The answer, in general, is "No". If you are lucky your set is polyhedron, whence it is enough to compute all the extreme points in the set and then compare function values.

It is clear that the solution must be an extreme point of the convex region if this is compact. In a minimal entropy problem this will typically mean that your solution is a discrete distribution with finite support.

I require a discrete distribution with finite support. But i do not know how to arrive at it preferably a closed form solution

Hi, could you please give us more details concerning the exact form of your function and also the constraint?

The answer, in general, is "No". If you are lucky your set is polyhedron, whence it is enough to compute all the extreme points in the set and then compare function values.

@Richard Epenoy. Hello, My objective function is

min sum (-Pi log(Pi)).

where the sum is over i=1....N. The constraint is of the form norm(P-X)0.

I guess your "

Hi, you may try to apply the KKT necessary optimality conditions to the problem and also analyze the second-order sufficient condition for optimality to try to prove that a local minimum of the problem necessarily satisfies norm(P-X)=epsilon.