Two-stage quadratic integer programs with stochastic right-hand sides

We consider two-stage quadratic integer programs with stochastic right-hand sides, and present an equivalent reformulation using value functions. We propose a two-phase solution approach. The first phase constructs value functions of quadratic integer programs in both stages. The second phase solves the reformulation using a global branch-and-bound algorithm or a level-set approach. We derive some basic properties of value functions of quadratic integer programs and utilize them in our algorithms. We show that our approach can solve instances whose extensive forms are hundreds of orders of magnitude larger than the largest quadratic integer programming instances solved in the literature.     

A Constraint Programming model for fast optimal stowage of container vessel bays

Container vessel stowage planning is a hard combinatorial optimization problem with both high economic and environmental impact. We have developed an approach that often is able to generate near-optimal plans for large container vessels within a few minutes. It decomposes the problem into a master planning phase that distributes the containers to bay sections and a slot planning phase that assigns containers of each bay section to slots. In this paper, we focus on the slot planning phase of this approach and present a Constraint Programming and Integer Programming model for stowing a set of containers in a single bay section. This so-called slot planning problem is NP-hard and often involves stowing several hundred containers. Using state-of-the-art constraint solvers and modeling techniques, however, we were able to solve 90% of 236 real instances from our industrial collaborator to optimality within 1 second. Thus, somewhat to our surprise, it is possible to solve most of these problems optimally within the time required for practical application.     

Inverse stochastic dominance constraints and rank dependent expected utility theory

We consider optimization problems with second order stochastic dominance constraints formulated as a relation of Lorenz curves. We characterize the relation in terms of rank dependent utility functions, which generalize Yaari's utility functions. We develop optimality conditions and duality theory for problems with Lorenz dominance constraints. We prove that Lagrange multipliers associated with these constraints can be identified with rank dependent utility functions. The problem is numerically tractable in the case of discrete distributions with equally probable realizations. Research supported by the NSF awards DMS-0303545, DMS-0303728, DMI-0354500 and DMI-0354678.     

An MPEC approach to second-best toll pricing

This paper addresses two second-best toll pricing problems, one with fixed and the other with elastic travel demands, as mathematical programs with equilibrium constraints. Several equivalent nonlinear programming formulations for the two problems are discussed. One formulation leads to properties that are of interest to transportation economists. Another produces an algorithm that is capable of solving large problems and easy to implement with existing software for linear and nonlinear programming problems. Numerical results using transportation networks from the literature are also presented.This research was partially supported by NSF grants DMI-9978642 and DMI-0300316.     

Mixed-Integer Cuts from Cyclic Groups

We analyze a separation procedure for Mixed-Integer Programs related to the work of Gomory and Johnson on interpolated subadditive functions. This approach has its roots in the Gomory-Johnson characterization on the master cyclic group polyhedron. To our knowledge, the practical benefit that can be obtained by embedding interpolated subadditive cuts in a cutting plane algorithm was not investigated computationally by previous authors. In this paper we compute, for the first time, the lower bound value obtained when adding (implicitly) all the interpolated subadditive cuts that can be derived from the individual rows of an optimal LP tableau, thus approximating the optimization over the intersection of the Gomory corner polyhedron with the LP relaxation of the original problem formulation. The computed bound is compared with that obtained when only Gomory mixed-integer cuts are used, on a very large test-bed of MIP instances.     

Perspective functions: Proximal calculus and applications in high-dimensional statistics

Perspective functions arise explicitly or implicitly in various forms in applied mathematics and in statistical data analysis. To date, no systematic strategy is available to solve the associated, typically nonsmooth, optimization problems. In this paper, we fill this gap by showing that proximal methods provide an efficient framework to model and solve problems involving perspective functions. We study the construction of the proximity operator of a perspective function under general assumptions and present important instances in which the proximity operator can be computed explicitly or via straightforward numerical operations. These results constitute central building blocks in the design of proximal optimization algorithms. We showcase the versatility of the framework by designing novel proximal algorithms for state-of-the-art regression and variable selection schemes in high-dimensional statistics.     

