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.

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We describe a new possibility to model disjunctive optimization problems as generalized semi-infinite programs. In contrast to existing methods in disjunctive programming, our approach does not expect any special formulation of the underlying logical expression. Applying existing lower-level reformulations for the corresponding semi-infinite program, we derive conjunctive nonlinear problems without any logical expressions, which can be locally solved by standard nonlinear solvers. Our preliminary numerical results on some small-scale examples indicate that our reformulation procedure is a reasonable method to solve disjunctive optimization problems.

We address the problem of determining convergent upper bounds in continuous nonconvex global minimization of box-constrained problems with equality constraints. These upper bounds are important for the termination of spatial branch-and-bound algorithms. Our method is based on the theorem of Miranda which helps to ensure the existence of feasible points in certain boxes. Then, the computation of upper bounds at the objective function over those boxes yields an upper bound for the globally minimal value. A proof of convergence is given under mild assumptions. An extension of our approach to problems including inequality constraints is possible.