An inverse problem In convex optimization

Abstract

This research aims mainly to solve an inverse problem arising in convex optimization. (P) n max x f(x) ; Ax = C(A), where f is a strictly increasing funnction with respect to each coordinate of the vector x, the Hessian matrix D2 x f is negative definite on the subspace {Dxf} ⊥, f is of class C 2 , A is an m × n matrix of rank m, C : R m×n ++ → R m ++ is homogeneous of degree one and x ∈ R n . We consider a maximization problem under m linear constraints, we characterize the solutions of this kind of problems and give necessary and sufficient conditions for a given function in R n to be the solution of a multi-constraints maximization problem.oblem.