ≤ b and identify the matrix A and the vector b as

Step 2 Use the command for executing linear programming program using simplex method as indicated below:

 clc
 clear all
 f=[-1;-2;-1];
 A=[2 1-1;
  2-1 5;
  4 1 1];
 b=[2;6;6];
 lb=zeros(3,1);
 Aeq=[];
 beq=[];
 options = optimset('LargeScale', 'off', 'Simplex', 'on');
 [x,fval,exitflag,output] = linprog (f,A,b,Aeq,beq,lb,[],[], optimset('Display','iter'))

This produces the solution or output as follows:

 Optimization terminated.
 x=
  0
  4
  2
 fval =
  -10
 exitflag =
  1
 output =
  iterations:3
  algorithm: 'medium scale: simplex'
  cgiterations: []
  message: 'Optimization terminated.'

17.4 Solution of LP Problems Using Interior Point Method

The solution of linear programming problems, based on the interior point method, using MATLAB is illustrated by the following example.

Example 17.3

Find the solution of the following linear programming problem using MATLAB (interior point method):

subject to

SOLUTION

Step 1 Express the objective function in the form f(x) = f ^T x and identify the vectors x and f as

Express the constraints in the form Ax ≤ b and identify the matrix A and the vector b as

Step 2 Use the command for executing linear programming program using interior point method as indicated below:

clc
clear all
f=[–1;–2;–1];
A=[2 1—1;
  2—1 5;
  4 1 1];
b=[2;6;6];
lb=zeros(3,1);
Aeq=[];
beq=[];
options = optimset('Display', 'iter');
[x,fval,exitflag,output] = linprog(f,A,b,Aeq,beq,lb,[],[],options)

This produces the solution or output as follows:

Iter 0: 1.03e+003 7.97e+000 1.50e+003 4.00e+002
Iter 1: 4.11e+002 2.22e–016 2.78e+002 4.72e+001
Iter 2: 1.16e–013 1.90e–015 2.85e+000 2.33e–001
Iter 3: 1.78e–015 1.80e–015 3.96e–002 3.96e–003
Iter 4: 7.48e–014 1.02e–015 1.99e–006 1.99e–007
Iter 5: 2.51e–015 4.62e–015 1.99e–012 1.98e–013
Optimization terminated.
x=
 0.0000
 4.0000
 2.0000
fval = -10.0000
exitflag = 1
output =
  iterations: 5
  algorithm: 'large-scale: interior point'
  cgiterations: 0
  message: 'Optimization terminated.'

17.5 Solution of Quadratic Programming Problems

Example 17.4

Find the solution of the following quadratic programming problem using MATLAB:

SOLUTION

Step 1 Express the objective function in the form f(x) = and identify the matrix H and vectors f and x:
Step 2 State the constraints in the form: Ax ≤ b and identify the matrix A and vector b:

Step 3 Use the command for executing quadratic programming as

 [x,fval] = quadprog(H,f,A,b)

which returns the solution vector x that minimizes

The MATLAB solution is given below:

 clear;clc;
 H = [2 —2;–2 4];
 f = [–4 0];
 A = [2 1;1 —4];
 b = [6; 0];
 [x,fval] = quadprog(H,f,A,b)
 Warning: Large-scale method does not currently solve this problem formulation, switching to medium-scale method.
 x=
  2.4615
  1.0769
 fval =
  -6.7692

17.6 Solution of One‐Dimensional Minimization Problems

The solution of one‐dimensional minimization problems, using the MATLAB program optimset, is illustrated by the following example.

Example 17.5

Find the minimum of the following function:

SOLUTION

Step 1 Write an M‐file objfun _. m for the objective function.

 function f= objfun(x)
 f= 0_.
65–(0_.
75/(1+x^2))– 0_.
65*x*atan(1/x);

Step 2 Invoke unconstrained optimization program (write this in new MATLAB file).

 clc
 clear all
 warning off
 options = optimset('LargeScale','off');
 [x,fval] = fminbnd(@objfun,0,0.5,options)

This produces the solution or output as follows:

 x=
 0.4809
 fval =
 -0.3100

17.7 Solution of Unconstrained Optimization Problems

The solution of multivariable unconstrained minimization problems using the MATLAB function fminunc is illustrated in this section.