Static arbitrage bounds on basket option prices

We consider the problem of computing upper and lower bounds on the price of an European basket call option, given prices on other similar options. Although this problem is hard to solve exactly in the general case, we show that in some instances the upper and lower bounds can be computed via simple closed-form expressions, or linear programs. We also introduce an efficient linear programming relaxation of the general problem based on an integral transform interpretation of the call price function. We show that this relaxation is tight in some of the special cases examined before.     

A solution approach for dynamic vehicle and crew scheduling

In this paper, we discuss the dynamic vehicle and crew scheduling problem and we propose a solution approach consisting of solving a sequence of optimization problems. Furthermore, we explain why it is useful to consider such a dynamic approach and compare it with a static one. Moreover, we perform a sensitivity analysis on our main assumption that the travel times of the trips are known exactly a certain amount of time before actual operation.We provide extensive computational results on some real-world data instances of a large public transport company in the Netherlands. Due to the complexity of the vehicle and crew scheduling problem, we solve only small and medium-sized instances with such a dynamic approach. We show that the results are good in the case of a single depot. However, in the multiple-depot case, the dynamic approach does not perform so well. We investigate why this is the case and conclude that the fact that the instance has to be split in several smaller ones, has a negative effect on the performance.     

Design and analysis of hybrid metaheuristics for the Reliability p-Median Problem

In the p-Median Problem, it is assumed that, once the facilities are opened, they may not fail. In practice some of the facilities may become unavailable due to several factors. In the Reliability p-Median Problem some of the facilities may not be operative during certain periods. The objective now is to find facility locations that are both inexpensive and also reliable. We present different configurations of two hybrid metaheuristics to solve the problem, a genetic algorithm and a scatter search approach. We have carried out an extensive computational experiment to study the performance of the algorithms and compare its efficiency solving well-known benchmark instances.     

Valid inequalities for MIPs and group polyhedra from approximate liftings

In this paper, we present an approximate lifting scheme to derive valid inequalities for general mixed integer programs and for the group problem. This scheme uses superadditive functions as the building block of integer and continuous lifting procedures. It yields a simple derivation of new and known families of cuts that correspond to extreme inequalities for group problems. This new approximate lifting approach is constructive and potentially efficient in computation. J.-P. P. Richard was supported by NSF grant DMI-348611.     

Optimal Consumption Portfolio and No-Arbitrage with Nonproportional Transaction Costs

In this paper we consider a finite-state financial market with non-proportional transaction cost and bid-ask spreads. The transaction cost consists of two parts: a fixed cost and a proportional cost to the size of transaction. We show that the existence of an optimal consumption policy implies that the market has no strong arbitrage; the opposite, however, is not true, i.e., no strong arbitrage does not imply the existence of an optimal consumption policy. This is in sharp contrast with the case of proportional transaction cost and other cases reported in the literature, where no strong arbitrage is equivalent to the existence of an optimal consumption policy. We also study the relationship between weak arbitrage and strong arbitrage. Different from the market with proportional transaction cost, we find that these two forms of arbitrage are equivalent unless the fixed cost is zero. A necessary and sufficient condition for the existence of an optimal consumption policy is also obtained. Supported by CAS, NSFC, RGC of Hong Kong and NSF under Grant No. DMI-0196084 and DMI-0200306.     

Formulations for a problem of petroleum transportation

Oil tankers play a fundamental role in every offshore petroleum supply chain and due to its high price, it is essential to optimize its use. Since this optimization requires handling detailed operational aspects, complete optimization models are typically intractable. Thus, a usual approach is to solve a tactical level model prior to optimize the operational details. In this case, it is desirable that tactical models are as precise as possible to avoid too severe adjustments in the next optimization level. In this paper, we study tactical models for a crude oil transportation problem by tankers. We did our work on the top of a previous paper found in the literature. The previous model considers inventory capacities and discrete lot sizes to be transported, aiming to meet given demands over a finite time horizon. We compare several formulations for this model using 50 instances from the literature and proposing 25 new harder ones. A column generation-based heuristic is also proposed to find good feasible solutions with less computational burden than the heuristics of the commercial solver used.     

Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications

We study sample approximations of chance constrained problems. In particular, we consider the sample average approximation (SAA) approach and discuss the convergence properties of the resulting problem. We discuss how one can use the SAA method to obtain good candidate solutions for chance constrained problems. Numerical experiments are performed to correctly tune the parameters involved in the SAA. In addition, we present a method for constructing statistical lower bounds for the optimal value of the considered problem and discuss how one should tune the underlying parameters. We apply the SAA to two chance constrained problems. The first is a linear portfolio selection problem with returns following a multivariate lognormal distribution. The second is a joint chance constrained version of a simple blending problem. B.K. Pagnoncelli’s research was supported by CAPES and FUNENSEG. S. Ahmed’s research was partly supported by the NSF Award DMI-0133943. A. Shapiro’s research was partly supported by the NSF Award DMI-0619977.     

Branch-and-cut for linear programs with overlapping SOS1 constraints

SOS1 constraints require that at most one of a given set of variables is nonzero. In this article, we investigate a branch-and-cut algorithm to solve linear programs with SOS1 constraints. We focus on the case in which the SOS1 constraints overlap. The corresponding conflict graph can algorithmically be exploited, for instance, for improved branching rules, preprocessing, primal heuristics, and cutting planes. In an extensive computational study, we evaluate the components of our implementation on instances for three different applications. We also demonstrate the effectiveness of this approach by comparing it to the solution of a mixed-integer programming formulation, if the variables appearing in SOS1 constraints ar bounded.     

Experimental analysis of heuristics for the bottleneck traveling salesman problem

In this paper we develop efficient heuristic algorithms to solve the bottleneck traveling salesman problem (BTSP). Results of extensive computational experiments are reported. Our heuristics produced optimal solutions for all the test problems considered from TSPLIB, JM-instances, National TSP instances, and VLSI TSP instances in very reasonable running time. We also conducted experiments with specially constructed ‘hard’ instances of the BTSP that produced optimal solutions for all but seven problems. Some fast construction heuristics are also discussed. Our algorithms could easily be modified to solve related problems such as the maximum scatter TSP and testing hamiltonicity of a graph.     

GRASP with hybrid heuristic-subproblem optimization for the multi-level capacitated minimum spanning tree problem

We propose a GRASP using an hybrid heuristic-subproblem optimization approach for the Multi-Level Capacitated Minimum Spanning Tree (MLCMST) problem. The motivation behind such approach is that to evaluate moves rearranging the configuration of a subset of nodes may require to solve a smaller-sized MLCMST instance. We thus use heuristic rules to define, in both the construction and the local search phases, subproblems which are in turn solved exactly by employing an integer programming model. We report numerical results obtained on benchmark instances from the literature, showing the approach to be competitive in terms of solution quality. The proposed GRASP have in fact improved the best known upper bounds for almost all of the considered instances.     

An Integrated Inventory-Routing System for Multi-item Joint Replenishment with Limited Vehicle Capacity

