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Algorithms for approximating the nondominated set of multiobjective optimization problems are reviewed. The approaches are categorized into general methods that are applicable under mild assumptions and, thus, to a wide range of problems, and into algorithms that are specifically tailored to structured problems. All in all, this survey covers 52 articles published within the last 41 years, that is, between 1979 and 2020. Summary of Contribution: In many problems in operations research, several conflicting objective functions have to be optimized simultaneously, and one is interested in finding Pareto optimal solutions. Because of the high complexity of finding Pareto optimal solutions and their usually very large number, however, the exact solution of such multiobjective problems is often very difficult, which motivates the study of approximation algorithms for multiobjective optimization problems. This research area uses techniques and methods from algorithmics and computing in order to efficiently determine approximate solutions to many well-known multiobjective problems from operations research. Even though approximation algorithms for multiobjective optimization problems have been investigated for more than 40 years and more than 50 research articles have been published on this topic, this paper provides the first survey of this important area at the intersection of computing and operations research. [ABSTRACT FROM AUTHOR]

This paper examines properties and interrelations of several concepts of generalized concavity. It shows that the class of functions that are both ‘generalized concave’ and ‘generalized convex’ is closely related to the class of monotone functions of a single variable. After excluding a certain small class of exceptions, the paper shows that, for arbitrary (perhaps not differentiable) functions, concave implies pseudoconcave, pseudoconcave implies strictly quasiconcave, and strictly quasiconcave implies quasiconcave. Several results characterizing the extreme values of generalized concave functions are given. [ABSTRACT FROM AUTHOR]

In 1958, Wagner and Whitin published a seminal paper on the deterministic uncapacitated lot-sizing problem, a fundamental model that is embedded in many practical production planning problems. In this paper, we consider a basic version of this model in which problem parameters are stochastic: the stochastic uncapacitated lot-sizing problem. We define the production-path property of an optimal solution for our model and use this property to develop a backward dynamic programming recursion. This approach allows us to show that the value function is piecewise linear and right continuous. We then use these results to show that a full characterization of the optimal value function can be obtained by a dynamic programming algorithm in polynomial time for the case that each nonleaf node contains at least two children. Moreover, we show that our approach leads to a polynomial-time algorithm to obtain an optimal solution to any instance of the stochastic uncapacitated lot-sizing problem, regardless of the structure of the scenario tree. We also show that the value function for the problem without setup costs is continuous, piecewise linear, and convex, and therefore an even more efficient dynamic programming algorithm can be developed for this special case. [ABSTRACT FROM AUTHOR]

This paper considers the provisioning of transmission line bandwidth on a private network with given traffic routing for the purpose of distribution of video-on-demand service with guaranteed end-to-end quality of service. We present an architecture for video-on-demand service delivery and model the assignment of bandwidth in the distribution network as a constrained, nonlinear optimization problem. To solve this optimization problem, we develop three new auction algorithm-based solution procedures. The optimization problem assumes that there is a functional relationship between the maximum acceptable end-to-end delay and the bandwidth requirement for the links in the distribution network. Absent reliable video traffic data models for MPEG-2 format, we sample a large number of DVD-recorded movies to form a basis for randomly generated aggregate traffic streams. The aggregate traffic streams are used in a simulation experiment to measure the maximum transmission buffer occupancy for each given traffic stream for different transmission rates. Based on this simulation experiment, we derive an empirical transmission line provisioning function that guarantees delivery of all video frames without frame loss within a maximum frame delay tolerance. To illustrate the effectiveness of the proposed solution procedure for the bandwidth assignment problem, we solve to near optimality 570 small problem instances under three demand structures and 100 large problem instances with uniformly distributed demand. [ABSTRACT FROM AUTHOR]

Researchers and practitioners frequently spend more time fine-tuning algorithms than designing and implementing them. This is particularly true when developing heuristics and metaheuristics, where the "right" choice of values for search parameters has a considerable effect on the performance of the procedure. When testing metaheuristics, performance typically is measured considering both the quality of the solutions obtained and the time needed to find them. In this paper, we describe the development of CALIBRA, a procedure that attempts to find the best values for up to five search parameters associated with a procedure under study. Because CALIBRA uses Taguchi's fractional factorial experimental designs coupled with a local search procedure, the best values found are not guaranteed to be optimal. We test CALIBRA on six existing heuristic-based procedures. These experiments show that CALIBRA is able to find parameter values that either match or improve the performance of the procedures resulting from using the parameter values suggested by their developers. The latest version of CALIBRA can be downloaded for free from the website that appears in the online supplement of this paper at http://or.pubs.informs.org/Pages.collect.html. [ABSTRACT FROM AUTHOR]