Example 17.6

Find the minimum of the Rosenbrock's parabolic valley function, given by Eq. (6.140), starting from initial point X ₁ = {−1.2 1.0}^T.

SOLUTION

Step 1 Write an M‐file objfun.m for the objective function.

function f= objfun (x)
f= 100* (x(2)-x(1) *x(1))^2+(1-x(1))^2;

Step 2 Invoke unconstrained optimization program (write this in new MATLAB file).

clc
clear all
warning off
x0 = [-1.2,1.0]; % Starting guess
fprintf ('The values of function value at starting pointn');
f=objfun(x0)
options = optimset('LargeScale', 'off');
[x, fval] = fminunc (@objfun,x0,options)

This produces the solution or output as follows:

The values of function value at starting point
f=
 24.2000
Optimization terminated: relative infinity-norm of gradient less than options.TolFun.
x=
 1.0000 1.0000
fval=
 2.8336e-011

17.8 Solution of Constrained Optimization Problems

The solution of multivariable minimization problems, with inequality and equality constraints, using the MATLAB function fmincon is illustrated in this section.

Example 17.7

Find the solution of Example 7.8 starting from the initial point X ₁ = {0.1 0.1 3.0}^T

SOLUTION

Step 1 Write an M‐file objfun.m for the objective function.
```
function f= objfun (x)
f= x(1)^3-6*x(1)^2+11*x(1)+x(3);
```

Step 2 Write an M‐file constraints.m for the constraints.

function [c, ceq] = constraints (x)
% Nonlinear inequality constraints
c = [x(1)^2+x(2)^2-x(3)^2;4-x(1)^2-x(2)^2-x(3)^2;x(3)-5; -x(1);-x(2);-x(3)];
% Nonlinear equality constraints
ceq = [];

Step 3 Invoke constrained optimization program (write this in new MATLAB file).

 clc
 clear all
 warning off
 x0 = [.1,.1, 3.0]; % Starting guess
 fprintf ('The values of function value and constraints at starting point');
 f=objfun (x0)
 [c, ceq] = constraints (x0)
 options = optimset ('LargeScale', 'off');
 [x, fval]=fmincon (@objfun, x0, [], [], [], [], [], [], @constraints, options)
 fprintf ('The values of constraints at optimum solution');
 [c, ceq] = constraints (x)
 % Check the constraint values at x

This Produces the Solution or Output as follows:

 The values of function value and constraints at starting point
 f=
 4.0410
 c=
 -8.9800
 -5.0200
 -2.0000
 -0.1000
 -0.1000
 -3.0000
 ceq =
 []
 Optimization terminated: first-order optimality measure less than options. TolFun and maximum constraint violation is less than options.TolCon.
 Active inequalities (to within options.TolCon = 1e-006): lower upper ineqlin ineqnonlin
 1
 2
 4
 x=
 0 1.4142 1.4142
 fval =
 1.4142
 The values of constraints at optimum solution
 c=
 -0.0000
 -0.0000
 -3.5858
  0
 -1.4142
 -1.4142
 ceq =
 []

Example 17.8

Find the solution of the welded beam problem described in Section 7.22.3 using the MATLAB function fmincon. Use the starting point .

