On the Optimality of Affine Decision Rules in Robust and Distributionally Robust Optimization

We propose tight conditions under which two-stage robust and distributionally robust optimization problems are optimally solved in affine decision rules. Contrary to previous work, our conditions do not impose any structure on the support of the uncertain problem parameters, and they ensure point-wise (as opposed to worst-case) optimality of affine decision rules. The absence of … Read more

A Primal-Dual Lifting Scheme for Two-Stage Robust Optimization

Two-stage robust optimization problems, in which decisions are taken both in anticipation of and in response to the observation of an unknown parameter vector from within an uncertainty set, are notoriously challenging. In this paper, we develop convergent hierarchies of primal (conservative) and dual (progressive) bounds for these problems that trade off the competing goals … Read more

Robust Dual Dynamic Programming

Multi-stage robust optimization problems, where the decision maker can dynamically react to consecutively observed realizations of the uncertain problem parameters, pose formidable theoretical and computational challenges. As a result, the existing solution approaches for this problem class typically determine subopti- mal solutions under restrictive assumptions. In this paper, we propose a robust dual dynamic programming … Read more

A Trust Region Method for the Solution of the Surrogate Dual in Integer Programming

We propose an algorithm for solving the surrogate dual of a mixed integer program. The algorithm uses a trust region method based on a piecewise affine model of the dual surrogate value function. A new and much more flexible way of updating bounds on the surrogate dual’s value is proposed, which numerical experiments prove to … Read more

Global Optimization of Generalized Semi-Infinite Programs via Restriction of the Right Hand Side

The algorithm proposed in [Mitsos Optimization 2011] for the global optimization of semi-infinite programs is extended to the global optimization of generalized semi-infinite programs (GSIP). No convexity or concavity assumptions are made. The algorithm employs convergent lower and upper bounds which are based on regular (in general nonconvex) nonlinear programs (NLP) solved by a (black-box) … Read more

Multi-Variate McCormick Relaxations

G. P. McCormick [Math Prog 1976] provides the framework for convex/concave relaxations of factorable functions, via rules for the product of functions and compositions of the form F(f(z)), where F is a univariate function. Herein, the composition theorem is generalized to allow multivariate outer functions F, and theory for the propagation of subgradients is presented. … Read more

Pessimistic Bi-Level Optimisation

Bi-level problems are optimisation problems in which some of the decision variables must optimise a subordinate (lower-level) problem. In general, the lower-level problem can possess multiple optimal solutions. One therefore distinguishes between optimistic formulations, which assume that the most favourable lower-level solution is implemented, and pessimistic formulations, in which the most adverse lower-level solution is … Read more

A New Approach to the Feasibility Pump in Mixed Integer Programming

The feasibility pump is a recent, highly successful heuristic for general mixed integer linear programming problems. We show that the feasibility pump heuristic can be interpreted as a discrete version of the proximal point algorithm. In doing so, we extend and generalize some of the fundamental results in this area to provide new supporting theory. … Read more

Boosting the Feasibility Pump

The Feasibility Pump (FP) has proved to be an effective method for finding feasible solutions to mixed integer programming problems. FP iterates between a rounding procedure and a projection procedure, which together provide a sequence of points alternating between LP feasible but fractional solutions, and integer but LP relaxed infeasible solutions. The process attempts to … Read more