ON A CLASS OF NONCONVEX PROBLEMS WHERE ALL LOCAL MINIMA ARE GLOBAL

We characterize a class of optimization problems having convex objective function and nonconvex feasible region with the property that all local minima are global. minx∈V f (x) g(x )=0 h(x) 0 where x ∈ R n , f : V → R with V ⊆ R n ,a ndg ,h are systems of equations and inequalities, respectively. We are interested in finding the global minimum ˆ x. Most Branch-and-Bound algorithms designed to solve this kind of problems need to compute the lower bound to the objective function value in each Branch- and-Bound region. The lower bound is calculated by solving a convex relaxation of the original problem, the reason for this being that in a convex problem all local minima are global ones; hence a local solver can be used to obtain a guaranteed lower bound to the objective function value of the original problem in the region of interest (SP99, Lib04, Lib05). However, the notion of globality of all local minima applies to many problem instances (Bon98), apart from the class of convex problems. This suggests the use of nonconvex relaxations (having the same local-to-global minimality property) which may be much tighter than an ordinary convex relaxations, and thus speed up considerably the global optimization software acting on the nonlinear problem. Here, we follow a topological approach. We prove that certain convex functions defined on nonconvex sets satisfying a special set of conditions have the desired property. There is some relation between this work and (Rap91).