SOLUTION

Step 1 Write an M‐file objfun.m for the objective function.

function f= Example_8_objfun(x)
f=1.10471*x(1)*x(1)*x(2)+0.04811*x(3)*x(4)*(14.0+x(2));
end

Step 2 Write an M‐file constraints.m for the constraints.

function [c, ceq] = Example_8_constraints(x)
P=6000;
E=30e6;
G=12e6;
smax=30000;
deltamax=0.25;
shearmax=13600;
L=14;
j1=2*x(1)*x(2)/sqrt(2);
j2=((x(2)^2)/12+((x(1)+x(3))/2)^2);
J=j1*j2;
S=6*P*L/(x(4)*x(3)*x(3));
K=(4*P*L^3)/(E*(x(3)^3)*x(4));
pc1=1-(x(3)/(2*L))*(E/(4*G))^(1/2);
pc2=4.013*((E*G*(x(3)^2 )*(x(4)^6)/36))^(1/2);
Pc=pc1*pc2/(L*L);
M=P*(L+x(2)/2);
R=((x(2)^2)/4+((x(1)+x(3))/2)^2)^0.5;
A=P/(x(1)*x(2)*1.414);
B=M*R/J;
T = (A^2+2*A*B*(x(2)/(2*R))+B^2)^0.5;
% Nonlinear inequality constraints
c = [T-shearmax;S-smax;x(1)-x(4);0.10471*x(1)*x(1)+0.04811*x(3)*x(4)*(14.0+x(2))-5.0;0.125-x(1);...
 K-deltamax;P-Pc];
% Nonlinear equality constraints
ceq = [];

Step 3 Invoke constrained optimization program (write this in new MATLAB file).

clc
clear all
warning off
x0 = [0.4;6;9;0.5]; % Starting guess
fprintf('The values of function value and constraints at starting point
');
f=Example_4_objfun(x0)
[c,ceq] = Example_8_constraints(x0)
lb=[0.1;0.1;0.1;0.1];
ub=[2;10;10;2];
options = optimset('LargeScale','off');
[x,fval]=fmincon(@Example_8_objfun,x0,[],[],[],[],lb,ub,@Example_4_constraints,options)
fprintf('The values of constraints at optimum solution
');
[c,ceq] = Example_8_constraints(x) % Check the constraint values at x

This produces the Solution or Output as follows:

The values of function value and constraints at starting point
f =
 5.3904
c =
 1.0e+04 *
 -0.5826
 -1.7556
 -0.0000
 -0.0001
 -0.0000
 -0.0000
 -4.8330
ceq =
 []
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the default value of the optimality tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.
 <stopping criteria details>
x =
 0.2444
 6.2177
 8.2915
 0.2444
fval =
 2.3810
The values of constraints at optimum solution
c =
 -0.0158
 -0.2319
 -0.0000
 -3.0229
 -0.1194
 -0.2342
 -0.0247
ceq =
 []

17.9 Solution of Binary Programming Problems

The MATLAB function bintprog can be used to solve a binary (or zero–one) programming problem. The following example illustrates the procedure.

Example 17.9

Find the solution of the following binary programming problem using the MATLAB function bintprog:

subject to

SOLUTION

Step 1 State the problem in the form required by the program bintprog:

Here

Step 2 The input is directly typed on the MATLAB command window and the program bintprog is called as indicated below:

 clear; clc;
f = [-5 -5 -8 2 4]′;
 A = [3 -6 7 -9 -9; 1 2 0 -1 -3];
b = [-10 0]′;
 x = bintprog (f, A, b,[])

Step 3 The output of the program is shown below:

 Optimization terminated.
 x =
  1
  1
  1
  1
  1

17.10 Solution of Multiobjective Problems

The MATLAB function fgoalattain can be used to solve a multiobjective optimization problem using the goal attainment method. The following example illustrates the procedure.

Example 17.10

Find the solution of the following three‐objective optimization problem using goal attainment method using the MATLAB function fgoalattain.

Minimize

subject to

Assume the initial design variables to be x ₁ = x ₂ = 0.1, the weights to be w ₁ = 0.2, w ₂ = 0.5, and w ₃ = 0.3, and the goals to be b ₁ = 5, b ₂ = −8, and b ₃ = 20.

