The robust bilevel continuous knapsack problem with uncertain coefficients in the follower’s objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower’s problem. More precisely, adopting the robust optimization approach and assuming that the follower’s profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower’s reaction from the leader’s perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader’s objective function.

ROmodel: modeling robust optimization problems in Pyomo

This paper introduces ROmodel, an open source Python package extending the modeling capabilities of the algebraic modeling language Pyomo to robust optimization problems. ROmodel helps practitioners transition from deterministic to robust optimization through modeling objects which allow formulating robust models in close analogy to their mathematical formulation. ROmodel contains a library of commonly used uncertainty sets which can be generated using their matrix representations, but it also allows users to define custom uncertainty sets using Pyomo constraints. ROmodel supports adjustable variables via linear decision rules. The resulting models can be solved using ROmodels solvers which implement both the robust reformulation and cutting plane approach. ROmodel is a platform to implement and compare custom uncertainty sets and reformulations. We demonstrate ROmodel’s capabilities by applying it to six case studies. We implement custom uncertainty sets based on (warped) Gaussian processes to show how ROmodel can integrate data-driven models with optimization.

A Toolkit for Robust Risk Assessment Using F-Divergences

This paper assembles a toolkit for the assessment of model risk when model uncertainty sets are defined in terms of an F-divergence ball around a reference model. We propose a new family of F-divergences that are easy to implement and flexible enough to imply convincing uncertainty sets for broad classes of reference models. We use our theoretical results to construct concrete examples of divergences that allow for significant amounts of uncertainty about lognormal or heavy-tailed Weibull reference models without implying that the worst case is necessarily infinitely bad. We implement our tools in an open-source software package and apply them to three risk management problems from operations management, insurance, and finance. This paper was accepted by Baris Ata, stochastic models and simulation.

Radius of Robust Feasibility for Mixed-Integer Problems

For a mixed-integer linear problem (MIP) with uncertain constraints, the radius of robust feasibility (RRF) determines a value for the maximal size of the uncertainty set such that robust feasibility of the MIP can be guaranteed. The approaches for the RRF in the literature are restricted to continuous optimization problems. We first analyze relations between the RRF of a MIP and its continuous linear (LP) relaxation. In particular, we derive conditions under which a MIP and its LP relaxation have the same RRF. Afterward, we extend the notion of the RRF such that it can be applied to a large variety of optimization problems and uncertainty sets. In contrast to the setting commonly used in the literature, we consider for every constraint a potentially different uncertainty set that is not necessarily full-dimensional. Thus, we generalize the RRF to MIPs and to include safe variables and constraints; that is, where uncertainties do not affect certain variables or constraints. In the extended setting, we again analyze relations between the RRF for a MIP and its LP relaxation. Afterward, we present methods for computing the RRF of LPs and of MIPs with safe variables and constraints. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 for computing the RRF. Summary of Contribution: Robust optimization is an important field of operations research due to its capability of protecting optimization problems from data uncertainties that are usually defined via so-called uncertainty sets. Intensive research has been conducted in developing algorithmically tractable reformulations of the usually semi-infinite robust optimization problems. However, in applications it also important to construct appropriate uncertainty sets (i.e., prohibiting too conservative, intractable, or even infeasible robust optimization problems due to the choice of the uncertainty set). In doing so, it is useful to know the maximal “size” of a given uncertainty set such that a robust feasible solution still exists. In this paper, we study one notion of “size”: the radius of robust feasibility (RRF). We contribute on the theoretical side by generalizing the RRF to MIPs as well as to include “safe” variables and constraints (i.e., where uncertainties do not affect certain variables or constraints). This allows to apply the RRF to many applications since safe variables and constraints exist in most applications. We also provide first methods for computing the RRF of LPs as well as of MIPs with safe variables and constraints. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 for computing the RRF.

Modeling Defender-Attacker Problems as Robust Linear Programs with Mixed-Integer Uncertainty Sets

We study a class of sequential defender-attacker optimization problems where the defender’s objective is uncertain and depends on the operations of the attacker, which are represented by a mixed-integer uncertainty set. The defender seeks to hedge against the worst possible data realization, resulting in a robust optimization problem with a mixed-integer uncertainty set, which requires the solution of a challenging mixed-integer problem, which can be seen as a saddle-point problem over a nonconvex domain. We study two exact solution algorithms and present two feature applications for which the uncertainty is naturally modeled as a mixed-integer set. Our computational experiments show that the considered algorithms greatly outperform standard algorithms both in terms of computational time and solution quality. Moreover, our results show that modeling uncertainty with mixed-integer sets, instead of approximating the data using convex sets, results in less conservative solutions, which translates to a lower cost for the defender to protect from uncertainty. Summary of Contribution: We consider a class of defender-attacker problems where the defender has to make operational decisions that depend on uncertain actions from an adversarial attacker. Due to the type of information available to the defender, neither probabilistic modeling, nor robust optimization methods with convex uncertainty sets, are well suited to address the defender’s decision-making problem. Consequently, we frame the defender’s problem as a class of robust optimization problems with a mixed-integer uncertainty sets, and devise two exact algorithms that solve this class of problems. A comprehensive computational study shows that for the considered applications, our algorithms improves the performance of existing robust optimization approaches that can be adapted to solve this class of problems. Moreover, we show how mixed-integer uncertainty sets can reduce the level of over-conservatism that is a known issue of robust optimization approaches.