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

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?

Answer 1

[Walter Roberson]

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.

[Ammy]

Thank you very much @Walter Roberson

[Walter Roberson]

You use gamultiobj() and look for Pareto fronts. You cannot combine two objectives with fminsearch, fmincon, ga, simulated annealing, patternsearch, or surrogate optimization.

However: for some situations it can make sense to take two objectives and weight them to create a biased combined single objective that is then the thing that is what is optimized.

For example hypothetically you could calculate

det(C) - rank(C) 

and maximize. It probably is not a good weighting though.

[Ammy]

Thank you very much.