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.
More recently, Feizollahi and Averbakh @cite_45 introduced the min-max regret quadratic assignment problem with interval flows, which is a generalization of the classical quadratic assignment problem in which material flows between facilities are uncertain and vary in given intervals. Although quadratic, the problem presents a structure that is very similar to the robust versions of combinatorial problems we address in this work. The authors proposed two mathematical formulations and adapted the logic-based Benders' algorithm of @cite_32 @cite_43 @cite_40 to solve them through the linearization of the corresponding master problems. They also developed a hybrid approach which combines Benders' decomposition with heuristics.
@cite_43 @cite_40 @cite_45 @cite_32
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The traveling salesman problem is one of the most famous combinatorial optimization problems and has been intensively studied. Many extensions to the basic problem have also been proposed, with the aim of making the resulting mathematical models as realistic as possible. We present a new extension to the basic problem, where travel times are specified as a range of possible values. This model reflects the intrinsic difficulties of estimating travel times in reality. We apply the robust deviation criterion to drive optimization over the interval data problem so obtained. Some interesting theoretical properties of the new optimization problems are identified and discussed, together with a new mathematical formulation and some exact and heuristic algorithms. Computational experiments are finally presented. Many real problems can be modelled as robust shortest path problems on digraphs with interval costs, where intervals represent uncertainty about real costs and a robust path is not too far from the shortest path for each possible configuration of the arc costs. We consider a generalization of the classical quadratic assignment problem, where material flows between facilities are uncertain, and only upper and lower bounds are known for each flow. The objective is to find a minmax regret solution. We present an exact Benders decomposition algorithm based on two developed mathematical programming formulations and on the developed linearizations of master problems, and a heuristic based on using tabu search in the context of a Benders decomposition framework. Then, we develop a hybrid Benders decomposition approach that allows us to combine the speed of heuristics with the rigor and precision of the exact Benders method. We discuss the results of extensive computational experiments. Abstract The robust spanning tree problem is a variation, motivated by telecommunications applications, of the classic minimum spanning tree problem. In the robust spanning tree problem edge costs lie in an interval instead of having a fixed value. Interval numbers model uncertainty about the exact cost values. A robust spanning tree is a spanning tree whose total cost minimizes the maximum deviation from the optimal spanning tree over all realizations of the edge costs. This robustness concept is formalized in mathematical terms and is used to drive optimization. This paper describes a new exact method, based on Benders decomposition, for the robust spanning tree problem with interval data. Computational results highlight the efficiency of the new method, which is shown to be very fast on all the benchmarks considered, and in particular on those that were harder to solve for the methods previously known.