Minimizing sums and products of linear fractional functions over a polytope

In this paper, we develop efficient deterministic algorithms for globally minimizing the sum and the product of several linear fractional functions over a polytope. We will show that an elaborate implementation of an outer approximation algorithm applied to the master problem generated by a parametric transformation of the objective function serves as an efficient method for calculating global minima of these nonconvex minimization problems if the number of linear fractional terms in the objective function is less than four or five. It will be shown that the Charnes–Cooper transformation plays an essential role in solving these problems. Also a simple bounding technique using linear multiplicative programming techniques has remarkable effects on structured problems. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 583–596, 1999     

Maximizing the sum of certain quasiconcave functions using generalized benders decomposition

In this paper we consider the problem of maximizing the sum of certain quasi-concave functions over a convex set. The functions considered belong to the classes of functions which are known as nonlinear fractional and binonlinear functions. Each individual function is quasi-concave but the sum is not. We show that this nonconvex programming problem can be solved using Generalized Benders Decomposition as developed by Geoffrion.     

Concave envelopes of monomial functions over rectangles

The construction of convex and concave envelopes of real‐valued functions has been of interest in mathematical programming for over 3 decades. Much of this interest stems from the fact that convex and concave envelopes can play important roles in algorithms for solving various discrete and continuous global optimization problems. In this article, we use a simplicial subdivision tool to present and validate the formula for the concave envelope of a monomial function over a rectangle. Potential algorithmic applications of this formula are briefly indicated. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

On the generation of deep disjunctive cutting planes

In this paper we address the question of deriving deep cuts for nonconvex disjunctive programs. These problems include logical constraints which restrict the variables to at least one of a finite number of constraint sets. Based on the works of Balas. Glover, and Jeroslow, we examine the set of valid inequalities or cuts which one may derive in this context, and defining reasonable criteria to measure depth of a cut we demonstrate how one may obtain the “deepest” cut. The analysis covers the case where each constraint set in the logical statement has only one constraint and is also extended for the case where each of these constraint sets may have more than one constraint.     

Sensor placement in active multistatic sonar networks

The idea of deploying noncollocated sources and receivers in multistatic sonar networks (MSNs) has emerged as a promising area of opportunity in sonar systems. This article is one of the first to address point coverage problems in MSNs, where a number of points of interest have to be monitored in order to protect them from hostile underwater assets. We consider discrete “definite range” sensors as well as various diffuse sensor models. We make several new contributions. By showing that the convex hull spanned by the targets is guaranteed to contain optimal sensor positions, we are able to limit the solution space. Under a definite range sensor model, we are able to exclude even more suboptimal solutions. We then formulate a nonlinear program and an integer nonlinear program to express the sensor placement problem. To address the nonconvex single‐source placement problem, we develop the Divide Best Sector (DiBS) algorithm, which quickly provides an optimal source position assuming fixed receivers. Starting with a basic implementation of DiBS, we show how incorporating advanced sector splitting methods and termination conditions further improve the algorithm. We also discuss two ways to use DiBS to find multiple source positions by placing sensors iteratively or simultaneously. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 287–304, 2017     

Capacity improvement,penalties, and the fixed charge transportation problem

Capacity improvement and conditional penalties are two computational aides for fathoming subproblems in a branch‐and‐bound procedure. In this paper, we apply these techniques to the fixed charge transportation problem (FCTP) and show how relaxations of the FCTP subproblems can be posed as concave minimization problems (rather than LP relaxations). Using the concave relaxations, we propose a new conditional penalty and three new types of capacity improvement techniques for the FCTP. Based on computational experiments using a standard set of FCTP test problems, the new capacity improvement and penalty techniques are responsible for a three‐fold reduction in the CPU time for the branch‐and‐bound algorithm and nearly a tenfold reduction in the number of subproblems that need to be evaluated in the branch‐and‐bound enumeration tree. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 341–355, 1999     

Approximating nondominated sets in continuous multiobjective optimization problems

Many important problems in Operations Research and Statistics require the computation of nondominated (or Pareto or efficient) sets. This task may be currently undertaken efficiently for discrete sets of alternatives or for continuous sets under special and fairly tight structural conditions. Under more general continuous settings, parametric characterisations of the nondominated set, for example through convex combinations of the objective functions or ε‐constrained problems, or discretizations‐based approaches, pose several problems. In this paper, the lack of a general approach to approximate the nondominated set in continuous multiobjective problems is addressed. Our simulation‐based procedure only requires to sample from the set of alternatives and check whether an alternative dominates another. Stopping rules, efficient sampling schemes, and procedures to check for dominance are proposed. A continuous approximation to the nondominated set is obtained by fitting a surface through the points of a discrete approximation, using a local (robust) regression method. Other actions like clustering and projecting points onto the frontier are required in nonconvex feasible regions and nonconnected Pareto sets. In a sense, our method may be seen as an evolutionary algorithm with a variable population size. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