This paper treats the discounted cost, optimal control problem for Markov processes with incomplete state information. The optimization approach for these partially observable Markov processes is a generalization of the well-known policy iteration technique for finding optimal stationary policies for completely observable Markov processes. The state space for the problem is the space of state occupancy probability distributions (the unit simplex). The development of the algorithm introduces several new ideas, including the class of finitely transient policies, which are shown to possess piecewise linear cost functions. The paper develops easily implemented approximations to stationary policies based on these finitely transient policies and shows that the concave hull of an approximation can be included in the well-known Howard policy improvement algorithm with subsequent convergence. The paper closes with a detailed example illustrating the application of the algorithm to the two-state partially observable Markov process. [ABSTRACT FROM AUTHOR]

Randomizing (stochastic) global optimization algorithms can be viewed as sampling procedures, where at each iteration a local optimum is sampled from the set of local optima. The effectiveness of such an algorithm depends crucially on the sampling distribution, i.e., on the probabilities of sampling the various local optima. There exists a very large body of research on stochastic global optimization. However, very little is known about the sampling distribution itself and, in particular, on the specific way in which it depends on the function being optimized, on the region in which the search takes place, and on the optimization routine. In this paper we set out to examine these issues in some detail and present some first results in this direction. The case analyzed in this paper is the optimization of a one-dimensional function using a randomizing version of binary search. We give an effective (computable in quadratic time) formula for the sampling distribution and obtain various interesting properties of this distribution which are relevant to the behavior of the algorithm in practice. We also discuss some issues related to adaptive search. [ABSTRACT FROM AUTHOR]

New lock box solution techniques have recently been suggested by Stone and Nauss-Markland. This paper briefly discusses these new methodologies and shows how the algorithms can be combined into a third solution procedure which exploits the computational efficiency of the Stone heuristic and retains the optimizing property of the Nauss-Markland algorithm. [ABSTRACT FROM AUTHOR]

This paper, which is presented in two parts, is a contribution to the theory of fractional programming, i.e. maximization of quotients subject to constraints. In Part 1, duality theory for linear and concave-convex fractional programs is developed and related to recent results by Bector, Craven-Mond, Jagannathan, Sharma-Swarup, et al. Basic duality theorems of linear, quadratic and convex programming are extended. In Part II Dinkelbach's algorithm solving fractional programs is considered. The rate of convergence as well as a priori and a posteriori error estimates are determined. In view of these results the stopping rule of the algorithm is changed. Also the starting rule is modified using duality as introduced in Part I. Furthermore a second algorithm is proposed. In contrast to Dinkelbach's procedure the rate of convergence is still controllable. Error estimates are obtained too. [ABSTRACT FROM AUTHOR]

Motivated by applications in freight handling and open-pit mining, Rhys, Balinski, and Picard studied the problems of selection and closure in papers published in Management Science in 1970 and 1976. They identified efficient algorithms based on linear programming and maximum-flow/minimum-cut procedures to solve these problems. This research has had major impact well beyond the initial applications, reaching across three decades and inspiring work on numerous applications and extensions. The extensions are nontrivial optimization problems that are of theoretical interest. The applications ranged from evolving technologies, image segmentation, revealed preferences, pricing, adjusting utilities for consistencies, just-in-time production, solving certain integer programs in polynomial time, and providing efficient 2-approximation algorithms for a wide variety of hard problems. A recent generalization to a convex objective function has even produced novel solutions to prediction and Bayesian estimation problems. This paper surveys the streams of research stimulated by these papers as an example of the impact of Management Science on the optimization field and an illustration of the far-reaching implications of good original research. [ABSTRACT FROM AUTHOR]

