How can I do this L1 integral minimization?

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Sijie Huang 2018-10-29

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编辑： Bruno Luong 2018-10-30

Greetings,

I have the following integral

where kdx = [pi/16, pi/2], and

I want to minimize the integral above and solving corresponding a_j. But I have no clue how do to it. Can anyone give me some hint?

Thanks.

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Bruno Luong 2018-10-29

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编辑：Bruno Luong 2018-10-29

在 MATLAB Online 中打开

The problem of linear L1 fit (your case); meaning

argmin_x | M*x - y |_l1
argmin sum abs(M*x - y)

can be reformulated and solved by linear programming (opt toolbox required) using slack variables trick as following

n = length(y);
Aeq = [M speye(n) -speye(n)];
Aeqpr=nonzeros(Aeq);
beq = y(:);
c = [zeros(1,size(M,2)) ones(1,2*n)];
LB = [-inf(1,size(M,2)) zeros(1,2*n)];
UB = [];
c = c(:);
LB = LB(:);
UB = UB(:);
x0 = zeros(size(c)); % guess vector
[sol, f, exitflag] = linprog(c,[],[], Aeq, beq, LB, UB, x0);
x = sol(1:size(M,2));