In this paper, we develop a mathematical programming approach for coordinating inventory and transportation decisions in an inbound commodity collection system. In particular, we consider a system that consists of a set of geographically dispersed suppliers that manufacture one or more non-identical items, and a central warehouse that stocks these items. The warehouse faces a constant and deterministic demand for the items from outside retailers. The items are collected by a fleet of vehicles that are dispatched from the central warehouse. The vehicles are capacitated, and must also satisfy a frequency constraint. Adopting a policy in which each vehicle always collects the same set of items, we formulate the inventory-routing problem of minimizing the long-run average inventory and transportation costs as a set partitioning problem. We employ a column generation approach to determine a lower bound on the total costs, and develop a branch-and-price algorithm that finds the optimal assignment of items to vehicles. We also propose greedy constructive heuristics, and develop a very large-scale neighborhood (VLSN) search algorithm to find near-optimal solutions for the problem. Computational tests are performed on a set of randomly generated problem instances.The work of this author was supported by a scholarship of the Faculty of Engineering of Ubonratchathani University, Ubonratchathani, Thailand., The work of this author was supported in part by the National Science Foundation under Grant No. DMI-0085682.     

Heterogeneous surface-to-air missile defense battery location: a game theoretic approach

In the context of an air defense missile-and-interceptor engagement, a challenge for the defender is that surface-to-air missile batteries often must be located to protect high-value targets dispersed over a vast area, subject to which an attacker may observe the disposition of batteries and subsequently develop and implement an attack plan. To model this scenario, we formulate a two-player, extensive form, three-stage, perfect information, zero-sum game that accounts for, respectively, a defender’s location of batteries, an attacker’s launch of missiles against targets, and a defender’s assignment of interceptor missiles from batteries to incoming attacker missiles. The resulting trilevel math programming formulation cannot be solved via direct optimization, and it is not suitable to solve via full enumeration for realistically-sized instances. We instead adapt the game tree search technique Double Oracle, within which we embed either of two alternative heuristics to solve an important subproblem for the attacker. We test and compare these solution methods to solve a designed set of 52 instances having parametric variations, from which we derive insights regarding the nature of the underlying problem. Enhancing the solution methods with alternative initialization strategies, our superlative methodology attains the optimal solution for over 75% of the instances tested and solutions within 3% of optimal, on average, for the remaining 25% of the instances, and it is promising for realistically-sized instances, scaling well with regard to computational effort.     

The asymptotic performance ratio of an on-line algorithm for uniform parallel machine scheduling with release dates

Jobs arriving over time must be non-preemptively processed on one of m parallel machines, each running at its own speed, so as to minimize a weighted sum of the job completion times. In this on-line environment, the processing requirement and weight of a job are not known before the job arrives. The Weighted Shortest Processing Requirement (WSPR) heuristic is a simple extension of the well known WSPT heuristic, which is optimal for the single machine problem without release dates. According to WSPR, whenever a machine completes a job, the next job assigned to it is the one with the least ratio of processing requirement to weight among all jobs available for processing at this point in time. We analyze the performance of this heuristic and prove that its asymptotic competitive ratio is one for all instances with bounded job processing requirements and weights. This implies that the WSPR algorithm generates a solution whose relative error approaches zero as the number of jobs increases. Our proof does not require any probabilistic assumption on the job parameters and relies extensively on properties of optimal solutions to a single machine relaxation of the problem. Research supported in part by ONR Contracts N00014-90-J-1649 and N00014-95-1-0232, NSF Contracts DDM-9322828, DMI-9732795, DMI-0085683 and DMI-0245352, NUS Academic Research Grant R314-000-046-112, and a research grant from the Natural Sciences and Research Council of Canada (NSERC).     

A facility neighborhood search heuristic for capacitated facility location with single-source constraints and flexible demand

We consider a generalization of the well-known capacitated facility location problem with single source constraints in which customer demand contains a flexible dimension. This work focuses on providing fast and practically implementable optimization-based heuristic solution methods for very large scale problem instances. We offer a unique approach that utilizes a high-quality efficient heuristic within a neighborhood search to address the combined assignment and fixed-charge structure of the underlying optimization problem. We also study the potential benefits of combining our approach with a so-called very large-scale neighborhood search (VLSN) method. As our computational test results indicate, our work offers an attractive solution approach that can be tailored to successfully solve a broad class of problem instances for facility location and similar fixed-charge problems.