A complementary programming problem is a linear programming problem with the additional restriction that xpxq=0 holds for each specified pair (xp, xq). This paper obtains constraints called C-cuts from the restriction xpxq=0, and uses them to facilitate the computation of the branch-and-bound procedure proposed in an earlier paper [Opns. Res. 19, 1523–1528 (1971)]. Some computational results are also reported. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for finding the K loopless paths that have the shortest lengths from one node to another node in a network. The significance of the new algorithm is that its computational upper bound increases only linearly with the value of K. Consequently, in general, the new algorithm is extremely efficient as compared with the algorithms proposed by Bock, Kantner, and Haynes [2], Pollack [7], [8], Clarke, Krikorian, and Rausan [3], Sakarovitch [9] and others. This paper first reviews the algorithms presently available for finding the K shortest loopless paths in terms of the computational effort and memory addresses they require. This is followed by the presentation of the new algorithm and its justification. Finally, the efficiency of the new algorithm is examined and compared with that of other algorithms. [ABSTRACT FROM AUTHOR]

The article presents comments on the management science paper "On Sequencing n Jobs on One Machine to Minimize the Number of Late Jobs," by William L. Maxwell. The paper focused on an integer programming formulation of a one-machine job shop problem. The author contends that Maxwell made in error in proving the validity of an optimal algorithm by applying cutting plane constraints to the problem. The author explains that the problem cannot be solved through traditional cutting plane procedures. The author provides a mathematical model in an attempt to correct the proof.

This paper addresses the optimal allocation of reliability among components that are to be assembled into a system. While it is a generally accepted notion that a component's cost is an increasing function of its reliability, most research to date adopts exponentially increasing, closed-form functions to relate cost and reliability. However, in practice such functions are often unknown or difficult to construct, and it is often more reasonable to describe cost-reliability relationships via discrete data sets. We consider such situations, where each component is available at several reliability levels with corresponding costs. The design optimization problem results in a nonlinear integer program. Because every system configuration has an equivalent representation as either a series connection of parallel subsystems or a parallel connection of series subsystems, we provide formulations, linear relaxations, and algorithms for these two. [ABSTRACT FROM AUTHOR]

Simulation and optimization are among the most commonly used elements in the OR toolkit. Often times, some of the data elements used to define an optimization problem are best described by random variables, yielding a stochastic program. If the distributions of the random variables cannot be specified precisely, one may have to resort to simulation to obtain observations of these random variables. In this paper, we present conditional stochastic decomposition (CSD), a method that may be construed as providing an algorithmic interface between simulation and optimization for the solution of stochastic linear programs with recourse. Derived from the concept of the stochastic decomposition of such problems, CSD uses randomly generated observations with a Benders decomposition of the problem. In this paper, our method is analytically verified and graphically illustrated. In addition, CSD is used to solve several test problems that have appeared in the literature. Our computational experience suggests that CSD may be particularly well suited for situations in which randomly generated observations are difficult to obtain. [ABSTRACT FROM AUTHOR]

Substantial simplification of existing processes and designs may be required before the potential benefits of modem manufacturing technology can be realized. This paper analyzes implementation problems associated with a flexible system that produces flat sheet-metal parts with interior holes. The paper makes three main contributions. First, we formulate the problem of selecting tooling and design standards as an optimization model and demonstrate that the model yields insight by applying it to one manufacturer's problem, thereby reducing substantially the required tooling. Second, we show that the model has a totally balanced constraint matrix, and hence, there are polynomial time algorithms for various versions of the problem. Third, we provide new algorithms with substantially improved performance bounds for two important versions of the problem. [ABSTRACT FROM AUTHOR]

We propose and study an iterative Forward And Backward (FAB) algorithm applicable to the problem of computing optimal search plans when the target's motion is modeled by a discrete space and time Markov chain. This paper generalizes the class of objective functions to which the method applies. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm using sensitivity analysis to solve a linear two-stage optimization problem. The underlying theory rests on a set of first order optimality conditions that parallel the Kuhn-Tucker conditions associated with a one-dimensional parametric linear program. The solution to the original problem is uncovered by systematically varying the parameter over the unit interval and solving the corresponding linear program. Finite convergence is established under nondegenerate assumptions. The paper also discusses other solution techniques including branch and bound and vertex enumeration and gives an example highlighting their computational and storage requirements. By these measures, the algorithm presented here has an overall advantage. Finally, a comparison is drawn between bicriteria and bilevel programming, and underscored by way of an example. [ABSTRACT FROM AUTHOR]

