Multiple initial points in fmincon optimization

Hi

Thank you for reading this question!

I am studying "Fmincon" in Matlab. Based on "Fmincon" document, I wrote a code, as shown below.

close all;
clear;
clc;
fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2; % objective function
x0 = [-1, 2; -1, 2; -1, 2]'; % initial points
Aeq = [0.5, 0.1, 0, 0, 0, 0; 0, 0, 2, 1, 0, 0; 0, 0, 0, 0, 0.2, 2]; % constraints
beq = [1; 1; 1]; % constraints
[x,fval] = fmincon(fun,x0,[],[],Aeq,beq)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

x = 2×3

1.5310 -0.6000 -1.1386 2.3448 2.2000 0.6139

fval = 0.2821

In this code, x0 contains six elements. Then, Aeq needs six columns of elements. Based on "Fmincon" document, Equality constraint equation can be written as below:

0.5*x(1)+0.1*x(2)+x(3)+x(4)+x(5)+x(6)=1 (1)

x(1)+x(2)+2*x(3)+x(4)+x(5)+x(6)=1 (2)

x(1)+x(2)+x(3)+x(4)+0.2*x(5)+2*x(6)=1 (3)

In fact, objective function only includes two variables that are x(1) and x(2). Therefore, how does this code ("Fmincon") work? I have not found the similar question in the community. Could anyone explain it or share the relevant link?

Many thanks in advance!

Answer 1

[Alan Weiss]

fmincon takes the size of the x0 input as determining the number of input variables. Your x0 has 6 elements, so fmincon thinks that you have a 6-variable problem.

You are attempting to vectorize fmincon. However, fmincon cannot be vectorized, meaning it does not accept multiple input points.

Alan Weiss

MATLAB mathematical toolbox documentation

[xin lin]

Hi Alan,

Thank you for your reply. This explanation is very helpful to me.

Best regards,

Xin

Answer 2

[Jon]

As @Alan Weiss points out it seems that you are misunderstanding how MATLAB interprets the 2 by 3 array you are inputting as x0. You are thinking that you are optimizing to find the optimum value of a two element vector, with elements x(1) and x(2), and you think that each column of x0 is a different initial guess.

Infact MATLAB only allows one initial guess. So it interprets your 6 element array of x0's as telling it that you are optimizing to find 6 values (that happen to be in a 2 by 3 array). It then uses linear indexing (numbering elements columwise) to find the initial values of x, so x(1) = x0(1,1), x(2) = x0(2,1), x(3) = x0(1,2), x(4) = x(2,2) etc. It doesn't mind that your objective function only uses the first two of these six elements, but uses all 6 when evaluating the constraints.

[xin lin]

Hi Jon,

Thanks for the detailed explanation. I previously misunderstood the definition of X0. Now, it should be fine.

Best regards,

Xin