A heuristic framework for a class of robust optimization problems is proposed.The heuristic framework explores dual information.The heuristic is successfully applied to solve two robust optimization problems.The heuristic is able to outperform a widely used 2-approximation procedure.A robust optimization version of the restricted shortest path problem is introduced. This work deals with a class of problems under interval data uncertainty, namely interval robust-hard problems, composed of interval data min-max regret generalizations of classical NP-hard combinatorial problems modeled as 0-1 integer linear programming problems. These problems are more challenging than other interval data min-max regret problems, as solely computing the cost of any feasible solution requires solving an instance of an NP-hard problem. The state-of-the-art exact algorithms in the literature are based on the generation of a possibly exponential number of cuts. As each cut separation involves the resolution of an NP-hard classical optimization problem, the size of the instances that can be solved efficiently is relatively small. To smooth this issue, we present a modeling technique for interval robust-hard problems in the context of a heuristic framework. The heuristic obtains feasible solutions by exploring dual information of a linearly relaxed model associated with the classical optimization problem counterpart. Computational experiments for interval data min-max regret versions of the restricted shortest path problem and the set covering problem show that our heuristic is able to find optimal or near-optimal solutions and also improves the primal bounds obtained by a state-of-the-art exact algorithm and a 2-approximation procedure for interval data min-max regret problems.
In this note, a 2-approximation method for minmax regret optimization problems is developed which extends the work of Kasperski and Zielinski [A. Kasperski, P. Zielinski, An approximation algorithm for interval data minmax regret combinatorial optimization problems, Information Processing Letters 97 (2006) 177-180] from finite to compact constraint sets. Preface. 1. Approaches to Handle Uncertainty In Decision Making. 2. A Robust Discrete Optimization Framework. 3. Computational Complexity Results of Robust Discrete Optimization Problems. 4. Easily Solvable Cases of Robust Discrete Optimization Problems. 5. Algorithmic Developments for Difficult Robust Discrete Optimization Problems. 6. Robust 1-Median Location Problems: Dynamic Aspects and Uncertainty. 7. Robust Scheduling Problems. 8. Robust Uncapacitated Network Design and International Sourcing Problems. 9. Robust Discrete Optimization: Past Successes and Future Challenges. 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. In order to find a robust solution under an unknown linear cost function it will be considered the minmax regret criterion. It is assumed the vector of costs can take on values from a given uncertainty set. The resulting optimization model has been extensively analyzed in the literature when the uncertain costs are modeled by closed intervals. Unfortunately, except for rare applications, this problem has NP-hard complexity which has led to the appearance of approximated methods seeking for good solutions in a short computational time. The use of "optimality" as an operational research criterion is insufficiently discriminating. Ample evidence exists that for many problems simple optimization (particularly profit maximization) does not represent the aims of management. In this paper we discuss the nature of the problem situations for which alternative decision criteria are more appropriate. In particular the structure of strategic planning problems is analysed. The provisional commitment involved in a plan (in contrast to the irrevocable commitment of a decision) leads to the development of a particular criterion, robustness—a measure of the flexibility which an initial decision of a plan maintains for achieving near-optimal states in conditions of uncertainty. The robustness concept is developed through the case study of a sequential factory location problem. The general problem of minimizing the maximal regret in combinatorial optimization problems with interval data is considered. In many cases, the minmax regret versions of the classical, polynomially solvable, combinatorial optimization problems become NP-hard and no approximation algorithms for them have been known. Our main result is a polynomial time approximation algorithm with a performance ratio of 2 for this class of problems.