How to solve multiobjective optimization problem?

Dear all,

How can I solve the attched multiobjective optimization problem?

[Walter Roberson]

possibly a change of variables so that the logs on the right hand side of the st equations became linear variables (might require introducing a new variable to link 1-v1 and v1 in linear ways)? If that can be done then all those log(constant) in the st equations become just numeric constants in an A*x<=b linear inequality setup and it could be handled as a single objective system.

You cannot use fmincon for this because min() is not differentiateable as required by most of the minimizers. ga() or patternsearch()

[Rajkumar Verma]

But, I want to solve this problem without making linear combination of log(1-v1) and log(v2). Because I need the values of v1 and v2 separately. Is there any way to solve without making linear combination?

Answer 1

[Alan Weiss]

If you have Global Optimization Toolbox you can use paretosearch or gamultiobj to find a Pareto set for your problem. See Multiobjective Optimization. If you need help after reading the examples in that section, ask again.

Alan Weiss

MATLAB mathematical toolbox documentation

[Walter Roberson]

There is only one objective, so paretosearch and gamultiobj are not appropriate.

[Rajkumar Verma]

Dear, I have written the following code.

clear all

clc

lb = [0,0,0,0,0];

Aeq = [1 1 1 0 0];

beq = 1;

nvars = length(lb);

fitnessfcn = @kur_MO;

Nonlcon = @NLcon;

A = [0 0 0 1 1];

b = [1];

ub = [];

[x,fval,exitflag,output,population,score]= gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq,lb,ub,Nonlcon);

function f = kur_MO(x)

f(1) = log(1-x(4));

f(2) = log(x(5));

end

function [C, Ceq] = NLcon(x)

C (1) = (x(1))*log(1.7/8)+(x(2))*log(0.8/8)+(x(3))*log(5.5/8)-log(1-x(4));

C (2) = (x(1))*log(5.6/8)+(x(2))*log(1.7/8)+(x(3))*log(4.5/8)-log(1-x(4));

C (3) = (x(1))*log(7.7/8)+(x(2))*log(6.5/8)+(x(3))*log(1.7/8)-log(1-x(4));

C (4) = (x(1))*log(1.2/8)+(x(2))*log(0.3/8)+(x(3))*log(5.1/8)-log(x(5));

C (5) = (x(1))*log(4.1/8)+(x(2))*log(1.2/8)+(x(3))*log(4.4/8)-log(x(5));

C (6) = (x(1))*log(7.5/8)+(x(2))*log(3.8/8)+(x(3))*log(1.2/8)-log(x(5));

Ceq = [];

end

but it is showing the error.

Error using constrValidate (line 59)

Constraint function must return real value.

Error in gacommon (line 125)

[LinearConstr, Iterate,nineqcstr,neqcstr,ncstr] =

constrValidate(NonconFcn, ...

Error in gamultiobj (line 289)

~,options] =

gacommon(nvars,fun,Aineq,bineq,Aeq,beq,lb,ub,nonlcon,[],options,user_options);

Error in MultGA (line 13)

[x,fval,exitflag,output,population,score]=

gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq,lb,ub,Nonlcon);

How can I resolve this error?

[Alan Weiss]

I have two suggestions:

1. Use the debugger and see why the nonlinear constraint function is returning a non-real value.
2. Set upper bounds on all variables of 1 - eps and lower bounds of eps, just in case gamultiobj is choosing a value of 0 or 1, which would cause this error.

Alan Weiss

MATLAB mathematical toolbox documentation

[Walter Roberson]

yes, exact 0 or 1 are causing the non-real problem

[Walter Roberson]

using gamultiobj is not appropriate for this problem. The task is not to find the Pareto front for the two logs. Your objective is a scalar, the min() of the two logs.

Answer 2

[Walter Roberson]

Is there any way to solve without making linear combination?

You could use ga with nonlinear inequalities.

There is another approach. You can do two seaches, one with the objective log(1-v1), and the other with objective log(v2). For each of those searches, the objective is continuous, so you can use fmincon() for each of them. Then take results that correspond to the search with the lower objective.

If you do this then you could use fmincon() with nonlinear constraints.

I have a suspicion that either the st on log(1-v1) or the st on log(v2) turn out to be equalities rather than inequalities, but I have not tested for that.

[Walter Roberson]

Separate fmincon works quite well, by the way. I did extensive search on the entire solution space and verified that the results from fmincon are within tolerance of the minimum possible.