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.
Later on, Pereira and Averbakh @cite_8 introduced the interval data min-max regret set covering problem. The authors proposed a mathematical formulation for the problem that is similar to the one proposed by @cite_43 for the interval data min-max regret traveling salesman problem. As in @cite_43 , the formulation has an exponential number of constraints. They also adapted the logic-based Benders' algorithm of @cite_32 @cite_43 @cite_40 to this problem and presented an extension of the method that aims at generating multiple Benders' cuts per iteration of the algorithm. Moreover, the work presents an exact approach that uses Benders' cuts in the context of a framework. Computational experiments showed that such approach, as well as the extended logic-based Benders' algorithm, outperforms the standard logic-based Benders' algorithm at solving the instances considered. This robust version of the set covering problem was also addressed in @cite_56 , where the authors propose scenario-based heuristics with path-relinking.
@cite_8 @cite_32 @cite_56 @cite_43 @cite_40
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We study the Set Covering Problem with uncertain costs. For each cost coefficient, only an interval estimate is known, and it is assumed that each coefficient can take on any value from the corresponding uncertainty interval, regardless of the values taken by other coefficients. It is required to find a robust deviation (also called minmax regret) solution. For this strongly NP-hard problem, we present and compare computationally three exact algorithms, where two of them are based on Benders decomposition and one uses Benders cuts in the context of a Branch-and-Cut approach, and several heuristic methods, including a scenario-based heuristic, a Genetic Algorithm, and a Hybrid Algorithm that uses a version of Benders decomposition within a Genetic Algorithm framework. 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. In this study, we investigate the minmax regret Robust Weighted Set Covering problem with Interval Data (RSCP), which is the robust counterpart of the Weighted Set Covering Problem (WSCP) where uncertain data are modeled using interval data. RSCP is NP-Hard and can provide foundations for solving several minmax regret covering problems. Moreover, RSCP encloses several challenges in terms of computer science and mathematical formulations. The challenges involve the development of innovative algorithms since solving the RSCP for a unique scenario implies to solve an NP-Hard problem. The exact algorithms in the literature rely on decomposition algorithms and cutting plane procedures since the mathematical formulation of RSCP contains an exponential number of constraints. In this study, scenario-based heuristics are used to generate additional cuts in order to reduce the average optimality gaps on the instances not solved to optimality. 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.