Hello everyone, I have a question in optimization

I would then suggest to solve 42 sub linear-programing problems by replacing the (1c) with one of those conditions. The sub problem can be solved with intlinprog, then we just take the argmin of those 42 problem solutions.

It must be more preditable and robust than GA.

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Bruno Luong 2020-11-21

编辑：Bruno Luong 2020-11-22

在 MATLAB Online 中打开

Here is a code for l^2-norm using QUADPROG for a toy example.

(this might be the last fish you might get from me).

Up to you to do l1 or linfinity norm (hint uing LINPROG or other tools we suggested you in other threads) .

EDIT: some bugs are found and corrected

N1=eye(2);
N0=[2; 4];
nh=-1000;
H1=[1 0; 0 1];
u1=[-0.7; 10];
R11=eye(2);
r00=[0; 0];
rm=2;
C11=eye(2);
c00=zeros(2,1);
cm=3;
% quadratic form and linear form where the sum is to be minimized
n = size(N1,2);
H = N1'*N1;
f = N0'*N1;
%
A = [-N1;  % constraint (a)
      H1]; % constraint (b)
b = [N0 - nh;
     u1];
% m slack variables to force L1-norm constraint (d)
m = size(C11,1);
As = [ C11 -eye(m);
      -C11 -eye(m)];
bs= [-c00;
     +c00];
Ad = [zeros(1,n) ones(1,m)];
bd = cm;
A = [A zeros(size(A,1),m);
     As;
     Ad];
b = [b;
     bs;
     bd];  
H(end+m,end+m) = 0;
f(end+m) = 0; % slack variables matches vector bounds, probably can be set the m-last elements to 1s
% Loop for constraint (c), decomposing
fmin = Inf;     
for i=1:size(R11,1)
    for s=[-1 1]     
        Ack = [-s*R11(i,:), zeros(1,m)];
        bck = s*r00(i)-rm;
        Ak = [A; Ack];
        bk = [b; bck];
        [xk, fk, flag] = quadprog(H, f, Ak, bk);
        if flag == -3
            fmin = -Inf;
            break
        elseif flag >= 0 && fk < fmin
            xmin = xk;
            fmin = fk;
        end
    end
    if flag < 0 && flag ~= -2 % not fesasible
        break
    end
end
if isfinite(fmin)
    w = xmin(1:n);
    obj = norm(N1*w + N0);
    tol = 1e-6;
    constraints = [all(N1*w + N0        >= nh-tol), ...	(a)
                   all(H1*w - u1        <= 0+tol),  ...	(b)
                   norm(R11*w+r00,Inf)  >= rm-tol,  ...	(c)
                   norm(C11*w+c00,1)    <= cm+tol   ...	(d)
                   ];
    fprintf('constraints satisfied = %s\n', mat2str(constraints));
    fprintf('solution = %s\n', mat2str(w));
    fprintf('minnorm = %g\n', obj);
    if ~all(constraints)
        fprintf('constraints not satisfied (bug?)\n')
    end
elseif fmin == -Inf
    fprintf('problem unbounded\n')
else
    error('No solution found')
end

Matt J 2020-11-23

编辑：Matt J 2020-11-23

so no the gradient does not use parethesis like your.

OK, but they should do it that way, no? They missed the opportunity to take advantage of the special form of the problem.

Bruno Luong 2020-11-23

Does it really matter for accuracy? At the cost of 2 matrix-vector product per gradient evaluation instead of one?

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