Showing papers in "Mathematics of Operations Research in 1983"
TL;DR: It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable.
Abstract: It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers.
1,256 citations
TL;DR: It is proved that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point and the algorithm is proven to converge to a global solution of the nonconvex program.
Abstract: This paper presents a branch-and-bound algorithm for minimizing the sum of a convex function in x, a convex function in y and a bilinear term in x and y over a closed set. Such an objective function is called biconvex with biconcave functions similarly defined. The feasible region of this model permits joint constraints in x and y to be expressed. The bilinear programming problem becomes a special case of the problem addressed in this paper. We prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point. The algorithm is proven to converge to a global solution of the nonconvex program. We discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs, for which the algorithm has been implemented.
566 citations
TL;DR: This paper studies two classes of sets, strongly and weakly convex sets, and studies a class of functions, denoted (rho)-convex, which satisfy for arbitrary points x 1 and x 2 and any value (lambda) [0, 1] the classical inequality of convex functions up to a term.
Abstract: In this paper we study two classes of sets, strongly and weakly convex sets. For each class we derive a series of properties which involve either the concept of supporting ball, an obvious extension of the concept of supporting hyperplane, or the normal cone to the set. We also study a class of functions, denoted ρ-convex, which satisfy for arbitrary points x1 and x2 and any value λ ∈ [0, 1] the classical inequality of convex functions up to a term ρ(1 − λ) λ‖x1 − x2‖2. Depending on the sign of the constant ρ the function is said to be strongly or weakly convex. We provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets. Finally, we give three applications: a separation theorem, a sufficient condition for global optimum of a nonconvex programming problem, and a sufficient geometrical condition for a set to be a manifold.
416 citations
TL;DR: The optimality of a particular control limit policy is proved directly, with heavy reliance on the change of variable formula for semimartingales, to minimize the expected discounted sum of holding costs and control costs over an infinite planning horizon.
Abstract: A controller continuously monitors a storage system, such as an inventory or bank account, whose content Z = {Zt,t≥0} fluctuates as a (μ, σ2) Brownian motion in the absence of control. Holding costs are incurred continuously at rate h(Zt). At any time, the controller may instantaneously increase the content of the system, incurring a proportional cost of r times the size of the increase, or decrease the content at a cost of l times the size of the decrease. We consider the case where h is convex on a finite interval [α, β] and h = ∞ outside this interval. The objective is to minimize the expected discounted sum of holding costs and control costs over an infinite planning horizon. It is shown that there exists an optimal control limit policy, characterized by two parameters a and b (α ≤ a < b ≤ β). Roughly speaking, this policy exerts the minimum amounts of control sufficient to keep Zt ∈ [a, b] for all t ≥ 0. Put another way, the optimal control limit policy imposes on Z a lower reflecting barrier at a an...
273 citations
TL;DR: This work proves the existence of an optimal control band policy and calculates explicitly the optimal values of the critical numbers ( q, Q, S) and aims to minimize expected discounted costs over an infinite planning horizon.
Abstract: Consider a storage system, such as an inventory or cash fund, whose content fluctuates as a (μ, σ2) Brownian motion in the absence of control. Holding costs are continuously incurred at a rate proportional to the storage level and we may cause the storage level to jump by any desired amount at any time except that the content must be kept nonnegative. Both positive and negative jumps entail fixed plus proportional costs, and our objective is to minimize expected discounted costs over an infinite planning horizon. A control band policy is one that enforces an upward jump to q whenever level zero is hit, and enforces a downward jump to Q whenever level S is hit (0 < q < Q < S). We prove the existence of an optimal control band policy and calculate explicitly the optimal values of the critical numbers (q, Q, S).
233 citations
TL;DR: An axiom of monotonicity is formulated and imposed with respect to changes in the number of agents, stating that if sacrifices have to be made to support an additional agent, then everybody should contribute.
Abstract: We reconsider the traditional problem of fair division. Division principles should be general enough to accommodate changes in what is to be divided as well as variations in the number of agents among whom the division is to take place. In the usual treatment of the question, this number is assumed to be fixed. Here, we explicitly allow it to vary. Agents are assumed to have von Neumann–Morgenstern utility functions and division problems are defined as subsets of the utility space. Our approach is axiomatic. Apart from the familiar axioms of Pareto-optimality, scale invariance, continuity and anonymity, we formulate and impose an axiom of monotonicity with respect to changes in the number of agents, stating that if sacrifices have to be made to support an additional agent, then everybody should contribute. There is a unique division principle that satisfies them all: it is the natural extension to the n-person case of the two-person solution proposed by Kalai and Smorodinsky.
230 citations
TL;DR: A new algorithm is described, and shown to have a better complexity estimate than existing methods and theoretical superiority, and empirical evidence supporting the theoretical superiority is presented.