Elimination methods for the flow-shop problem have been investigated by several authors using special elimination conditions to construct a set of dominant schedules and then enumerate them in order to find an optimum. The few papers reporting computational experience with elimination algorithms indicate that the approach may be an efficient method for obtaining optimal schedules. This note builds a simple model in order to help provide some perspective on these computational results and to describe how elimination algorithms may behave in moderate-sized problems. [ABSTRACT FROM AUTHOR]

This paper considers the static flowshop-scheduling problem with the objective of minimizing, as a cost function, the mean job-completion time. Within the more general framework of combinatorial optimization problems, it defines a heuristic search technique--an approach that has been successful in the past in obtaining near-optimal solutions for problems that could not be solved exactly, either for lack of theory or because of exorbitant computational requirements. The paper presents a two-phase algorithm: The first phase searches among schedules with identical processing orders on all machines; the second refines the schedule by allowing passing. Results of computer study are presented for a large ensemble of pseudorandom problems, and for two particular problems previously cited in the literature. The method is shown to provide solutions that are exceptionally low in cost, and superior to those provided by sampling techniques in the cases for which comparison is possible. Computation time is also discussed and is given in machine-independent terms. [ABSTRACT FROM AUTHOR]

The problem of determining consistent and realistic reorder intervals in complex production-distribution environments was formulated as a large scale, nonlinear, integer programming problem by Maxwell and Muckstadt (1985). They show how the special structure of the problem permits its solution by a standard network flow algorithm. In this paper, we review the Maxwell-Muckstadt model, provide necessary and sufficient conditions that characterize the solution, and show that the optimal partition of nodes in the production-distribution network is invariant to an arbitrary scaling of the set-up and holding cost parameters. We consider two capacitated versions of the model: one with a single constrained work center, and the other with multiple constrained work centers. For single constraint problems, the invariance corollary provides a simple closed-form solution. For the multiple work center problem, the invariance corollary is exploited in the development of a Lagrange multiplier method of solution. The technique is illustrated by means of a small example problem and a problem taken from a real industrial setting. [ABSTRACT FROM AUTHOR]

The solution technique developed in this paper integrates the simplex search algorithm into the recursive calculations of dynamic programming in order to mitigate the "curse-of-dimensionality." Since differentiation is not required as a means of optimization procedure, it is expected that this technique will have an important impact on planning problems, such as the multiple resource allocation problems which involve nondifferentiable and/or highly nonlinear performance functions. Applications of the technique to very complex problems seem to confirm the anticipated polynomial average performance. [ABSTRACT FROM AUTHOR]

This paper compares solution concepts associated with three location problems on a network: (i) a single-facility distance minimization problem; (ii) a two-facility spatial competition problem; and (iii) a single-facility locational voting problem. It is shown that the three problems have a common global solution when the network exhibits a property of symmetry around a point. For more general networks, this ceases to be true for global solutions but an identity holds for local solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the modeling and resolution of the energy generation planning problem of a large hydroelectric system for a horizon up to two years, with emphasis on hydro plant generation, hydro system aggregation, and objective function modeling. The problem has been formulated as a deterministic discrete-time optimal control problem, with linear transition equations, a highly nonlinear objective, and bounds on both the state and the control variables. The solution algorithm is of the reduced gradient type, with the control variables nonbasic to provide good conditioning. Because of the linear relationship between the state and control variables, the set of feasible directions at a point is a polyhedral convex cone and the bounds both on the state and control variables can be accounted for by means of an algorithm for the orthogonal projection of the gradient on the cone. Some computational experience with this large-scale non-linear program is given, followed by a discussion of implementation. [ABSTRACT FROM AUTHOR]

This paper describes the integration of a network optimization algorithm with a decision and reformation system for the Graduate School of Management at UCLA. This optimization algorithm plays an important role in the determination of the annual teaching schedule by assigning faculty to courses and other activities on the basis of their responses to course preference questionnaires. In addition, the network structure provides an ideal framework for a compact information system for the department. The system can also provide some assistance in determining faculty recruiting priorities or in studying the impacts of changes in student enrollments. Our actual experiences in implementing and using this system are described. [ABSTRACT FROM AUTHOR]