SOLUTION

Step 1 Create an m‐file for the objective functions and save it as fgoalattain _{_} obj _. m

 function f = fgoalattainobj(x)
 f(1) = (x(1)-2)^2/2+(x(2)+1)^2/13+3
 f(2) = (x(1)+x(2)-3)^2/175+(2*x(2)-x(1))^2/17-13
 f(3) = (3*x(1)-2*x(2)+4)^2/8+(x(1)-x(2)+1)^2/27+15

Step 2 Create an m‐file for the constraints and save it as fgoalattain _{_} con _. m

 function [c ceq] = fgoalattaincon(x)
c= [- 4- x(1); …
x(1)- 4; …
- 4- x(2); …
x(2)- 4; …
x(2)+4*x(1)- 4; …
- 1- x(1); …
 x(1)- 2- x(2)]
 ceq = [];

Step 3 Create an m‐file for the main program and save it as fgoalattain _{_} main _. m

 clc; clear all;
 x0 = [0_.
1 0_.
1]
 weight = [0_.
2 0.5
 0_.
3]
 goal = [5 -8 20]
 x,fval,attainfactor,exitflag] = fgoalattain (@fgoalattainobj, x0,goal,weight,[],[],[],[],[],[],@fgoalattaincon)

Step 4Run the program fgoalattain _{_} main _. m to obtain the following result:

Initial design vector:	`0.1,0.1`
Initial objective values:	`4.8981 ‐12.9546 17.1383`
Constraints at initial design:	`‐4.1000`
	`‐3.9000`
	`‐4.1000`
	`‐3.9000`
	`‐3.5000`
	`‐1.1000`
	`‐2.0000`
Optimum design vector:	`0.8308 0.6769`
Optimum objective values:	`3.8999 ‐12.9712 18.3498`
Constraints at optimum design:	`‐4.8308`
	`‐3.1692`
	`‐4.6769`
	`‐3.3231`
	`‐0.0000`
	`‐1.8308`
	`‐1.8462`

Example 17.11

Find the solution of the following two‐objective optimization problem using weighting method using the MATLAB function fmincon.

function Example_11_weight

clear,clc;warning off;

%%%%%%%%initial point
x0=[2 2 4 1 4 5];
lb=[0 0 1 0 1 0];
ub=[10 10 5 6 5 10];
a=0;b=0;
  
f01=obj1(x0);
f02=obj2(x0);
[cons0,ceq0]=cons(x0);
%%%%%%%%% weighting method
options=optimset('Display','iter');
[xw,fw]=fmincon(@objweight,x0,[],[],[],[],lb,ub,@cons,options)
  
%%%%%%%%%%
fv1=obj1(xw,a,b);
fv2=obj2(xw,a,b);
[consv,ceqv]=cons(xw);
  
fprintf('************** Example 10 (Weighted method) ****************
');
fprintf('************************************************************
');
fprintf(' x0(1) x0(2) x0(3) x0(4) x0(5) x0(6) 
');
fprintf(' %4.4f %4.4f %4.4f %4.4f %4.4f %4.4f
', x0(1),x0(2),x0(3),x0(4),x0(5),x0(6));
fprintf('************************************************************
');
fprintf(' f1(x0) f2(x0)
');
fprintf(' %8.4f %8.4f
', f01, f02);
fprintf('************************************************************
');
fprintf(' g1(x0) g2(x0) g3(x0) g4(x0) g5(x0) g6(x0)
');
fprintf(' %4.4f %4.4f %4.4f %4.4f %4.4f %4.4f
',cons0(1),cons0(2),cons0(3),cons0(4),cons0(5),cons0(6));
fprintf('************************************************************
');
fprintf('************************************************************
');
fprintf(' x*(1) x*(2) x*(3) x*(4) x*(5) x*(6) 
');
fprintf(' %4.4f %4.4f %4.4f %4.4f %4.4f %4.4f
', xw(1),xw(2),xw(3),xw(4),xw(5),xw(6));
fprintf('************************************************************
');
fprintf(' f1(x*) f2(x*)
');
fprintf(' %8.4f %8.4f
', fv1, fv2);
fprintf('************************************************************
');
fprintf(' g1(x*) g2(x*) g3(x*) g4(x*) g5(x*) g6(x*)
');
fprintf(' %4.4f %4.4f %4.4f %4.4f %4.4f %4.4f
',consv(1),consv(2),consv(3),consv(4),consv(5),consv(6));
end
  
