Hacky unconstrained vs. constrained numerical optimization

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Evan 2017-5-1

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/338207-hacky-unconstrained-vs-constrained-numerical-optimization

Short version: Why is unconstrained optimization with a constraint "hacked" into the objective function working while constrained optimization isn't?

Long version:

I'm trying to find a p x n subspace U which explains an arbitrary amount of variance in my m x p dataset X. The objective function is:

    projection = U'*X';
    fracVar = sum(var(projection'))/totalVar;
    fVal = (targetVar - fracVar)^2;

I tried using `fmincon()` with a nonlinear equality constraint that U must have orthogonal columns of unit norm:

temp = norm(U'*U - eye(size(U,2)));

I can not get this to work. It fails in different ways, including not getting anywhere near the desired fraction of variance and failing to meet the constraints.

However, if I use `fminsearch()` instead and change the objective to:

    projection = orth(U)'*X';
    fracVar = sum(var(projection'))/totalVar;
    fVal = abs(targetVar - fracVar);

and then use `orth()` again on the result, it seems to work perfectly. I thought this would be the "wrong" way as I'm basically hacking the constraint into the objective.

What is an explanation for this? Is there a "correct" approach to this problem?