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Generating random starting points in n-dimension for fmincon algorithm - uniform distribution of space

user1704
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user1704 Published in May 24, 2018, 11:21 pm

I am using fmincon in Matlab to minimize a smooth yet highly non-linear objective function. My problem is in n-dimension, with n>250 in some instances. I have found that using different starting points improves dramatically the solution, as can be expected from an objective function with multiple local minima. I have used haltonset to generate quasi random starting point in the n dimensional hypercube, which I then rescale into the n-dimensional hyperspace defined by the lower and upper bounds of my problem.

I am however unsure of whether I am covering the space of possible solutions (defined by lower and upper bounds) in a sufficiently uniform manner. It could come from 2 issues: 1) the initial generation of the quasi random points is not "dense" enough, 2) the rescaling could create holes in the hyperspace because of bounds that are highly different

Is there a way that I can ensure that my starting points cover my space adequately (maybe specify the maximum distance between my starting points in n-dimension).

Thank you for your help

See below code I am using:

objective_list = containers.Map('KeyType','int32','ValueType','any');
initial_pt_list = containers.Map('KeyType','int32','ValueType','any');
optimal_pt_list = containers.Map('KeyType','int32','ValueType','any');

net_size=100;
[variable_size, ~]=size(initial_pt);


p=haltonset(variable_size,'Skip',1e3,'Leap',1e2);
p = scramble(p,'RR2');
X0 = net(p,net_size);

for net0 = 1:1:net_size

    temp_unit_vector= X0(net0,:);    
    temp_initial_pt = lower_b + temp_unit_vector'.*(upper_b - lower_b);
    initial_pt_list(net0)=temp_initial_pt;

end
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