New complexity results and algorithms for min-max-min robust combinatorial optimization

In this work we investigate the min-max-min robust optimization problem applied to combinatorial problems with uncertain cost-vectors which are contained in a convex uncertainty set. The idea of the approach is to calculate a set of k feasible solutions which are worst-case optimal if in each possible scenario the best of the k solutions would be implemented. It is known that the min-max-min robust problem can be solved efficiently if k is at least the dimension of the problem, while it is theoretically and computationally hard if k is small. While both cases are well studied in the literature nothing is known about the intermediate case, namely if k is smaller than but close to the dimension of the problem. We approach this open question and show that for a selection of combinatorial problems the min-max-min problem can be solved exactly and approximately in polynomial time if some problem specific values are fixed. Furthermore we approach a second open question and present the first implementable algorithm with oracle-pseudopolynomial runtime for the case that k is at least the dimension of the problem. The algorithm is based on a projected subgradient method where the projection problem is solved by the classical Frank-Wolfe algorithm. Additionally we derive a branch & bound method to solve the min-max-min problem for arbitrary values of k and perform tests on knapsack and shortest path instances. The experiments show that despite its theoretical impact the projected subgradient method cannot compete with an already existing method. On the other hand the performance of the branch & bound method scales very well with the number of solutions. Thus we are able to solve instances where k is above some small threshold very efficiently

A linear programming based heuristic framework for min-max regret combinatorial optimization problems with interval costs

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

Min-ordering and max-ordering scalarization methods for multi-objective robust optimization

Several robustness concepts for multi-objective uncertain optimization have been developed during the last years, but not many solution methods. In this paper we introduce two methods to find min–max robust efficient solutions based on scalarizations: the min-ordering and the max-ordering method. We show that all point-based min–max robust weakly efficient solutions can be found with the max-ordering method and that the min-ordering method finds set-based min–max robust weakly efficient solutions, some of which cannot be found with formerly developed scalarization based methods. We then show how the scalarized problems may be approached for multi-objective uncertain combinatorial optimization problems with special uncertainty sets. We develop compact mixed-integer linear programming formulations for multi-objective extensions of bounded uncertainty (also known as budgeted or Γ-uncertainty). For interval uncertainty, we show that the resulting problems reduce to well-known single-objective problems

Reactive max-min ant system: An experimental analysis of the combination with K-OPT local searches

Ant colony optimization (ACO) is a stochastic search method for solving NP-hard problems. The exploration versus exploitation dilemma rises in ACO search.Reactive max-min ant system algorithm is a recent proposition to automate the exploration and exploitation.It memorizes the search regions in terms of reactive heuristics to be harnessed after restart, which is to avoid the arbitrary exploration later.This paper examined the assumption that local heuristics are useless when combined with local search especially when it applied for combinatorial optimization problems with rugged fitness landscape.Results showed that coupling reactive heuristics with k-Opt local search algorithms produces higher quality solutions and more robust search than max-min ant system algorithm.Well-known combinatorial optimization problems are used in experiments, i.e. traveling salesman and quadratic assignment problems. The benchmarking data for both problems are taken from TSPLIB and QAPLIB respectively

Min-Max-Min Robustness for Combinatorial Problems with Discrete Budgeted Uncertainty

We consider robust combinatorial optimization problems with cost uncertainty where the decision maker can prepare K solutions beforehand and chooses the best of them once the true cost is revealed. Also known as min-max-min robustness (a special case of K-adaptability), it is a viable alternative to otherwise intractable two-stage problems. The uncertainty set assumed in this paper considers that in any scenario, at most Γ of the components of the cost vectors will be higher than expected, which corresponds to the extreme points of the budgeted uncertainty set. While the classical min-max problem with budgeted uncertainty is essentially as easy as the underlying deterministic problem, it turns out that the min-max-min problem is N P-hard for many easy combinatorial optimization problems, and not approximable in general. We thus present an integer programming formulation for solving the problem through a row-and-column generation algorithm. While exact, this algorithm can only cope with small problems, so we present two additional heuristics leveraging the structure of budgeted uncertainty. We compare our row-and-column generation algorithm and our heuristics on knapsack and shortest path instances previously used in the scientific literature and find that the heuristics obtain good quality solutions in short computational times

Algoritmos para el problema de diseño de rutas de vehículos con incertidumbre usando modelo Minmax Regret

103 p.El presente trabajo presenta el problema de diseño robusto de rutas para vehículos, asumiendo incertidumbre intervalar y asumiendo el criterio min-max regret para su resolución. En la literatura de optimización robusta no se ha hecho referencia a este enfoque, por lo cual este trabajo presenta la formulación matemática del problema de diseño de rutas de vehículos robusto. Esta formulación matemática resultante, como es de esperar, presenta una gran complejidad, debido a la complejidad intrínseca del problema de diseño de rutas de vehículos y del criterio min-max regret. Además en este trabajo se presenta un método exacto para la resolución de este nuevo problema, usando la Descomposición de Benders, la cual fue aplicada para resolver el problema del vendedor viajero, como se presenta en la literatura. También, se presentan los resultados de resolver este problema usando las heurísticas propuestas por (Zielinski, 2008) para este tipo de problemas. Por último, se presenta la aplicación de las metaheurísticas Simulated Annealing y Tabu Search aplicadas al problema de diseño de rutas de vehículos robusto. Estas metaheurísticas fueron escogidas debido a la gran cantidad de aplicaciones en las cuales se han usado para resolver problemas de optimización combinatoria y en especial para el problema de diseño de rutas de vehículos con buenos resultados. Debido a que este problema es nuevo, las instancias a utilizar fueron construidas en base a las instancias comúnmente utilizadas para el problema de diseño de rutas de vehículos. Estas instancias fueron transformadas y con ellas se obtuvieron todos los resultados de este trabajo. Los resultados obtenidos concuerdan con lo presentado en la literatura asociada a la optimización robusta y refuerzan el hecho de que la complejidad de los problemas robustos recae fuertemente en la naturaleza del problema clásico.El solver utilizado para resolver los problemas matemáticos fue CPLEX y el lenguaje de programación fue JAVA. Esta combinación presentó algunos problemas de estabilidad numérica en la resolución de los modelos matemáticos por lo que los resultados resultaron difíciles de interpretar./ABSTRACT: This work presents the robust vehicle routing problem, assuming intervalar uncertainty and using the min-max regret criterion for his resolution. In robust optimization literature has not referred to this approach, so this work presents the mathematical formulation of the robust vehicle routing problem. The resulting mathematical formulation, as expected, is a highly complex due to the inherent complexity of the vehicle routing problem itself and min-max regret criterion. Additionally, this work presents an exact method to solve this new problem, using the Benders Decomposition, which was applied to solve the traveling salesman problem, as presented in the literature. Also, the results are presented to solve this problem by using heuristics proposed by Kasperski & Zielinski for such problems.Finally, we present the application of the etaheuristics Simulated Annealing and Tabu Search applied to the robust vehicle routing problem. These metaheuristics were chosen because of the large number of applications which have been used to solve combinatorial optimization problems and particularly for the vehicle routing problem with good performance. Because this problem is new, the instances used were builted using very known instances for the vehicle routing problem. The results agree with those reported in the literature associated with the robust optimization and demonstrate the fact that the complexity of the robust problems relies heavily on the nature of the classic problem.The solver used to solve mathematical problems was CPLEX and the language programming used was Java. This combination presented some problems of numerical stability in solving mathematical models so that the results were difficult to interpret