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.
Given a set of items, a set of scenarios, and a knapsack of fixed capacity, a nonnegative weight is associated with each item; and a value is associated with each item under each scenario. The max-min Knapsack MNK problem is defined as filling the knapsack with a selected set of items so that the minimum total value gained under all scenarios is maximized. The MNK problem is a generalization of the conventional knapsack problem to situations with multiple scenarios. This extension significantly enlarges its scope of applications, especially in the application of recent robust optimization developments. In this paper, the MNK problem is shown to be strongly NP-hard for an unbounded number of scenarios and pseudopolynomially solvable for a bounded number of scenarios. Effective lower and upper bounds are generated by surrogate relaxation. The ratio of these two bounds is shown to be bounded by a constant for situations where the data range is limited to be within a fixed percentage from its mean. This result leads to an approximation algorithm for MNK in the special case. A branch-and-bound algorithm has been implemented to efficiently solve the MNK problem to optimality. Extensive computational results are presented. This paper investigates, for the first time in the literature, the approximation of min–max (regret) versions of classical problems like shortest path, minimum spanning tree, and knapsack. For a constant number of scenarios, we establish fully polynomial-time approximation schemes for the min–max versions of these problems, using relationships between multi-objective and min–max optimization. Using dynamic programming and classical trimming techniques, we construct a fully polynomial-time approximation scheme for min–max regret shortest path. We also establish a fully polynomial-time approximation scheme for min–max regret spanning tree and prove that min–max regret knapsack is not at all approximable. For a non-constant number of scenarios, in which case min–max and min–max regret versions of polynomial-time solvable problems usually become strongly NP-hard, non-approximability results are provided for min–max (regret) versions of shortest path and spanning tree. 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.