Searching for Minimum of a function

Hello,

I must find the minimum of the following fuction but i'm having hard time. Some help would appreciated!

f(x) = x1^2 + x2^2 - 2x1 - 18x2

2x1 + 3x2 - 6 <= 0

x1 + 4x2 - 5 <= 0

x1, x2 >= 0

[Dyuman Joshi]

look into fmincon

[Iliana Zheleva]

Ahh I looked at that but i don't get it. Don't have much experience with MatLab but an assignment is due few days from now.

[Jan]

"I don't get it" does not allow to clarify the detail, you do not understand. Please post, what you have tried so far and ask a specific question.

[Matt J]

See also quadprog.

Answer 1

[Sam Chak]

Hi @Iliana Zheleva

You should analyze the function so that you get to see where the solution might be and it allows you to guess the starting point of the optimization. This works well if the number of variables is lesser than or equal to 2.

[x1, x2] = meshgrid(-10:20/40:10, -5:25/40:20);
f  = x1.^2 + x2.^2 - 2*x1 - 18*x2;
g1 = 2*x1 - 3*x2 - 6;
g2 = x1 + 4*x2 - 5;

To show the objective function and the constraints in 3D:

figure(1);
surf(x1, x2, f)
hold on
surf (x1, x2, g1)
surf (x1, x2, g2)
hold off
title('3D display of obj fcn f(x_{1}, x_{2}) and the constraints g_{1} and g_{2}');
xlabel({'$x_{1}$'}, 'Interpreter', 'latex')
ylabel({'$x_{2}$'}, 'Interpreter', 'latex')

To show the contours of the objective function and the constraints:

figure(2);
[c] = contour(x1, x2, f, '-');
clabel(c);
hold on
[c] = contour(x1, x2, g1, ':');
clabel(c);
[c] = contour(x1, x2, g2, ':');
clabel(c);
hold off
axis square
title('Contour of obj fcn f(x_{1}, x_{2}) and the constraints g_{1} and g_{2}');
xlabel({'$x_{1}$'}, 'Interpreter', 'latex')
ylabel({'$x_{2}$'}, 'Interpreter', 'latex')

If you follow the first example in fmincon, it looks very easy because there are only a few lines. The "hardest part" is that you need to rewrite the linear inequality constraints

in the matrix form :

fun = @(x) x(1)^2 + x(2)^2 - 2*x(1) - 18*x(2);
x0  = [initial_x1, initial_x2];
A   = [a1 a2; a3 a4];
b   = [b1; b2];
lb  = [0, 0];       % lower bounds set to 0 since the problem requires to search x1 ≥ 0 and x2 ≥ 0
ub  = [10, 20];     % upper bounds, makes to algorithm to search in 0 ≤ x1 ≤ 10 & 0 ≤ x2 ≤ 20 region
x   = fmincon(fun, x0, A, b, [], [], lb, ub)

[Iliana Zheleva]

Thank you so much <3