Maximize 3x+y with constraints in Matlab

Question

I need to maximize the equation 3x+y in matlab with the following constraints:

2x+y<=6, x+3y<=9, and x,y>=0

I am having a really hard time figuring out

Answer 1

As @Franck mentioned, you can in general use fmincon to solve optimization problems. However, as your problem is simply a linear programming problem, the solution is much simpler (and guaranteed to be optimal) :

f = -[3 1]; % Note the minus as we want maximization
A = [2 1; 1 3];
b = [6; 9];
LB = [0 0];

[X, FVAL] = linprog(f,A,b,[],[],LB)

Will give:

X =

    3.0000
    0.0000


FVAL =

   -9.0000

Hence the optimum is found at point (3,0) and the resulting value is 9.

Try help linprog to read more about this very usefull function.

Answer 2

Create the following files and run maximize_stuff:

maximize_stuff.m:

function [] = maximize_stuff()

x0 = [2 2]; % fmincon starts at X0 and finds a minimum X
[x,fval] = fmincon('objfun',x0,[],[],[],[],[0;0],[Inf;Inf],'constraint');
fval = -fval; % Because we want to find the maximum, not the minimum

x
fval

end

objfun.m

function f=objfun(x)    
f = 3*x(1) + x(2);
f = -f; % Because we want to find the maximum, not the minimum
end

constraint.m :

function [c,ceq]=constraint(x)

c1 = 2 * x(1) + x(2) - 6; 
c2= x(1) + 3*x(2) - 9;
c = [c1;c2];
ceq = [];

end

It should return:

>> maximize_stuff

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 function tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.

<stopping criteria details>

Active inequalities (to within options.TolCon = 1e-06):
  lower      upper     ineqlin   ineqnonlin
    2                                1

x =

    3.0000         0


fval =

    9.0000

You can verify the results http://www.wolframalpha.com/input/?i=2x%2By%3C%3D6%3B+x%2B3y%3C%3D9%3B+x%3E%3D0%3By%3E%3D0%3B :

A very good tutorial: http://www.math.colostate.edu/~gerhard/classes/331/lab/fmincon.html

fmincon is called as follows:

• with linear inequality constraints Ax£b only (as in linprog): [x,fval]=fmincon('objfun',x0,A,b)

• with linear inequality constraints and linear equality constraints Aeq·x=beq only: [x,fval]=fmincon('objfun',x0,A,b,Aeq,beq)

• with linear inequality and equality constraints, and in addition a lower bound of the form x³lb only: [x,fval]=fmincon('objfun',x0,A,b,Aeq,beq,lb) If only a subset of the variables has a lower bound, the components of lb corresponding to variables without lower bound are -Inf. For example, if the variables are (x,y), and x³1 but y has no lower bound, then lb=[1;-Inf].

• with linear inequality and equality constraints and lower as well as an upper bound of the form x£ub only: [x,fval]=fmincon('objfun',x0,A,b,Aeq,beq,lb,ub) If only a subset of the variables has an upper bound, the components of ub corresponding to variables without upper bound are Inf. For example, if the variables are (x,y) and x£1 but y has no lower bound, then lb=[1;Inf].

• with linear inequality and equality constraints, lower and upper bounds, and nonlinear inequality and equality constraints: [x,fval]=fmincon('objfun',x0,A,b,Aeq,beq,lb,ub,'constraint') The last input argument in this call is the name of a function file (denoted constraint in these notes and saved as constraint.m in the working directory), in which the nonlinear constraints are coded.