Is a global optimal solution of a convex problem always unique?

Question

I do not have a specific problem. Could a convex optimization problem (not strictly convex) have alternate solutions?

is the constant function convex ? hence the answer is no, we need the convex objective function being strictly convex around the minimum for the minimization problem having a unique solution — reuns, Mar 16 '16 at 23:04

score 2 · Accepted Answer · answered Mar 17 '16 at 14:54

2

Yes. Take for instance $$\inf_x \max\{|x|-1,0\}$$ The minimum value is clearly zero, but $$\mathop{\textrm{argmin}}_x \max\{|x|-1,0\}=[1,-1]$$ Notice that the set of optimal points is an interval. In the general, multivariate case, the set of optimal points for a convex optimization model is always a convex set (including possibly the empty set or a singleton).

answered Mar 17 '16 at 14:54

Michael Grant

19,450

You said Yes but then you said it is a convex set. So there could be many solutions? – Turbo Apr 29 '18 at 04:04
The OP asked if a convex optimization problem can have multiple solutions. Hence I said yes... – Michael Grant Apr 30 '18 at 15:51

Is a global optimal solution of a convex problem always unique?

1 Answers1