%%%%%%%% f01*, f02*
function f1=obj1(x,a,b);
f1=-25*(x(1)-2)^2-(x(2)-2)^2-(x(3)-1)^2-(x(4)-4)^2-(x(5)-1)^2;
end
%%%%%%%%%%%%%%%%%% f(2)
function f2=obj2(x,a,b);
f2=x(1)^2+x(2)^2+x(3)^2+x(4)^2+x(5)^2+x(6)^2;
end
%%%%%%%%%%%%%%%%%%%%%
function fw=objweight(x,a,b)
fw=obj1(x)+obj2(x);
end
%%%%%%%%%%%%%%
function [c,ceq]=cons(x)
c(1)=-x(1)-x(2)+2;
c(2)=x(1)+x(2)-6;
c(3)=-x(1)+x(2)-2;
c(4)=x(1)-3*x(3)-2;
c(5)=(x(3)-3)^2+x(4)^2-4;
c(6)=-(x(5)-3)^2-x(6)+4;
ceq=[];
end

Output:

              First-order Norm of 
Iter F-count    f(x)   feasibility optimality  Step
 0     7  3.900000e+01 0.000e+00 8.055e+00
 1    14 -1.667902e+01 0.000e+00 4.478e+01 3.986e+00
 2    21 -2.398118e+01 0.000e+00 4.714e+01 4.400e-01
 3    28 -5.751137e+01 0.000e+00 6.378e+01 1.488e+00
 4    35 -9.288998e+01 0.000e+00 7.769e+01 5.882e-01
 5    42 -9.908576e+01 0.000e+00 2.942e+01 9.312e-02
 6    49 -1.010258e+02 0.000e+00 2.062e+01 1.158e+00
 7    56 -1.015468e+02 0.000e+00 1.062e+00 2.490e-01
 8    63 -1.014473e+02 0.000e+00 6.428e-01 2.364e-01
 9    70 -1.018941e+02 0.000e+00 1.398e-01 7.860e-02
10    77 -1.020607e+02 0.000e+00 3.594e-02 3.037e-02
11    84 -1.020632e+02 0.000e+00 2.002e-02 4.278e-03
12    91 -1.020632e+02 0.000e+00 6.791e-03 8.403e-03
13    98 -1.020643e+02 0.000e+00 4.064e-05 3.039e-04

Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in
feasible directions, to within the default value of the optimality tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.
 <stopping criteria details>
xw =
  0.0000  2.0000  1.0000  0.0000  4.9343  0.2586
fw =
 -102.0643
************** Example 10 (Weighted method) ***************
 x0(1)   x0(2)  x0(3)  x0(4)  x0(5)   x0(6)
 2.0000  2.0000  4.0000  1.0000  4.0000  5.0000
********************************************************
 f1(x0)    f2(x0)
 -27.0000   66.0000
********************************************************
 g1(x0)   g2(x0)   g3(x0)  g4(x0)  g5(x0)  g6(x0)
 -2.0000  -2.0000  -2.0000  -12.0000  -2.0000  -2.0000
********************************************************
x*(1)  x*(2)  x*(3)  x*(4)  x*(5)  x*(6)
0.0000  2.0000  1.0000  0.0000  4.9343  0.2586
********************************************************
 f1(x*)    f2(x*)
 -131.4784   29.4141
********************************************************
g1(x*)    g2(x*)   g3(x*)  g4(x*)  g5(x*)  g6(x*)
 -0.0000  -4.0000  -0.0000  -5.0001  -0.0002  -0.0001

References and Bibliography

1 (2003). Optimization Toolbox for use with MATLAB^R, User's Guide, Version 2. The MathWorks.
2 Foster, N. (2016). MATLAB Optimization Functions and Examples. ISBN: 9781520265025.
3 Lopez, C.P. (2014). MATLAB Optimization Techniques. Apress. ISBN: 9781484202920.