The paper presents a means of attempting to account for overhead, as well as function evaluations, in evaluating unconstrained optimization techniques. Criteria are established which compare techniques on classes of algorithms. While not completely machine independent, and certainly not programmer independent, the new method eliminates much of the machine dependency of earlier criteria. The criteria are applied to three known and one new quasi-Newton algorithm, with interesting results. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for solving resource-constrained network scheduling problems, a general class of problems that includes the classical job-shop-scheduling problem. It uses Lagrange multipliers to dualize the resource constraints, forming a Lagrangian problem in which the network constraints appear explicitly, while the resource constraints appear only in the Lagrangian function. Because the network constraints do not interact among jobs, the problem of minimizing the Lagrangian decomposes into a subproblem for each job. Algorithms are presented for solving these subproblems. Minimizing the Lagrangian with fixed multiplier values yields a lower bound on the cost of an optimal solution to the scheduling problem. The paper gives procedures for adjusting the multipliers iteratively to obtain strong bounds, and it develops a branch-and-bound algorithm that uses these bounds in the solution of the scheduling problem. Computational experience with this algorithm is discussed. [ABSTRACT FROM AUTHOR]

This paper presents new types of cutting-plane algorithms for integer-constrained optimization problems. The underlying idea of the method of enumerative cuts is to make use of an enumeration procedure in order to construct cutting planes that can be made arbitrarily deep. In recent publications, BALLAS and YOUNG have established in somewhat different contexts that valid cutting planes can be generated from the intersection of (basic) rays with adequately constructed hyperspheres. This paper introduces a family of polyhedra to replace the sphere; these polyhedra are not tangential to the hypersphere and produce cutting planes with particular characteristics that are discussed. Constructive procedures are sketched for generating step-by-step cutting planes that become deeper and deeper at every step, until the deepest cut is obtained. This construction often relies on a partial enumeration scheme of the set of integer solutions. [ABSTRACT FROM AUTHOR]

This paper studies the close relation between the Gass-Saaty parametric programming algorithm and the `primal-dual' procedures recently exploited by DANTZIG, FORD, AND FULKERSON lit is shown that the two procedures are equivalent The possibility of eliminating the two-phase character of the simplex method using these techniques is discussed Finally, the application of the techniques to problems with special structure is considered [ABSTRACT FROM AUTHOR]

Randomly-generated linear programming problems of three different types and five different sizes were solved by the criss-cross method and by the simplex method. One hundred problems of each type and size were solved, and the results are overwhelmingly favorable to the criss-cross method. An improvement to the criss-cross method used in these tests is given, and the extension of the results of the paper to variations of the criss-cross and simplex methods is discussed. [ABSTRACT FROM AUTHOR]

The paper considers a class of optimization problems. The problems are linear programming problems: maximize ex subject to Ax = b with the additional constraint that x must also be an extreme point of a second convex polyhedron Dx = d. X ≥ 0. A cutting-plane algorithm for solving such problems is presented. Two numerical examples are also included. [ABSTRACT FROM AUTHOR]

The author responds to the paper "The Loading Problem," by Eilon and Christofides. He focuses on suggesting a solution to the problem. The recommended solution involves a heuristic algorithm that allegedly requires fewer computations than the one proposed by Eilon and Christofides. It is suggested that the simpler algorithm offered by the author reaches mathematical optimization as often as the aforementioned algorithm. It is further proposed that the algorithm presented by Eilon and Christofides cannot be used to solve multi-dimensional programming problems. The author goes on to discuss a numerical analysis that can be used for the allocation of industrial resources. Related mathematical models are discussed in detail. A table comparing the results of the two algorithms is also presented.

