André B. Chassein

A new bound for the midpoint solution in minmax regret optimization with an application to the robust shortest path problem

Minmax regret optimization aims at finding robust solutions that perform best in the worst-case, compared to the respective optimum objective value in each scenario. Even for simple uncertainty sets like boxes, most polynomially solvable optimization problems have strongly NP-complete minmax regret counterparts. Thus, heuristics with performance guarantees can potentially be of great value, but only few such guarantees exist.

Inverse eccentric vertex problem on networks

This paper addresses the problem of optimally modifying the edge lengths such that a prespecified vertex becomes the furthest vertex from a given fixed vertex in the perturbed network. We call this problem the inverse eccentric vertex problem. We show that the problem is

On the recoverable robust traveling salesman problem

NP

A bicriteria approach to robust optimization

NP-complete even on cactus graphs. However, if the underlying graph is a cycle or a tree, we develop efficient algorithms with linear time complexity.

Minmax regret combinatorial optimization problems with ellipsoidal uncertainty sets

We investigate the inverse convex ordered 1-median problem on unweighted trees under the cost functions related to the Chebyshev norm and the Hamming distance. By the special structure of the problem under Chebyshev norm, we deduce the so-called maximum modification to modify the edge lengths of the tree. Additionally, the cost function of the problem receives only finite values under the bottleneck Hamming distance. Therefore, we can find the optimal cost of the problem by applying binary search. It is shown that both of the problems, under Chebyshev norm and under the bottleneck Hamming distance, can be solved in O(n2log n) time in all situations, with or without essential topology changes. Here, n is the number of vertices of the tree. Finally, we prove that the problem under weighted sum Hamming distance is NP-hard.

Alternative formulations for the ordered weighted averaging objective

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal values. A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately. While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared.

The quadratic shortest path problem: complexity, approximability, and solution methods

We discuss the problem of evaluating a robust solution . To this end, we first give a short primer on how to apply robustification approaches to uncertain optimization problems using the assignment problem and the knapsack problem as illustrative examples. As it is not immediately clear in practice which such robustness approach is suitable for the problem at hand, we present current approaches for evaluating and comparing robustness from the literature, and introduce the new concept of a scenario curve. Using the methods presented in this chapter, an easy guide is given to the decision maker to find, solve and compare the best robust optimization method for his purposes.

Variable-sized uncertainty and inverse problems in robust optimization

The classic approach in robust optimization is to optimize the solution with respect to the worst case scenario. This pessimistic approach yields solutions that perform best if the worst scenario happens, but also usually perform bad for an average case scenario. On the other hand, a solution that optimizes the performance of this average case scenario may lack in the worst-case performance guarantee.In practice it is important to find a good compromise between these two solutions. We propose to deal with this problem by considering it from a bicriteria perspective. The Pareto curve of the bicriteria problem visualizes exactly how costly it is to ensure robustness and helps to choose the solution with the best balance between expected and guaranteed performance.In this paper we consider linear programming problems with uncertain cost functions. Building upon a theoretical observation on the structure of Pareto solutions for these problems, we present a column generation approach that requires no direct solution of the computationally expensive worst-case problem. In computational experiments we demonstrate the effectiveness of both the proposed algorithm, and the bicriteria perspective in general. HighlightsWe introduce a bicriteria optimization problem for robust optimization.We define the AC-WC curve as the Pareto front of this problem.We show how column generation can be used to efficiently compute the AC-WC curve.We compare two common robustness concepts with the AC-WC curve.