557 research outputs found
The robust shortest path problem with interval data via Benders decomposition
Abstract.: 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. In this paper we discuss the application of a Benders decomposition approach to this problem. Computational results confirm the efficiency of the new algorithm. It is able to clearly outperform state-of-the-art algorithms on many classes of networks. For the remaining classes we identify the most promising algorithm among the others, depending of the characteristics of the network
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
Minmax regret combinatorial optimization problems: an Algorithmic Perspective
Candia-Vejar, A (reprint author), Univ Talca, Modeling & Ind Management Dept, Curico, Chile.Uncertainty in optimization is not a new ingredient. Diverse models considering uncertainty have been developed over the last 40 years. In our paper we essentially discuss a particular uncertainty model associated with combinatorial optimization problems, developed in the 90's and broadly studied in the past years. This approach named minmax regret (in particular our emphasis is on the robust deviation criteria) is different from the classical approach for handling uncertainty, stochastic approach, where uncertainty is modeled by assumed probability distributions over the space of all possible scenarios and the objective is to find a solution with good probabilistic performance. In the minmax regret (MMR) approach, the set of all possible scenarios is described deterministically, and the search is for a solution that performs reasonably well for all scenarios, i.e., that has the best worst-case performance. In this paper we discuss the computational complexity of some classic combinatorial optimization problems using MMR. approach, analyze the design of several algorithms for these problems, suggest the study of some specific research problems in this attractive area, and also discuss some applications using this model
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 asà 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 CapÃtulos 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 CapÃtulo 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 CapÃtulo
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 capÃtulo, usaremos poliedros generales
para este cometido. Para analizar este primer bloque de aplicaciones, en el CapÃtulo 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
CapÃtulos 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
capÃtulos 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 CapÃtulo 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 CapÃtulo 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
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
Decomposition Algorithms in Stochastic Integer Programming: Applications and Computations.
In this dissertation we focus on two main topics. Under the first topic, we develop a new framework for stochastic network interdiction problem to address ambiguity in the defender risk preferences. The second topic is dedicated to computational studies of two-stage stochastic integer programs. More specifically, we consider two cases. First, we develop some solution methods for two-stage stochastic integer programs with continuous recourse; second, we study some computational strategies for two-stage stochastic integer programs with integer recourse. We study a class of stochastic network interdiction problems where the defender has incomplete (ambiguous) preferences. Specifically, we focus on the shortest path network interdiction modeled as a Stackelberg game, where the defender (leader) makes an interdiction decision first, then the attacker (follower) selects a shortest path after the observation of random arc costs and interdiction effects in the network. We take a decision-analytic perspective in addressing probabilistic risk over network parameters, assuming that the defender\u27s risk preferences over exogenously given probabilities can be summarized by the expected utility theory. Although the exact form of the utility function is ambiguous to the defender, we assume that a set of historical data on some pairwise comparisons made by the defender is available, which can be used to restrict the shape of the utility function. We use two different approaches to tackle this problem. The first approach conducts utility estimation and optimization separately, by first finding the best fit for a piecewise linear concave utility function according to the available data, and then optimizing the expected utility. The second approach integrates utility estimation and optimization, by modeling the utility ambiguity under a robust optimization framework following \cite{armbruster2015decision} and \cite{Hu}. We conduct extensive computational experiments to evaluate the performances of these approaches on the stochastic shortest path network interdiction problem. In third chapter, we propose partition-based decomposition algorithms for solving two-stage stochastic integer program with continuous recourse. The partition-based decomposition method enhance the classical decomposition methods (such as Benders decomposition) by utilizing the inexact cuts (coarse cuts) induced by a scenario partition. Coarse cut generation can be much less expensive than the standard Benders cuts, when the partition size is relatively small compared to the total number of scenarios. We conduct an extensive computational study to illustrate the advantage of the proposed partition-based decomposition algorithms compared with the state-of-the-art approaches. In chapter four, we concentrate on computational methods for two-stage stochastic integer program with integer recourse. We consider the partition-based relaxation framework integrated with a scenario decomposition algorithm in order to develop strategies which provide a better lower bound on the optimal objective value, within a tight time limit
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