Inverse Optimization with Noisy Data

Inverse optimization refers to the inference of unknown parameters of an optimization problem based on knowledge of its optimal solutions. This paper considers inverse optimization in the setting where measurements of the optimal solutions of a convex optimization problem are corrupted by noise. We first provide a formulation for inverse optimization and prove it to be NP-hard. In contrast to existing methods, we show that the parameter estimates produced by our formulation are statistically consistent. Our approach involves combining a new duality-based reformulation for bilevel programs with a regularization scheme that smooths discontinuities in the formulation. Using epi-convergence theory, we show the regularization parameter can be adjusted to approximate the original inverse optimization problem to arbitrary accuracy, which we use to prove our consistency results. Next, we propose two solution algorithms based on our duality-based formulation. The first is an enumeration algorithm that is applicable to settings where the dimensionality of the parameter space is modest, and the second is a semiparametric approach that combines nonparametric statistics with a modified version of our formulation. These numerical algorithms are shown to maintain the statistical consistency of the underlying formulation. Lastly, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics

Solving a type of biobjective bilevel programming problem using NSGA-II

AbstractThis paper considers a type of biobjective bilevel programming problem, which is derived from a single objective bilevel programming problem via lifting the objective function at the lower level up to the upper level. The efficient solutions to such a model can be considered as candidates for the after optimization bargaining between the decision-makers at both levels who retain the original bilevel decision-making structure. We use a popular multiobjective evolutionary algorithm, NSGA-II, to solve this type of problem. The algorithm is tested on some small-dimensional benchmark problems from the literature. Computational results show that the NSGA-II algorithm is capable of solving the problems efficiently and effectively. Hence, it provides a promising visualization tool to help the decision-makers find the best trade-off in bargaining

On SOCP-based disjunctive cuts for solving a class of integer bilevel nonlinear programs

We study a class of integer bilevel programs with second-order cone constraints at the upper-level and a convex-quadratic objective function and linear constraints at the lower-level. We develop disjunctive cuts (DCs) to separate bilevel-infeasible solutions using a second-order-cone-based cut-generating procedure. We propose DC separation strategies and consider several approaches for removing redundant disjunctions and normalization. Using these DCs, we propose a branch-and-cut algorithm for the problem class we study, and a cutting-plane method for the problem variant with only binary variables. We present an extensive computational study on a diverse set of instances, including instances with binary and with integer variables, and instances with a single and with multiple linking constraints. Our computational study demonstrates that the proposed enhancements of our solution approaches are effective for improving the performance. Moreover, both of our approaches outperform a state-of-the-art generic solver for mixed-integer bilevel linear programs that is able to solve a linearized version of our binary instances.Comment: arXiv admin note: substantial text overlap with arXiv:2111.0682

Hierarchical decision making with supply chain applications

Hierarchical decision making is a decision system, where multiple decision makers are involved and the process has a structure on the order of levels. It gains interest not only from a theoretical point of view but also from real practice. Its wide applications in supply chain management are the main focus of this dissertation.The first part of the work discusses an application of continuous bilevel programming in a remanufacturing system. Under intense competitive pressures to lower production costs, coupled with increasing environmental concerns, used products can often be collected via customer returns to retailers in supply chains and remanufactured by producers, in orderto bring them back into “as-new” condition for resale. In this part, hierarchical models are developed to determine optimal decisions involving inventory replenishment, retail pricingand collection price for returns. Based on the simplified assumption of a single manufacturer and a single retailer dealing with a single recoverable item under deterministic conditions,all of these decisions are examined in an integrated manner. Models depicting decentralized, as well as centralized policies are explored. Analytical results are derived and detailed sensitivity analysis is performed via an extensive set of numerical computations.In the second part of this dissertation, a discrete bilevel problem is illustrated by investigating a biofuel production problem. The issues of governmental incentives, industry decisions of price, and farm management of land are incorporated. While fixed costs are natural components of decision making in operations management, such discrete phenomena have not received sufficient research attention in the current literature on bilevel programming, due to a variety of theoretical and algorithmic difficulties. When such costs are taken into account, it is not easy to derive optimality conditions and explore convergence properties due to discontinuities and the combinatorial nature of this problem, which is NP-hard. In order to solve this problem, a derivative-free search technique is used to arrive at a solution to this bilevel problem. A new heuristic methodology is developed, which integrates sensitivity analysis and warm-starts to improve the efficiency of the algorithm.Ph.D., Decision Sciences -- Drexel University, 201

A parametric integer programming algorithm for bilevel mixed integer programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.Comment: 11 page