Vector input for an objective function using MultiStart?

Hi All,

Assume I have three rows of the experimental data A[1x13], B[1x13], and C[1x13], and I'd like to find out the best fit for A vs. B & C using the following known equation:

A=x(1)*B+x(2)*C^2 ----- Eq(1)

A,B,C are known values, each of them is a [1x13] vector. I just need the optimization to find out the two coefficients x(1) and x(2). Specifically, I'd like to use MultiStart & Optimization. My real function that describes A vs. B & C is a lot more complex than Eq(1), and my real function has more dimensions than 3D listed here (eg. A vs B,C,D,E,etc.) But I just want to learn where I did wrong using a simple example.

I wrote a simple function that aims to return the optimized x(1) and x(2):

function [x fval R_2] = test(A,B,C,P)

myfunction=@(x,B,C) (x(1).*B+x(2).*C.^2)

fun=@(x)((myfunction([x(1),x(2)],B,C)-A).^2); opts = optimset('Algorithm','interior-point');

problem = createOptimProblem('fmincon','x0',randn(1,2),... 'objective',fun,'lb',[0 0],'ub',... [10000 30],'options',opts);

rs = RandomStartPointSet('NumStartPoints',P); % P= number of iterations

ms = MultiStart('UseParallel','always','Display','iter');

[x fval] = run(ms,problem,rs)

However, I get the error similar to "Inputs must be a scalar and a square matrix". And I read about how to write a scalar objective function, it seems my input for a scalar objective function can be vector. So I am lost.

Can anybody help? Thank you.

Answer 1

[Alan Weiss]

I believe that you settled on a tool before thinking about your problem. It seems to me that you have a linear problem, so should not be using an iterative solution.

Make a matrix M whose columns are composed of the transpose of B and the transpose of C.^2:

M = [B',(C.^2)'];

In this notation, your equations are:

 M*x = A'

Then your least-squares solution vector x is given by

 x = M\(A')

This reasoning extends to any number of dimensions.

Alan Weiss

MATLAB mathematical toolbox documentation