Thresholded Covering Algorithms for Robust and Max-Min Optimization

The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? Feige et al. and Khandekar et al. considered the k-robust model where the possible outcomes tomorrow are given by all demand-subsets of size k, and gave algorithms for the set cover problem, and the Steiner tree and facility location problems in this model, respectively. In this paper, we give the following simple and intuitive template for k-robust problems: "having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat". In this paper we show that this template gives us improved approximation algorithms for k-robust Steiner tree and set cover, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios (except for multicut) are almost best possible. As a by-product of our techniques, we also get algorithms for max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?".Comment: 24 page

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

Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets

Consider the following problem: given a set system (U,I) and an edge-weighted graph G = (U, E) on the same universe U, find the set A in I such that the Steiner tree cost with terminals A is as large as possible: "which set in I is the most difficult to connect up?" This is an example of a max-min problem: find the set A in I such that the value of some minimization (covering) problem is as large as possible. In this paper, we show that for certain covering problems which admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system I is given by a p-system or q-knapsacks or both. This result is similar to results for constrained maximization of submodular functions. Although many natural covering problems are not even approximately submodular, we show that one can use properties of the online algorithm as a surrogate for submodularity. Moreover, we give stronger connections between max-min optimization and two-stage robust optimization, and hence give improved algorithms for robust versions of various covering problems, for cases where the uncertainty sets are given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and http://arxiv.org/abs/0912.1045 appeared in ICALP 201

On the approximability of robust spanning tree problems

In this paper the minimum spanning tree problem with uncertain edge costs is discussed. In order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution. The min-max, min-max regret and 2-stage min-max versions of the problem are discussed. The complexity and approximability of all these problems are explored. It is proved that the min-max and min-max regret versions with nonnegative edge costs are hard to approximate within $O(\log^{1-\epsilon} n)$ for any $\epsilon>0$ unless the problems in NP have quasi-polynomial time algorithms. Similarly, the 2-stage min-max problem cannot be approximated within $O(\log n)$ unless the problems in NP have quasi-polynomial time algorithms. In this paper randomized LP-based approximation algorithms with performance ratio of $O(\log^2 n)$ for min-max and 2-stage min-max problems are also proposed

