On solving generalized convex MINLP problems using supporting hyperplane techniques

Abstract

Solution methods for convex mixed integer nonlinear programming (MINLP) problems have, usually, proven convergence properties if the functions involved are differentiable and convex. For other classes of convex MINLP problems fewer results have been given. Classical differential calculus can, though, be generalized to more general classes of functions than differentiable, via subdifferentials and subgradients. In addition, more general than convex functions can be included in a convex problem if the functions involved are defined from convex level sets, instead of being defined as convex functions only. The notion generalized convex, used in the heading of this paper, refers to such additional properties. The generalization for the differentiability is made by using subgradients of Clarke’s subdifferential. Thus, all the functions in the problem are assumed to be locally Lipschitz continuous. The generalization of the functions is done by considering quasiconvex functions. Thus, instead of differentiable convex functions, nondifferentiable $f^{\circ }$-quasiconvex functions can be included in the actual problem formulation and a supporting hyperplane approach is given for the solution of the considered MINLP problem. Convergence to a global minimum is proved for the algorithm, when minimizing an $f^{\circ }$-pseudoconvex function, subject to $f^{\circ }$-pseudoconvex constraints. With some additional conditions, the proof is also valid for $f^{\circ }$-quasiconvex functions, which sums up the properties of the method, treated in the paper. The main contribution in this paper is the generalization of the Extended Supporting Hyperplane method in Eronen et al. (J Glob Optim 69(2):443–459, 2017) to also solve problems with $f^{\circ }$-pseudoconvex objective function.