A specially structured nonlinear integer resource allocation problem

We present an algorithm for solving a specially structured nonlinear integer resource allocation problem. This problem was motivated by a capacity planning study done at a large Health Maintenance Organization in Texas. Specifically, we focus on a class of nonlinear resource allocation problems that involve the minimization of a convex function over one general convex constraint, a set of block diagonal convex constraints, and bounds on the integer variables. The continuous variable problem is also considered. The continuous problem is solved by taking advantage of the structure of the Karush‐Kuhn‐Tucker (KKT) conditions. This method for solving the continuous problem is then incorporated in a branch and bound algorithm to solve the integer problem. Various reoptimization results, multiplier bounding results, and heuristics are used to improve the efficiency of the algorithms. We show how the algorithms can be extended to obtain a globally optimal solution to the nonconvex version of the problem. We further show that the methods can be applied to problems in production planning and financial optimization. Extensive computational testing of the algorithms is reported for a variety of applications on continuous problems with up to 1,000,000 variables and integer problems with up to 1000 variables. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 770–792, 2003.     

Scheduling target illuminators in naval battle-group anti-air warfare

We schedule a set of illuminators (homing devices) to strike a set of targets using surface-to-air missiles in a naval battle. The task is viewed as a production floor shop scheduling problem of minimizing the total weighted flow time, subject to time-window job availability and machine downtime side constraints. A simple algorithm based on solving assignment problems is developed for the case when all the job processing times are equal and the data are all integer. For the general case of scheduling jobs with unequal processing times, we develop two alternate formulations and analyze their relative strengths by comparing their respective linear programming relaxations. We select the better formulation in this comparison and exploit its special structures to develop several effective heuristic algorithms that provide good-quality solutions in real time; this is an essential element for use by the Navy. © 1995 John Wiley & Sons, Inc.     

Multi‐item capacitated lot‐sizing problems with setup times and pricing decisions

We study a multi‐item capacitated lot‐sizing problem with setup times and pricing (CLSTP) over a finite and discrete planning horizon. In this class of problems, the demand for each independent item in each time period is affected by pricing decisions. The corresponding demands are then satisfied through production in a single capacitated facility or from inventory, and the goal is to set prices and determine a production plan that maximizes total profit. In contrast with many traditional lot‐sizing problems with fixed demands, we cannot, without loss of generality, restrict ourselves to instances without initial inventories, which greatly complicates the analysis of the CLSTP. We develop two alternative Dantzig–Wolfe decomposition formulations of the problem, and propose to solve their relaxations using column generation and the overall problem using branch‐and‐price. The associated pricing problem is studied under both dynamic and static pricing strategies. Through a computational study, we analyze both the efficacy of our algorithms and the benefits of allowing item prices to vary over time. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

A novel modeling approach for express package carrier planning

Express package carrier networks have large numbers of heavily‐interconnected and tightly‐constrained resources, making the planning process difficult. A decision made in one area of the network can impact virtually any other area as well. Mathematical programming therefore seems like a logical approach to solving such problems, taking into account all of these interactions. The tight time windows and nonlinear cost functions of these systems, however, often make traditional approaches such as multicommodity flow formulations intractable. This is due to both the large number of constraints and the weakness of the linear programming (LP) relaxations arising in these formulations. To overcome these obstacles, we propose a model in which variables represent combinations of loads and their corresponding routings, rather than assigning individual loads to individual arcs in the network. In doing so, we incorporate much of the problem complexity implicitly within the variable definition, rather than explicitly within the constraints. This approach enables us to linearize the cost structure, strengthen the LP relaxation of the formulation, and drastically reduce the number of constraints. In addition, it greatly facilitates the inclusion of other stages of the (typically decomposed) planning process. We show how the use of templates, in place of traditional delayed column generation, allows us to identify promising candidate variables, ensuring high‐quality solutions in reasonable run times while also enabling the inclusion of additional operational considerations that would be difficult if not impossible to capture in a traditional approach. Computational results are presented using data from a major international package carrier. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Optimization methods for large-scale multiechelon stockage problems

I examine the problem of determining inventory stockage levels and locations of different parts in a multiechelon system. This stockage problem is complicated by parts commonality—each part may be used by several different end items. Stockage levels and locations of each part affect the availability of end items that use the part, since an end item will be out of service if it requires a part that is not available. Of course, if the part is available at another nearby location, then the end item will be out of service for a shorter period of time. An essential feature of any model for this problem is constraints on operational availability of the end items. Because these constraints would involve nonconvex functions if the stockage levels were allowed to vary continuously, I formulate a 0–1 linear optimization model of the stockage problem. In this model, each part can be stocked at any of a number of prespecified levels at each echelon. The model is to minimize stockage cost of the selected items subject to the end-item availability constraints and limits on the total weight, volume, and number of different parts stocked at each echelon. Advantages and disadvantages of different Lagrangian relaxations and the simplex method with generalized upper-bounding capability are discussed for solving this stockage model.     