Contributions to robust and bilevel optimization models for decision-making

Los problemas de optimización combinatorios han sido ampliamente estudiados en la literatura especializada desde mediados del siglo pasado. No obstante, en las últimas décadas ha habido un cambio de paradigma en el tratamiento de problemas cada vez más realistas, en los que se incluyen fuentes de aleatoriedad e incertidumbre en los datos, múltiples criterios de optimización y múltiples niveles de decisión. Esta tesis se desarrolla en este contexto. El objetivo principal de la misma es el de construir modelos de optimización que incorporen aspectos inciertos en los parámetros que de nen el problema así como el desarrollo de modelos que incluyan múltiples niveles de decisión. Para dar respuesta a problemas con incertidumbre usaremos los modelos Minmax Regret de Optimización Robusta, mientras que las situaciones con múltiples decisiones secuenciales serán analizadas usando Optimización Binivel. En los Capítulos 2, 3 y 4 se estudian diferentes problemas de decisión bajo incertidumbre a los que se dará una solución robusta que proteja al decisor minimizando el máximo regret en el que puede incurrir. El criterio minmax regret analiza el comportamiento del modelo bajo distintos escenarios posibles, comparando su e ciencia con la e ciencia óptima bajo cada escenario factible. El resultado es una solución con una eviciencia lo más próxima posible a la óptima en el conjunto de las posibles realizaciones de los parámetros desconocidos. En el Capítulo 2 se estudia un problema de diseño de redes en el que los costes, los pares proveedor-cliente y las demandas pueden ser inciertos, y además se utilizan poliedros para modelar la incertidumbre, permitiendo de este modo relaciones de dependencia entre los parámetros. En el Capítulo 3 se proponen, en el contexto de la secuenciación de tareas o la computación grid, versiones del problema del camino más corto y del problema del viajante de comercio en el que el coste de recorrer un arco depende de la posición que este ocupa en el camino, y además algunos de los parámetros que de nen esta función de costes son inciertos. La combinación de la dependencia en los costes y la incertidumbre en los parámetros da lugar a dependencias entre los parámetros desconocidos, que obliga a modelar los posibles escenarios usando conjuntos más generales que los hipercubos, habitualmente utilizados en este contexto. En este capítulo, usaremos poliedros generales para este cometido. Para analizar este primer bloque de aplicaciones, en el Capítulo 4, se analiza un modelo de optimización en el que el conjunto de posibles escenarios puede ser alterado mediante la realización de inversiones en el sistema. En los problemas estudiados en este primer bloque, cada decisión factible es evaluada en base a la reacción más desfavorable que pueda darse en el sistema. En los Capítulos 5 y 6 seguiremos usando esta idea pero ahora se supondrá que esa reacción a la decisión factible inicial está en manos de un adversario o follower. Estos dos capítulos se centran en el estudio de diferentes modelos binivel. La Optimización Binivel aborda problemas en los que existen dos niveles de decisión, con diferentes decisores en cada uno ellos y la decisión se toma de manera jerárquica. En concreto, en el Capítulo 5 se estudian distintos modelos de jación de precios en el contexto de selección de carteras de valores, en los que el intermediario nanciero, que se convierte en decisor, debe jar los costes de invertir en determinados activos y el inversor debe seleccionar su cartera de acuerdo a distintos criterios. Finalmente, en el Capítulo 6 se estudia un problema de localización en el que hay distintos decisores, con intereses contrapuestos, que deben determinar secuencialmente la ubicación de distintas localizaciones. Este modelo de localización binivel se puede aplicar en contextos como la localización de servicios no deseados o peligrosos (plantas de reciclaje, centrales térmicas, etcétera) o en problemas de ataque-defensa. Todos estos modelos se abordan mediante el uso de técnicas de Programación Matemática. De cada uno de ellos se analizan algunas de sus propiedades y se desarrollan formulaciones y algoritmos, que son examinados también desde el punto de vista computacional. Además, se justica la validez de los modelos desde un enfoque de las aplicaciones prácticas. Los modelos presentados en esta tesis comparten la peculiaridad de requerir resolver distintos problemas de optimización encajados.Combinatorial optimization problems have been extensively studied in the specialized literature since the mid-twentieth century. However, in recent decades, there has been a paradigm shift to the treatment of ever more realistic problems, which include sources of randomness and uncertainty in the data, multiple optimization criteria and multiple levels of decision. This thesis concerns the development of such concepts. Our objective is to study optimization models that incorporate uncertainty elements in the parameters de ning the model, as well as the development of optimization models integrating multiple decision levels. In order to consider problems under uncertainty, we use Minmax Regret models from Robust Optimization; whereas the multiplicity and hierarchy in the decision levels is addressed using Bilevel Optimization. In Chapters 2, 3 and 4, we study di erent decision problems under uncertainty to which we give a robust solution that protects the decision-maker minimizing the maximum regret that may occur. This robust criterion analyzes the performance of the system under multiple possible scenarios, comparing its e ciency with the optimum one under each feasible scenario. We obtain, as a result, a solution whose e ciency is as close as possible to the optimal one in the set of feasible realizations of the uncertain parameters. In Chapter 2, we study a network design problem in which the costs, the pairs supplier-customer, and the demands can take uncertain values. Furthermore, the uncertainty in the parameters is modeled via polyhedral sets, thereby allowing relationships among the uncertain parameters. In Chapter 3, we propose time-dependent versions of the shortest path and traveling salesman problems in which the costs of traversing an arc depends on the relative position that the arc occupies in the path. Moreover, we assume that some of the parameters de ning these costs can be uncertain. These models can be applied in the context of task sequencing or grid computing. The incorporation of time-dependencies together with uncertainties in the parameters gives rise to dependencies among the uncertain parameters, which require modeling the possible scenarios using more general sets than hypercubes, normally used in this context. In this chapter, we use general polyhedral sets with this purpose. To nalize this rst block of applications, in Chapter 4, we analyze an optimization model in which the set of possible scenarios can be modi ed by making some investments in the system. In the problems studied in this rst block, each feasible decision is evaluated based on the most unfavorable possible reaction of the system. In Chapters 5 and 6, we will still follow this idea, but assuming that the reaction to the initial feasible decision will be held by a follower or an adversary, instead of assuming the most unfavorable one. These two chapters are focused on the study of some bilevel models. Bilevel Optimization addresses optimization problems with multiple decision levels, di erent decision-makers in each level and a hierarchical decision order. In particular, in Chapter 5, we study some price setting problems in the context of portfolio selection. In these problems, the nancial intermediary becomes a decisionmaker and sets the transaction costs for investing in some securities, and the investor chooses her portfolio according to di erent criteria. Finally, in Chapter 6, we study a location problem with several decision-makers and opposite interests, that must set, sequentially, some location points. This bilevel location model can be applied in practical applications such as the location of semi-obnoxious facilities (power or electricity plants, waste dumps, etc.) or interdiction problems. All these models are stated from a Mathematical Programming perspective, analyzing their properties and developing formulations and algorithms, that are tested from a computational point of view. Furthermore, we pay special attention to justifying the validity of the models from the practical applications point of view. The models presented in this thesis share the characteristic of involving the resolution of nested optimization problems.Premio Extraordinario de Doctorado U