Problems

17.1 Find the solution of the optimization problem formulated in Problem 1.32(a) using MATLAB.
17.2 Find the solution of the optimization problem formulated in Problem 1.33(a) using MATLAB.
17.3 Find the solution of Problem 1.31 using MATLAB.
17.4 Find the solution of Problem 1.2(a) using MATLAB.
17.5 Find the solution of Problem 1.2(b) using MATLAB.
17.6 Solve Example 3.7 using MATLAB (simplex method).
17.7 Solve Problem 3.12 using MATLAB (simplex method).
17.8 Solve Problem 3.24 using MATLAB (simplex method).
17.9 Find the optimal solution of the LP problem stated in Problem 3.45 using MATLAB (simplex method).
17.10 Find the optimal solution of the LP problem described in Problem 3.101 using MATLAB.
17.11 Solve the LP problem stated in Problem 4.9 using MATLAB (interior method).
17.12 Solve the LP problem stated in Problem 4.12 using MATLAB (interior method).
17.13 Solve the LP problem stated in Problem 4.13 using MATLAB (interior method).
17.14 Solve the LP problem stated in Problem 4.36 using MATLAB (interior method).
17.15 Solve the LP problem stated in Problem 4.37 using MATLAB (interior method).
17.16 Solve the following quadratic programming problem using MATLAB:
17.17 Solve the following quadratic programming problem using MATLAB:
17.18 Solve the following quadratic programming problem using MATLAB:
17.19 Solve the following quadratic programming problem using MATLAB:
17.20 Write a computer program, in the form of a subroutine, to implement the Fibonacci method.
17.21 Write a computer program, in the form of a subroutine, to implement the golden section method.
17.22 Write a computer program, in the form of a subroutine, to implement the quadratic interpolation method.
17.23 Write a computer program, in the form of a subroutine, to implement the cubic interpolation method.
17.24 Write a computer program, in the form of a subroutine, to implement the secant method.
17.25 Find the maximum of the function given by Eq. (4) in Problem 5.5 using MATLAB. Assume the bounds on λ as 0 and 3.
17.26 Find the minimum of the function f (λ) given in Problem 5.16, in the range 0–5, using MATLAB.
17.27 Find the minimum of f(x) = x(x − 1.5) in the interval (0, 1) using MATLAB.
17.28 Find the minimum of the function in the range (0, 10) using MATLAB.
17.29 Find the minimum of the function f(x) = x ³ + x ² − x − 2 in the interval −4 and 4 using MATLAB.
17.30 Find the minimum of the function in the interval −4 and 4 using MATLAB.
17.31 Write a computer program to implement Powell's method using the golden section method of one‐dimensional search.
17.32 Write a computer program to implement the Davidon–Fletcher–Powell method using the cubic interpolation method of one‐dimensional search. Use a finite‐difference scheme to evaluate the gradient of the objective function.
17.33 Write a computer program to implement the BFGS method using the cubic interpolation method of one‐dimensional minimization. Use a finite‐difference scheme to evaluate the gradient of the objective function.
17.34 Write a computer program to implement the steepest descent method of unconstrained minimization with the direct root method of one‐dimensional search.
17.35 Write a computer program to implement the Marquardt method coupled with the direct root method of one‐dimensional search.
17.36 Find the minimum of the quadratic function given by Eq. (6.141) starting from the solution X ₁ = {0, 0}^T using MATLAB.
17.37 Find the minimum of the Powell's quartic function given by Eq. (6.142) starting from the solution X ₁ = {3, −1, 0, 1}^T using MATLAB.
17.38 Find the minimum of the Fletcher and Powell's helical valley function given by Eq. (6.143) starting from the solution X ₁ = {−1, 0, 0}^T using MATLAB.
17.39 Find the minimum of the nonlinear function given by Eq. (6.144) starting from the solution X ₁ = {0, 1, 2}^T using MATLAB.
17.40 Find the minimum of the Wood's function given by Eq. (6.149) starting from the solution X ₁ = {−3, −1, −3, −1}^T using MATLAB.
17.41 Write a computer program to implement the interior penalty function method with the DFP method of unconstrained minimization and the cubic interpolation method of one‐dimensional search.
17.42 Write a computer program to implement the exterior penalty function method with the BFGS method of unconstrained minimization and the direct root method of one‐dimensional search.
17.43 Write a computer program to implement the augmented Lagrange multiplier method with a suitable method of unconstrained minimization.
17.44 Write a computer program to implement the sequential linear programming method.
17.45 Find the solution of the welded beam design problem formulated in Section 7.22.3 using the MATLAB function fmincon with the starting point X ₁ = {0.4, 6.0, 9.0, 0.5}^T
17.46 Find the solution of the following problem (known as Rosen–Suzuki problem) using the MATLAB function fmincon with the starting point X ₁ = {0, 0, 0, 0}^T:
Minimize