Discrete equal‐capacity p‐median problem

This paper examines the discrete equal‐capacity p‐median problem that seeks to locate p new facilities (medians) on a network, each having a given uniform capacity, in order to minimize the sum of distribution costs while satisfying the demand on the network. Such problems arise, for example, in local access and transport area telecommunication network design problems where any number of a set of p facility units can be constructed at the specified candidate sites (hence, the net capacity is an integer multiple of a given unit capacity). We develop various valid inequalities, a separation routine for generating cutting planes that are specific members of such inequalities, as well as an enhanced reformulation that constructs a partial convex hull representation that subsumes an entire class of valid inequalities via its linear programming relaxation. We also propose suitable heuristic schemes for this problem, based on sequentially rounding the continuous relaxation solutions obtained for the various equivalent formulations of the problem. Extensive computational results are provided to demonstrate the effectiveness of the proposed valid inequalities, enhanced formulations, and heuristic schemes. The results indicate that the proposed schemes for tightening the underlying relaxations play a significant role in enhancing the performance of both exact and heuristic solution methods for this class of problems. © 2000 John & Sons, Inc. Naval Research Logistics 47: 166–183, 2000.     

A theoretical and computational comparison of “equivalent” mixed-integer formulations

This paper provides a theoretical and computational comparison of alternative mixed integer programming formulations for optimization problems involving certain types of economy-of-scale functions. Such functions arise in a broad range of applications from such diverse areas as vendor selection and communications network design. A “nonstandard” problem formulation is shown to be superior in several respects to the traditional formulation of problems in this class.     

Geometry and resolution of duality gaps

In this study we interpret the exterior penalty function method as a generalized lagrangian metliod which fills duality gaps in nonconvex problems. Geometry and resolution of these gaps from a duality point of view are highlighted.     

Multiobjective fractional programming duality theory

Results of Geoffrion for efficient and properly efficient solutions of multiobjective programming problems are extended to multiobjective fractional programming problems. Duality relationships are given for these problems where the functions are generalized convex or invex.     

Formulations for dynamic lot sizing with service levels

In this article, we study deterministic dynamic lot‐sizing problems with a service‐level constraint on the total number of periods in which backlogs can occur over a finite planning horizon. We give a natural mixed integer programming formulation for the single item problem (LS‐SL‐I) and study the structure of its solution. We show that an optimal solution to this problem can be found in \begin{align*}\mathcal O(n^2\kappa)\end{align*} time, where n is the planning horizon and \begin{align*}\kappa=\mathcal O(n)\end{align*} is the maximum number of periods in which demand can be backlogged. Using the proposed shortest path algorithms, we develop alternative tight extended formulations for LS‐SL‐I and one of its relaxations, which we refer to as uncapacitated lot sizing with setups for stocks and backlogs. {We show that this relaxation also appears as a substructure in a lot‐sizing problem which limits the total amount of a period's demand met from a later period, across all periods.} We report computational results that compare the natural and extended formulations on multi‐item service‐level constrained instances. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

基于松弛序列凸优化的轮式机器人协同轨迹规划

针对威胁环境下的多智能体协同轨迹规划问题,以轮式机器人为对象,研究了基于序列凸优化方法的协同轨迹规划方法.首先通过对轮式机器人模型的分析,给出单独轮式机器人实际物理约束,同时以状态量、控制量加权为性能指标,考虑运动学方程、避障避碰约束、个体物理性能约束、终端约束,建立多轮式机器人协同轨迹规划问题;其次,对运动学方程、避...     

Efficient distributions of arms‐control inspection effort

A rule that constrains decision‐makers is enforced by an inspector who is supplied with a fixed level of inspection resources—inspection personnel, equipment, or time. How should the inspector distribute its inspection resources over several independent inspectees? What minimum level of resources is required to deter all violations? Optimal enforcement problems occur in many contexts; the motivating application for this study is the role of the International Atomic Energy Agency in support of the Treaty on the Non‐Proliferation of Nuclear Weapons. Using game‐theoretic models, the resource level adequate for deterrence is characterized in a two‐inspectee problem with inspections that are imperfect in the sense that violations can be missed. Detection functions, or probabilities of detecting a violation, are assumed to be increasing in inspection resources, permitting optimal allocations over inspectees to be described both in general and in special cases. When detection functions are convex, inspection effort should be concentrated on one inspectee chosen at random, but when they are concave it should be spread deterministicly over the inspectees. Our analysis provides guidance for the design of arms‐control verification operations, and implies that a priori constraints on the distribution of inspection effort can result in significant inefficiencies. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Evaluation of costs of ordering policies in large machine repair problems

This article examines a version of the machine repair problem where failures may be irreparable. Since the number of machines in the system keeps decreasing, we impose a fixed state-dependent ordering policy of the type often encountered in inventory models. Although the system is Markovian, the number of states becomes very large. The emphasis of the article, therefore, is on deriving computationally tractable formulas for the steady-state probabilities, the long-run average cost per unit time, and the vector of expected discounted costs. When the state space is so large that exact computations may be infeasible, we propose approximations which are relatively quick and simple to compute and which yield very accurate results for the test problems examined.