We consider a quadratic minmax problem with coupled inner constraints and propose a method to compute a class of stationary points. To motivate the need to compute such stationary points, we first show that they are meaningful, in the sense that they can be locally optimal for our problem under suitablenon-degeneracy conditions. Then based on a suitable log barrier function, we build an infeasible interior point-type single loop method (which does not explicitly distinguish between the outer and inner problem) and prove that a non-degenerate stationary point is an attraction point as the algorithm moves along the designed central path. We show in particular that our method is polynomial in the special case where the inner feasible set of our constrained minmax problem is independent from outer variables. Our numerical experiments, on both synthetic data and a class of min-cost flow problems, showcase the behavior of our method and how it outperforms existing algorithms from the literature in terms of the quality of the computed stationary points.

Many mathematical models are based on the coupling of two or more optimization problems. This paper surveys possibilities to couple two optimization problems and discusses how solutions of the different models are interrelated with each other. The considered pairs stem from the fields of standard and generalized Nash equilibrium problems, optimistic and pessimistic bilevel problems, saddle problems, standard and generalized semi-infinite problems, robust optimization, Lagrange duality, bicriteria optimization problems, minimax problems, decomposition, and two-stage stochastic optimization. Some connections to vector optimization and variational inequalities are discussed as well.

Graphs of minimal point mappings of parametric optimization problems appear in the definition of feasible sets of bilevel optimization problems and of semi-infinite optimization problems, and the intersection of multiple such graphs defines (generalized) Nash equilibria. This paper shows how minimal point graphs of nonconvex parametric optimization problems can be enclosed with the help of well structured problems with additional parameters. This enclosure coincides with the minimal point graph under mild assumptions. We specify our results to the setting of generalized Nash equilibrium problems. This well structured formulation of the enclosure makes it accessible to approximations by branch-and-bound methods. We provide corresponding numerical results in a separate paper.

We study the supportedness of nondominated points of multiobjective optimization problems, that is, whether they can be obtained via weighted sum scalarization. One key question is how supported points behave under an efficiency-preserving transformation of the original problem. Under a differentiability assumption, we characterize the transformations that preserve both efficiency and supportedness as the component-wise transformations with strictly increasing and convex components. In addition, we consider transformations that can render originally unsupported points supported in the transformed problem. This enables algorithms to find nondominated points by applying the weighted sum scalarization to a transformed problem.

This paper systematically surveys useful properties of the epigraph reformulation for optimization problems, and complements them by some new results. We focus on the complete compatibility of the original formulation and the epigraph reformulation with respect to solvability and unsolvability, the compatibility with respect to some, but not all, basic constraint qualifications, the formulation of first-order optimality conditions for problems with max-type objective function, and the interpretation of feasibility and optimality cuts along epigraphs in the framework of cutting plane methods. Finally we introduce a generalized epigraph reformulation which is particularly useful for treating nonsmooth summands of objective and constraint functions independently in the reformulation.

Finding good feasible points is crucial in mixed-integer programming. For this purpose we combine a sufficient condition for consistency, called granularity, with the moment-/sum-of-squares-hierarchy from polynomial optimization. If the mixed-integer problem is granular, we obtain feasible points by solving continuous polynomial problems and rounding their optimal points. The moment-/sum-of-squares-hierarchy is hereby used to solve those continuous polynomial problems, which generalizes known methods from the literature. Numerical examples from the MINLPLib illustrate our approach.

The sharp local minimality of feasible points of nonlinear optimization problems is known to possess a characterization by a strengthened version of the Karush-Kuhn-Tucker conditions, as long as the Mangasarian-Fromovitz constraint qualification holds. This strengthened condition is not easy to check algorithmically since it involves the topological interior of some set. In this article, we derive an algorithmically tractable version of this condition, called strong Karush-Kuhn-Tucker condition. We show that the Guignard constraint qualification is the weakest condition under which a feasible point is a strong Karush-Kuhn-Tucker point for every continuously differentiable objective function possessing the point as a sharp local minimizer. As an application, our results yield an algebraic characterization of strict local minimizers of linear programs with cardinality constraints.

This paper treats the solution of nonlinear optimization problems involving discrete decision variables, also known as generalized disjunctive programming (GDP) or mixed-integer nonlinear programming (MINLP) problems, that arise in process engineering. The key idea is to eliminate the discrete decision variables by adding a set of continuous variables and constraints that represent the discrete decision space of the optimization problem. With such a reformulation, we are able to apply solution algorithms for purely continuous nonlinear optimization problems to efficiently calculate local minima of GDP or MINLP problems. In this contribution, we propose different alternatives to reformulate GDP/MINLP problems as continuous optimization problems. We furthermore investigate theoretical properties of the different reformulations with regard to their numerical solution. The proposed formulations are illustrated and analyzed on the basis of optimization problems dealing with process engineering applications involving stationary as well as dynamic process models.

This paper introduces a spatial branch-and-bound method for the computation of the set of all 6-Nash equilibria of continuous box-constrained nonconvex Nash equilibrium problems with an approximation guarantee. Thereby, the existence of 6-Nash equilibria is not assumed, but the algorithm is also able to detect their absence. We explain appropriate discarding and fathoming techniques, provide a termination proof for a prescribed approximation tolerance, and report our computational experience.

In their seminal work "A General Theory of Equilibrium Selection in Games" [The MIT Press, Cambridge 1988] J. C. Harsanyi and R. Selten introduce the notion of payoff dominance to explain how players select some solution of a Nash equilibrium problem from a set of nonunique equilibria. We formulate this concept for generalized Nash equilibrium problems, relax payoff dominance to the more widely applicable requirement of payoff nondominatedness, and show how different characterizations of generalized Nash equilibria yield different semi -infinite optimization problems for the computation of payoff nondominated equilibria. Since all these problems violate a standard constraint qualification, we also formulate regularized versions of the optimization problems. Under additional assumptions we state a nonlinear cutting algorithm and provide numerical results for a multi -agent portfolio optimization problem.

Inspired by the recently introduced branch-and-bound method for continuous multiobjective optimization problems from G. Eichfelder, P. Kirst, L. Meng, O. Stein [A general branch-and-bound framework for continuous global multiobjective optimization. J Glob Optim. 2021;80:195-227], we study for a general class of branch-and-bound methods in which sense the generated terminal enclosure and the terminal provisional nondominated set approximate the nondominated set when the termination accuracy is driven to zero. Our convergence analysis of the enclosures relies on constructions from the above paper, but is self-contained and also covers the mixed-integer case. The analysis for the provisional nondominated set is based on general convergence properties of the epsilon-nondominated set, and hence it is also applicable to other algorithms which generate such points. Furthermore, we discuss post-processing steps for the terminal enclosure and provide numerical illustrations for the cases of two and three objective functions.