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How to define an objective function that maximize one thing and minimize the other at the same time?

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I want to define an objective function that maximize the determinant and minimize the rank at the same time.
A=rand(5);
B=rand(5);
X=A+k*B;
maximize @(k) det(X)
minimize @(k) rank(X)
Is there anyway to joint both in one way?

Accepted Answer

Walter Roberson
Walter Roberson on 4 Aug 2021
Any matrix that does not have full rank will have a determinant of 0. If there is a k in range that makes X have rank less than maximum then the determinant for that will be 0.
It can make sense to look for the smallest rank that k can drive X but all such cases will have det 0, and any full rank matrix with positive det would have larger det. That is an incompatible goal with low rank.
Exception: it is hypothetically possible that det(X) is negative for all k that do not drive X singular. In such a case both goals can be realized by looking for the k that generates the lowest rank, since det 0 would be greater than any negative det.
But I doubt that situation applies in practice.
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