A framework of concepts is developed which helps to unify a substantial portion of the literature on large-scale mathematical programming. These concepts fall into two categories. The first category consists of problem manipulations that can be used to derive what are often referred to as "master" problems; the principal manipulations discussed are Projection, Inner Linearization, and Outer Linearization. The second category consists of solution strategies that can be used to solve the master problems, often with the result that "subproblems" arise which can then be solved by specialized algorithms. The Piecewise, Restriction, and Relaxation strategies are the principal ones discussed. Numerous algorithms found in the literature are classified according to the manipulation/strategy pattern they can be viewed as using, and the usefulness of the framework is demonstrated by using it (see Part II of this paper) to rederive a representative selection of algorithms. The material presented is listed in the following order: The first section is introductory in nature, and discusses types of large-scale problems, the scope of discussion and the literature, and the notation used. The second section, entitled "Problem Manipulation: Source of 'Master' Problems" covers the subjects of projection, inner linearization and outer linearization. The third section, "Solution Strategies: Source of 'Subproblems'," discusses piecewise strategy, restriction and relaxation. The fourth section is entitled "Synthesizing Known Algorithms from Manipulations and Strategies," and is followed by a concluding section and an extensive bibliography. [ABSTRACT FROM AUTHOR]

In this paper the Fiacco-McCormick SUMT technique is embedded in a class of inside-out algorithms. Convergence is demonstrated for the nonlinear programming problem under fairly general conditions and the algorithms are interpreted in a geometric structure. [ABSTRACT FROM AUTHOR]

The usefulness of integer programming as a tool of capital budgeting hinges on the development of an efficient solution technique. An algorithm based on partial enumeration has been developed by E. L. Lawler and M. D. Bell for solving integer linear programs with 0-1 decision variables; however their algorithm is not general enough to deal with all problems in which the objective function is quadratic. This paper extends Lawler and Bell's method so that it can be generally applied to integer quadratic programs. The new algorithm is illustrated by examples from capital budgeting. [ABSTRACT FROM AUTHOR]

This paper considers the multiarmed bandit problem with multiple simultaneous arm pulls and the additional restriction that we do not allow recourse to arms that were pulled at some point in the past but then discarded. This additional restriction is highly desirable from an operational perspective, and we refer to this problem as the "irrevocable multiarmed bandit" problem. We observe that natural modifications to well-known heuristics for multiarmed bandit problems that satisfy this irrevocability constraint have unsatisfactory performance and, thus motivated, introduce a new heuristic: the "packing" heuristic. We establish through numerical experiments that the packing heuristic offers excellent performance, even relative to heuristics that are not constrained to be irrevocable. We also provide a theoretical analysis that studies the "price" of irrevocability, i.e., the performance loss incurred in imposing the constraint we propose on the multiarmed bandit model. We show that this performance loss is uniformly bounded for a general class of multiarmed bandit problems and indicate its dependence on various problem parameters. Finally, we obtain a computationally fast algorithm to implement the packing heuristic; the algorithm renders the packing heuristic computationally cheaper than methods that rely on the computation of Gittins indices. [ABSTRACT FROM AUTHOR]

In this paper we consider min-max convex semi-infinite programming. To solve these problems we introduce a unified framework concerning Remez-type algorithms and integral methods coupled with penalty and smoothing methods. This framework subsumes well-known classical algorithms, but also provides some new methods with interesting properties. Convergence of the primal and dual sequences are proved under minimal assumptions. [ABSTRACT FROM AUTHOR]

Assembly line balancing problems are concerned with the optimization of manufacturing processes. In this paper we consider the so-called simple assembly line balancing problem with the objective of minimizing the number of used workstations. This problem is denoted by SALB-1 in the literature. For tackling this problem, we present a so-called Beam-ACO approach. This technique results from hybridizing the metaheuristic ant colony optimization with beam search. The experimental results show that our algorithm is a state-of-the-art method for this problem. It can solve 263 of 269 existing benchmark instances to optimality. [ABSTRACT FROM AUTHOR]

This paper solves FC_2TDC, the two-staged fixed-orientation two-dimensional cutting stock problem. FC_2TDC is a variant of the constrained two-dimensional cutting problem where each piece is produced, in the final cutting pattern, by at most two cuts, and each piece has a fixed orientation. It is solved by an algorithm that combines a strip-generation procedure with beam search (BS). BS, which is a truncated branch-andbound, investigates a subset of elite nodes and permanently fathoms the others. It chooses the elite nodes based on an evaluation operator. The importance of the evaluation operator on the performance of BS is highlighted by comparing a local beam search (LBS) that uses a priority cost-evaluation operator to a global beam search (GBS) that adopts a global cost-evaluation operator. Computational results, based on several medium and large instances, further show that LBS provides good solutions, whereas GBS gives (near) optimal solutions within reasonable computational time. [ABSTRACT FROM AUTHOR]

