Algorithm Engineering in Robust Optimization

Robust optimization is a young and emerging field of research having received a considerable increase of interest over the last decade. In this paper, we argue that the the algorithm engineering methodology fits very well to the field of robust optimization and yields a rewarding new perspective on both the current state of research and open research directions. To this end we go through the algorithm engineering cycle of design and analysis of concepts, development and implementation of algorithms, and theoretical and experimental evaluation. We show that many ideas of algorithm engineering have already been applied in publications on robust optimization. Most work on robust optimization is devoted to analysis of the concepts and the development of algorithms, some papers deal with the evaluation of a particular concept in case studies, and work on comparison of concepts just starts. What is still a drawback in many papers on robustness is the missing link to include the results of the experiments again in the design

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

On the recoverable robust traveling salesman problem

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal values. A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately. While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared

The resource constrained shortest path problem with uncertain data: a robust formulation and optimal solution approach

International audienceThe Resource Constrained Shortest Path Problem (RCSP P) models several applications in the fields of transportation and communications. The classical problem supposes that the resource consumptions and the costs are certain and looks for the cheapest feasible path. These parameters are however hardly known with precision in real applications, so that the deterministic solution is likely to be infeasible or suboptimal. We address this issue by considering a robust counterpart of the RCSP P. We focus here on resource variation and model its variability through the uncertainty set defined by Bertismas and Sim (2003,2004), which can model the risk aversion of the decision maker through a budget of uncertainty. We solve the resulting problem to optimality through the well-known three phase approach dealing with bounds computation, network reduction and gap closing. In particular, we compute robust bounds on the resource consumption and cost by solving the robust shortest path problem and the dual robust Lagrangian relaxation, respectively. Dynamic programming is used to close the duality gap. Upper and lower bounds are used to reduce the dimension of the network and incorporated in the dynamic programming in order to fathom unpromising states. An extensive computational phase is carried out in order to asses the behavior of the defined strategy comparing its performance with the state-of-the-art. The results highlight the effectiveness of our approach in solving to optimality * 1 benchmark instances for RCSP P when Γ is not too large, tailored for the robust counterpart. For larger values of Γ, we show that the most efficient method combines deterministic preprocesing with the iterative algorithm from Bertsimas and Sim (2003). We also illustrate the failure probability of the robust solutions through Monte Carlo sampling

Nature-inspired Methods for Stochastic, Robust and Dynamic Optimization

Nature-inspired algorithms have a great popularity in the current scientific community, being the focused scope of many research contributions in the literature year by year. The rationale behind the acquired momentum by this broad family of methods lies on their outstanding performance evinced in hundreds of research fields and problem instances. This book gravitates on the development of nature-inspired methods and their application to stochastic, dynamic and robust optimization. Topics covered by this book include the design and development of evolutionary algorithms, bio-inspired metaheuristics, or memetic methods, with empirical, innovative findings when used in different subfields of mathematical optimization, such as stochastic, dynamic, multimodal and robust optimization, as well as noisy optimization and dynamic and constraint satisfaction problems

On recoverable and two-stage robust selection problems with budgeted uncertainty

In this paper the problem of selecting p out of n available items is discussed, such that their total cost is minimized. We assume that the item costs are not known exactly, but stem from a set of possible outcomes modeled through budgeted uncertainty sets, i.e., the interval uncertainty sets with an additional linear (budget) constraint, in their discrete and continuous variants. Robust recoverable and two-stage models of this selection problem are analyzed through an in-depth discussion of variables at their optimal values. Polynomial algorithms for both models under continuous budgeted uncertainty are proposed. In the case of discrete budgeted uncertainty, compact mixed integer formulations are constructed and some approximation algorithms are proposed. Polynomial combinatorial algorithms for the adversarial and incremental problems (the special cases of the considered robust models) under both discrete and continuous budgeted uncertainty are constructed

Networks, Uncertainty, Applications and a Crusade for Optimality

In this thesis we address a collection of Network Design problems which are strongly motivated by applications from Telecommunications, Logistics and Bioinformatics. In most cases we justify the need of taking into account uncertainty in some of the problem parameters, and different Robust optimization models are used to hedge against it. Mixed integer linear programming formulations along with sophisticated algorithmic frameworks are designed, implemented and rigorously assessed for the majority of the studied problems. The obtained results yield the following observations: (i) relevant real problems can be effectively represented as (discrete) optimization problems within the framework of network design; (ii) uncertainty can be appropriately incorporated into the decision process if a suitable robust optimization model is considered; (iii) optimal, or nearly optimal, solutions can be obtained for large instances if a tailored algorithm, that exploits the structure of the problem, is designed; (iv) a systematic and rigorous experimental analysis allows to understand both, the characteristics of the obtained (robust) solutions and the behavior of the proposed algorithm