subject to
17.47 Find the solution of the following problem using the MATLAB function fmincon with the starting point X ₁ = {0.5, 1.0}^T:
Minimize

subject to
17.48 Find the solution of the following problem using the MATLAB function fmincon with the starting point: X ₁ = {0.5, 1.0, 1.0}:

subject to
17.49 Find the solution of the following problem using the MATLAB function fmincon with the starting point: X ₁ = {1.0, 1.0}^T:

subject to
17.50 Find the solution of the following binary programming problem using the MATLAB function bintprog:

where
17.51 Find the solution of the following binary programming problem using the MATLAB function bintprog:

where
17.52 Consider the following two‐objective optimization problem:
Find X = {x ₁ x ₂ x ₃ x ₄ x ₅ x ₆}^T

to minimize

subject to

Find the minima of the individual objective functions under the stated constraints using the MATLAB function fmincon.
17.53 Find the solution of the two‐objective optimization problem stated in Problem 17.52 using the weighting function method with the weights w ₁ = w ₂ = 1. Use the MATLAB function fmincon for the solution.
17.54 Find the solution of the two‐objective optimization problem stated in Problem 17.52 using the global criterion method with p = 2. Use the MATLAB function fmincon for the solution.
17.55 Find the solution of the two‐objective optimization problem stated in Problem 17.52 using the bounded objective function method. Take the lower and upper bounds on f ₂ as 80 and 120% of the optimum value found in Problem 17.52. Use the MATLAB function fmincon for the solution.
17.56 Find the solution of the two‐objective optimization problem stated in Problem 17.52 using the goal attainment method. Use the MATLAB function fgoalattain for the solution. Use suitable goals for the objectives.
17.57 Consider the following three‐objective optimization problem:
Find X = {x ₁ x ₂}^T to minimize

subject to

Find the minima of the individual objectives under the stated constraints using the MATLAB function fmincon.
17.58 Find the solution of the three‐objective problem stated in Problem 17.57 using the weighting function method with the weights w ₁ = w ₂ = w ₃ = 1. Use the MATLAB function fmincon for the solution.
17.59 Find the solution of the multiobjective problem stated in Problem 17.57 using the goal attainment method.
Use the MATLAB function fgoalattain for the solution. Use suitable goals for the objectives.

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Table of Contents for 17 Solution of Optimization Problems Using MATLAB

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17.1 Introduction

17.2 Solution of General Nonlinear Programming Problems

17.3 Solution of Linear Programming Problems

17.4 Solution of LP Problems Using Interior Point Method

17.5 Solution of Quadratic Programming Problems

17.6 Solution of One‐Dimensional Minimization Problems

17.7 Solution of Unconstrained Optimization Problems

17.8 Solution of Constrained Optimization Problems

17.9 Solution of Binary Programming Problems

17.10 Solution of Multiobjective Problems

References and Bibliography

Problems

Note

Table of Contents for
17 Solution of Optimization Problems Using MATLAB