On the Minimization of the Sum of Nonconvex Functions with Applications to Mathematical Programming

We study the minimization problem of the sum of two functions in which one of them is nonconvex and nonsmooth and the other is differentiable with a Lipschitz continuous gradient (and possibly nonconvex too). By assuming that the nonconvex nonsmooth function is strongly quasiconvex in the sense of Polyak, we first provide interesting necessary optimality conditions and then we implement the proximal gradient algorithm. As a consequence, new and useful information regarding the point obtained by the stopping criteria as well as for the limit point of the generated sequence under the standard Polyak-Kurdyka-ojasiewicz property is provided. Moreover, we apply these results to different problems from nonconvex optimization as the minimization of the sum of nonconvex functions including difference of convex programming and quadratic fractional programming problems. Finally, examples of strongly quasiconvex functions which satisfy the Polyak-Kurdyka-ojasiewicz property as well as numerical illustrations are presented.