Leveraging Decision Diagrams to Solve Two-stage Stochastic Programs with Binary Recourse and Logical Linking Constraints

Two-stage stochastic programs with binary recourse are challenging to solve and efficient solution methods for such problems have been limited. In this work, we generalize an existing binary decision diagram-based (BDD-based) approach of Lozano and Smith (Math. Program., 2018) to solve a special class of two-stage stochastic programs with binary recourse. In this setting, the first-stage decisions impact the second-stage constraints. Our modified problem extends the second-stage problem to a more general setting where logical expressions of the first-stage solutions enforce constraints in the second stage. We also propose a complementary problem and solution method which can be used for many of the same applications. In the complementary problem we have second-stage costs impacted by expressions of the first-stage decisions. In both settings, we convexify the second-stage problems using BDDs and parametrize either the arc costs or capacities of these BDDs with first-stage solutions depending on the problem. We further extend this work by incorporating conditional value-at-risk and we propose, to our knowledge, the first decomposition method for two-stage stochastic programs with binary recourse and a risk measure. We apply these methods to a novel stochastic dominating set problem and present numerical results to demonstrate the effectiveness of the proposed methods

Robust network design under polyhedral traffic uncertainty

Ankara : The Department of Industrial Engineering and The Institute of Engineering and Science of Bilkent Univ., 2007.Thesis (Ph.D.) -- Bilkent University, 2007.Includes bibliographical references leaves 160-166.In this thesis, we study the design of networks robust to changes in demand estimates. We consider the case where the set of feasible demands is defined by an arbitrary polyhedron. Our motivation is to determine link capacity or routing configurations, which remain feasible for any realization in the corresponding demand polyhedron. We consider three well-known problems under polyhedral demand uncertainty all of which are posed as semi-infinite mixed integer programming problems. We develop explicit, compact formulations for all three problems as well as alternative formulations and exact solution methods. The first problem arises in the Virtual Private Network (VPN) design field. We present compact linear mixed-integer programming formulations for the problem with the classical hose traffic model and for a new, less conservative, robust variant relying on accessible traffic statistics. Although we can solve these formulations for medium-to-large instances in reasonable times using off-the-shelf MIP solvers, we develop a combined branch-and-price and cutting plane algorithm to handle larger instances. We also provide an extensive discussion of our numerical results. Next, we study the Open Shortest Path First (OSPF) routing enhanced with traffic engineering tools under general demand uncertainty with the motivation to discuss if OSPF could be made comparable to the general unconstrained routing (MPLS) when it is provided with a less restrictive operating environment. To the best of our knowledge, these two routing mechanisms are compared for the first time under such a general setting. We provide compact formulations for both routing types and show that MPLS routing for polyhedral demands can be computed in polynomial time. Moreover, we present a specialized branchand-price algorithm strengthened with the inclusion of cuts as an exact solution tool. Subsequently, we compare the new and more flexible OSPF routing with MPLS as well as the traditional OSPF on several network instances. We observe that the management tools we use in OSPF make it significantly better than the generic OSPF. Moreover, we show that OSPF performance can get closer to that of MPLS in some cases. Finally, we consider the Network Loading Problem (NLP) under a polyhedral uncertainty description of traffic demands. After giving a compact multicommodity formulation of the problem, we prove an unexpected decomposition property obtained from projecting out the flow variables, considerably simplifying the resulting polyhedral analysis and computations by doing away with metric inequalities, an attendant feature of most successful algorithms on NLP. Under the hose model of feasible demands, we study the polyhedral aspects of NLP, used as the basis of an efficient branch-and-cut algorithm supported by a simple heuristic for generating upper bounds. We provide the results of extensive computational experiments on well-known network design instances.Altın, AyşegülPh.D