使用fmincon函数MATLAB进行定义的步长变化

Question

I'm using the optimization toolbox "fmincon" of MATLAB, but I'm having the next problem:

I have 6 parameters to vary, a couple of them most vary in even numbers, from 4 to 16 (this values can vary, but always will vary in even numbers). So let's define them like this:

x1=[4:2:16];
x2=[4:2:16];

Another couple of variables must change between 300 and 1500, in steps of 100, I mean:

x3=[300:100:1500];
x4=[300:100:1500];

The last couple just vary between 4 and 6, like these:

x5=4:6;
x6=4:6;

The restriction of the parameters are these:

x1<=x2
x3<=x4
x5<=x6

A very important thing here is that the variation that makes fmincon can't make little changes, I mean, the first value of x1 which is 4 , can't be 4.0000000001 , because in my objective function that changes will make no difference; and there is my problem, because the steps are too little, so the variation will make nothing, and the algorithm stops, saying that there's no variation of the objective function.

I have set in the fmincon , DiffMinChange=1 , and that works for the first iteration, and them, it starts to make too little steps. This is the initial configuration for fmincon :

options1 = optimset('Display','iter',...
    'Algorithm','sqp','PlotFcns',@optimplotfval,...
    'MaxIter',400,'MaxFunEvals',2000,'DiffMinChange',1);

The initial restrictions are:

A=[1 -1 0 0 0 0;0 0 1 -1 0 0;0 0 0 0 1 -1];
b=[0;0;0];

To be more clear, what I'm looking for is make 3 ranges, lets defined like these:

R1=[x1:2:x2];
R2=[x3:100:x4];
R3=[x5:x6];

EDIT 1: You may to know that each evaluating of the objective function will take about 2-3 hours.

As you can see, finally what I'm looking for is the variation of an interval, for that reason the limit in the beginning can't be bigger than the limit in the top, otherwise the rank will be empty.

Answer 1

Well, this doesn't address how to make fmincon respect these constraints, but just a potential thought...

Given the ranges you want, you have a 7x7x13x13x3x3 space of potential variable combinations, which is a total of ~75000 combinations, before you restrict to x1<=x2, x3<=x4, x5<=x6. That's not a huge space - why not just evaluate the objective function at every combination of the parameters, and then use min to find the minimum of your objective function?

Answer 2

Since fmincon is an optimizer for continuous problems, and your problem is discrete, there is a misfit between problem and solution approach, see for instance wiki .

You could try a discrete optimization solver, such as branch and bound . Since the number of points in your feassible region is quite limited, a brute force approach could also work (depends on the time your objective function needs). Something like this should work for your objective function fun :

lb = [4 4 300 300 4 4]; % lower bound
st = [2 2 100 100 1 1]; % step size
ub = [16 16 1500 1500 6 6]; % upoper bound

% init the best solution as infinity
bestF = inf;
bestX = nan(6,1);

% construct all permutations
for idx = 1:numel(lb)
    % construct range
    nextRange = (lb(idx):st(idx):ub(idx))';
    if (idx==1)
        permutations = nextRange;
    else
        a = repmat(permutations,numel(nextRange),1);
        b = repmat(nextRange,1,size(permutations,1))';
        permutations = [a,b(:)];
    end
    % remove permutations that do not satisfy constraints
    if (idx==2)
        permutations(permutations(:,1) > permutations(:,2),:) = [];
    end
    if (idx==4)
        permutations(permutations(:,3) > permutations(:,4),:) = [];
    end
    if (idx==6)
        permutations(permutations(:,5) > permutations(:,6),:) = [];
    end
end

N = size(permutations,1);

assert(N == 7*4 * 13*7 * 6)

for idx = 1:N
    candX = permutations(idx,:)';
    % evaluate ...
    candF = fun(candX);
    % ... and if improvement, update best
    if (candF < bestF)
        bestF = candF;
        bestX = candX;
    end
end

% results
bestF
bestX

The number of permutations equals 15288, so if your objective function fun needs .1 seconds (which is quite a lot ;)), you have to wait for 25 minutes for an answer.

Answer 3

I've been reading in a lot of forums, and I found a very interesting solution, I tried and actually it works pretty well. What I found is that there are some differences between the objective functions. What I been trying here is using fmincon, but this function only works for objective functions that changes in all the range, in other words, is differenciable in all the range. But here there is a difference because this function only works with some specific values, and the objective function will be the same if the variation is not significant; in other words, is not differenciable in all the range. What I found, is that there's a function in MATLAB called "patter search", and I tried and I got wonderfull results, It quite similar to fmincon, but it works different. I recomend it for this kind of problems.