Consider the linear complementarity problem given in the system: (1) W = MZ + q (2) W ≥ 0, Z ≥ 0 (I) (3) Z[sup T] W = 0 where, W, Z and q are vectors of dimension n. M is a matrix of order n x n and Z[sup T] is the transpose of Z. Any (Z, W) satisfying (1), (2), and (3) is a complementary feasible solution to system (I). In the literature, a class of matrices is defined such that if M belongs to this class, then existence of a feasible solution to system (I) implies the existence of a complementary feasible solution to system (I) with W = 0. In this paper, a new class of matrices M is developed. It is shown that membership of a matrix M in M is equivalent to the property; for any q existence of a feasible solution to system (I) implies the existence of a complementary feasible solution to system (I) for that q with W = 0. This new class of matrices is not contained in any one of the known classes, namely, copositive plus, positive definite or semidefinite, P-matrices, P-matrices, Z-class, etc. [ABSTRACT FROM AUTHOR]

Model reference adaptive search (MRAS) for solving global optimization problems works with a parameterized probabilistic model on the solution space and generates at each iteration a group of candidate solutions. These candidate solutions are then used to update the parameters associated with the probabilistic model in such a way that the future search will be biased toward the region containing high-quality solutions. The parameter updating procedure in MRAS is guided by a sequence of implicit probabilistic models we call reference models. We provide a particular algorithm instantiation of the MRAS method, where the sequence of reference models can be viewed as the generalized probability distribution models for estimation of distribution algorithms (EDAs) with proportional selection scheme. In addition, we show that the model reference framework can also be used to describe the recently proposed cross-entropy (CE) method for optimization and to study its properties. Hence, this paper can also be seen as a study on the effectiveness of combining CE and EDAs. We prove global convergence of the proposed algorithm in both continuous and combinatorial domains, and we carry out numerical studies to illustrate the performance of the algorithm. [ABSTRACT FROM AUTHOR]

The constrained two-dimensional cutting (C_TDC) problem consists of determining a cutting pattern of a set of n small rectangular piece types on a rectangular stock plate S with length L and width W, to maximize the sum of the profits of the pieces to be cut. Each piece type i, i = 1, ..., n, is characterized by a length li, a width wi, a profit (or weight) ci, and an upper demand value bi. The upper demand value is the maximum number of pieces of type i that can be cut on S. In this paper, we study the two-staged C_TDC problem, noted C_2TDC. It is a classical variant of the C_TDC where each piece is produced, in the final cutting pattern, by at most two cuts. We solve the C_2TDC problem using an exact algorithm that is mainly based on a bottom-up strategy. We introduce new lower and upper bounds and propose new strategies that eliminate several duplicate patterns. We evaluate the performance of the proposed exact algorithm on problem instances extracted from the literature and compare it to the performance of an existing exact algorithm. [ABSTRACT FROM AUTHOR]

Concavity cuts play an important role in several algorithms for concave minimization, such as pure cutting plane algorithms, conical algorithms, and branch-and-bound algorithms. For concave quadratic minimization problems Konno et al. (1998) have demonstrated that the lower the rank of the problem, i.e., the smaller the number of nonlinear variables, the deeper the concavity cuts usually turn out to be. In this paper we examine the case where the number of nonlinear variables of a concave minimization problem is large, but most of the objective value of a good solution is determined by a small number of variables only. We will discuss ways to exploit such a situation to derive deep cutting planes. To this end we apply concepts usually applied for efficiently solving low-rank concave minimization problems. [ABSTRACT FROM AUTHOR]

In this paper we propose to modify the big square small square global optimization search in the plane with a big triangle small triangle approach. The triangulation of the feasible region is obtained by using Voronoi diagrams. The resulting algorithm was tested on the obnoxious facility location and the attraction-repulsion Weber problems with excellent results. [ABSTRACT FROM AUTHOR]

