A Survey on Bilevel Optimization Under Uncertainty

Bilevel optimization is a very active field of applied mathematics. The main reason is that bilevel optimization problems can serve as a powerful tool for modeling hierarchical decision making processes. This ability, however, also makes the resulting problems challenging to solve—both in theory and practice. Fortunately, there have been significant algorithmic advances in the field … Read more

Confidence Interval Software for Multi-stage Stochastic Programs

When the uncertainty is explicitly modeled in an optimization problem, it is often necessary to use samples to compute a solution, which gives rise to a need to compute confidence intervals around the objective function value that is obtained. In this paper we describe software that implements well-known methods for two stage problems and we … Read more

Γ-Robust Linear Complementarity Problems

Complementarity problems are often used to compute equilibria made up of specifically coordinated solutions of different optimization problems. Specific examples are game-theoretic settings like the bimatrix game or energy market models like for electricity or natural gas. While optimization under uncertainties is rather well-developed, the field of equilibrium models represented by complementarity problems under uncertainty … Read more

Risk management for forestry planning under uncertainty in demand and prices.

The forest-harvesting and road-construction planning problem basically consists of managing land designated for timber production and divided into harvest cells. For each time period in the given time horizon one must decide which cells to cut and what access roads to build in order to maximize expected net profit under a risk manageable scheme to … Read more

Satisficing Models under Uncertainty

Satisficing, as an approach to decision-making under uncertainty, aims at achieving solutions that satisfy the problem’s constraints as well as possible. Mathematical optimization problems that are related to this form of decision-making include the P-model of Charnes and Cooper (1963). In this paper, we propose a general framework of satisficing decision criteria, and show a … Read more

The Decision Rule Approach to Optimization under Uncertainty: Methodology and Applications

Dynamic decision-making under uncertainty has a long and distinguished history in operations research. Due to the curse of dimensionality, solution schemes that naively partition or discretize the support of the random problem parameters are limited to small and medium-sized problems, or they require restrictive modeling assumptions (e.g., absence of recourse actions). In the last few … Read more

Selected Topics in Robust Convex Optimization

Robust Optimization is a rapidly developing methodology for handling optimization problems affected by non-stochastic “uncertain-but-bounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of {\sl robust counterpart} of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links … Read more