Deterministic upper bounds for spatial branch-and-bound methods in global minimization with nonconvex constraints

Abstract

We discuss some difficulties in determining valid upper bounds in spatial branch-and-bound methods for global minimization in the presence of nonconvex constraints. In fact, two examples illustrate that standard techniques for the construction of upper bounds may fail in this setting. Instead, we propose to perturb infeasible iterates along Mangasarian-Fromovitz directions to feasible points whose objective function values serve as upper bounds. These directions may be calculated by the solution of a single linear optimization problem per iteration. Preliminary numerical results indicate that our enhanced algorithm solves optimization problems where a standard branch-and-bound method does not converge to the correct optimal value.

Convergent upper bounds in global minimization with nonlinear equality constraints

“…Yet, as n − m variables have to be fixed using heuristic approaches and some parameters have to be carefully chosen, verification of feasible points is not guaranteed in general. A deterministic method to calculate convergent upper bounds for v * and ensure the termination of branch-and-bound algorithms for purely inequality constrained problems is presented in [21]. In the context of semiinfinite programming a deterministic upper bounding procedure is derived in [31,32] by exploiting a certain Slater condition.…”

“…In this section we present a general spatial branch-and-bound framework to solve global minimization problems and then discuss its main difficulties in the presence of nonconvex equality constraints. The framework and our notation are based on [21]. It is primarily designed to approximate the minimal value v * of problem P(B).…”

“…However, as the problem P(B) is in general nonconvex, finding a feasible point may be as difficult as solving the problem itself. Therefore, there is no guarantee for the occurrence of feasible points within current branch-and-bound algorithms [21]. This is crucial, though, since a branch-and-bound algorithm requires convergent valid upper bounds in the termination criterion.…”

“…This is crucial, though, since a branch-and-bound algorithm requires convergent valid upper bounds in the termination criterion. To cope with this, in the literature sometimes it is suggested to accept ε f -feasible solutions [13], but [21] shows that instead of upper bounds this may generate arbitrarily bad lower bounds. We shall discuss this issue in more detail in Sect.…”

“…The idea behind this is that the small distance between approximately feasible points and the feasible set leads to a small distance in function values as well. However, the function value of an ε ffeasible solution is not guaranteed to be a valid upper bound for v * , as exemplarily shown in [21] for purely inequality constrained problems. In fact, for any ε f examples can be created where the objective value of an ε f -feasible point is arbitrarily much smaller than v * .…”