Condition numbers based on the "distance to ill-posedness" ρ(d) have been shown to play a crucial role in the theoretical complexity of solving convex optimization models. In this paper, we present two algorithms and corresponding complexity analysis for computing estimates of ρ(d) for a finite-dimensional convex feasibility problem P(d) in standard primal form: find x that satisfies Ax = b, x ∈ C[subx], where d = (A, b) is the data for the problem P(d). Under one choice of norms for the m- and n-dimensional spaces, the problem of estimating ρ(d) is hard (co-NP complete even when C[subx] = ℜ[supn][sub+]). However, when the norms are suitably chosen, the problem becomes much easier: We can estimate ρ(d) to within a constant factor of its true value with complexity bounds that are linear in ln(C(d)) (where C(d) is the condition number of the data d for P(d), plus other quantities that arise naturally in consideration of the problem P(d). The first algorithm is an interior-point algorithm, and the second algorithm is a variant of the ellipsoid algorithm. The main conclusion of this work is that when the norms are suitably chosen, computing an estimate of the condition measures of P(d) is essentially not much harder than computing a solution of P(d) itself. [ABSTRACT FROM AUTHOR]

This paper considers the shortest path problem with waiting costs (SPWC) as an extension to the shortest path problem with time windows. The problem consists of finding the minimum cost path in a network, where cost and time are two independent quantities associated with a path. Path feasibility is constrained by time windows at each node and a linear cost penalty is imposed for each unit of time spent waiting along the path. Starting from a known, nonlinear, integer programming formulation, we propose two alternative formulations for which algorithms already exist. First, we indicate how to transform the SPWC into a shortest path problem with time windows and linear node costs. Second, we prove that the SPWC can also be formulated as a two-resource generalized shortest path problem with resource constraints. Computational results used to compare these alternative formulations are presented. [ABSTRACT FROM AUTHOR]

In this paper we present some special procedures for the numerical solution of the optimal scheduling problem of a multi-item single machine. We study the infinite horizon case when the optimization criterion is the average cost. We establish the solution of the problem in terms of viscosity solutions of the Quasi-Variational Inequality (QVI) system associated to the problem. The existence of solution of the QVI and the uniqueness of the optimal average cost are proved. A method of discretization and a computational procedure are described. They allow us to compute the solution in a short time and with precision of order k. We obtain an estimate for the discretization error and develop an algorithm that converges in a finite number of steps. In our method the nodes of the triangulation mesh are joined by segments of trajectories of the original system. This feature allows us to obtain the k-order precision which, in general, is impossible to obtain by usual methods. [ABSTRACT FROM AUTHOR]

A companion paper (Part I) considers the problem of minimizing the weighted earliness and tardiness of jobs scheduled on a single machine around a common due date, d, which is unrestrictively late. This paper (Part II) considers the problem of minimizing the unweighted earliness and tardiness of jobs, allowing the possibility that d is early enough to constrain the scheduling decision. We describe several optimality conditions. The recognition version of the problem is shown to be NP-complete in the ordinary sense, confirming a well known conjecture. Moreover, this complexity definition is precise, since we describe a dynamic programming algorithm which runs in pseudopolynomial time. This algorithm is also extremely efficient computationally, providing an improvement over earlier procedures, of almost two orders of magnitude in the size of instance that can be solved. Finally, we describe a special case of the problem which is polynomially solvable. [ABSTRACT FROM AUTHOR]

This paper analyzes the computational complexity of the project sequencing problem. Specifically, we prove that the general decision problem is NP-complete, but note that an important special case is solvable in polynomial time. [ABSTRACT FROM AUTHOR]

The surveillance search problem is a generalization of the detection search problem studied by Brown and Stone, among others. In the surveillance version of the problem, the search terminates only if the time and location of a detection satisfy certain problem specific conditions; the objective is to maximize the expected value of a payoff received during and at the end of the search. For instance, we may wish to maximize the probability of finding the target in a specified subset of the state space or at a specified time. Consequently, we might detect the target several times before the search terminates. A search strategy, i.e., an allocation of the available search effort, must account for this possibility. In this paper, we formulate the surveillance search problem for applications in which the target moves according to a (not necessarily time-homogeneous) Markov process. We then derive a set of necessary conditions for the optimality of a search strategy, end develop algorithms, based on the algorithm of Brown for the detection search problem, to find strategies that satisfy these necessary conditions for problems in which the search effort is infinitely divisible. [ABSTRACT FROM AUTHOR]