Abstract: The computational complexity of problems relating to the enumeration of all the vertices of a convex polyhedron defined by linear inequalities is examined. Several published approaches are evaluated in this light. A new algorithm is described, and shown to have a better complexity estimate than existing methods. Empirical evidence supporting the theoretical superiority is presented. Finally vertex enumeration is discussed when the space containing the polyhedra is of fixed dimension and only the size of the inequality system is permitted to vary.
184 citations
TL;DR: It is shown how dynamic programming techniques can be used to construct pseudopolynomial time optimization algorithms and fully polynomial time approximation schemes for the partially ordered knapsack problem and how this approach can be adapted to the case of in-trees and to a related tree partitioning problem arising in integrated circuit design.
Abstract: Let G be an acyclic directed graph with weights and values assigned to its vertices. In the partially ordered knapsack problem we wish to find a maximum-valued subset of vertices whose total weight does not exceed a given knapsack capacity, and which contains every predecessor of a vertex if it contains the vertex itself. We consider the special case where G is an out-tree. Even though this special case is still NP-complete, we observe how dynamic programming techniques can be used to construct pseudopolynomial time optimization algorithms and fully polynomial time approximation schemes for it. In particular, we show that a nonstandard approach we call “left-right” dynamic programming is better suited for this problem than the standard “bottom-up” approach, and we show how this “left-right” approach can also be adapted to the case of in-trees and to a related tree partitioning problem arising in integrated circuit design. We conclude by presenting complexity results which indicate that similar success cannot be expected with either problem when the restriction to trees is lifted.
183 citations
TL;DR: This paper considers the sequential uncooperative problem in which two decision makers wish to maximize their own objective functions over a feasible region defined by interactive strategy sets and proposes a grid search algorithm which exhibits the desirable property of monotonicity.
Abstract: The conflict that naturally arises in a hierarchical system can often be modeled as a multistage optimization problem. This paper considers the sequential uncooperative problem in which two decision makers wish to maximize their own objective functions over a feasible region defined by interactive strategy sets. The resultant problem is known as the bilevel program. Such programs are inherently nonconvex and resistant to standard nonlinear programming solution techniques such as piecewise linearization and convex underestimating envelopes. Alternatively, a grid search algorithm is offered which exhibits the desirable property of monotonicity. The algorithm is based on two sets of necessary conditions previously developed and combined here to provide an operational check for stationarity and local optimality. Potential solutions are obtained from a parameterized master program whose feasible region approximates that of the original problem. As the one-dimensional parameter is varied over the unit interval ...
132 citations
TL;DR: Some general theorems on the existence of minimal points of compact sets in linear spaces with respect to closed convex orderings are proved and compared to the axiom of choice.
Abstract: We prove some general theorems on the existence of minimal points of compact sets in linear spaces with respect to closed convex orderings. These results are compared to the axiom of choice.
121 citations
TL;DR: An O(n(log n)3(log log n)2) algorithm for the problem of finding a point (x, y) in the plane that minimizes the maximal weighted distance to a point in a set of n given points is presented.
Abstract: We present an O(n(log n)3(log log n)2) algorithm for the problem of finding a point (x, y) in the plane that minimizes the maximal weighted distance to a point in a set of n given points. The algorithm can be extended to higher dimensional spaces. For any fixed dimension our bound is o(n1+ϵ) for any ϵ > 0.
TL;DR: The uncapacitated plant location problem is considered as a node-packing problem where necessary and sufficient conditions for trivial facets along with necessary conditions for nontrivial facets are derived.
Abstract: The uncapacitated plant location problem is considered as a node-packing problem. For this problem, several valid inequalities and facets are discussed. Necessary and sufficient conditions for trivial facets along with necessary conditions for nontrivial facets are derived. In addition, all of the facets for the case of three plants and three or more destinations are identified.
TL;DR: This work presents an algorithm which finds all autonomous subsets of graphs and posets with p points in O(p3) time and O( p2) space and reveals rather strong connections between graph decomposition, homomorphisms of graphs
Abstract: Many combinatorial (optimization) problems of graphs (e.g., finding maximal independent sets or maximum matchings) and acyclic networks (e.g., finding shortest paths or maximum flows) can be solved by means of decomposition of the graph (network) into autonomous subsets. (Given a relation R on A, a subset B of A is called autonomous, if for each α ∈ A\B the following holds: if (α, β0) ∈ R for some β0 ∈ B, then (α, β) ∈ R for all β ∈ B; if (β0, α) ∈ R for some β0 ∈ B, then (β, α) ∈ R for all β ∈ B). We present an algorithm which finds all autonomous subsets of graphs and posets with p points in O(p3) time and O(p2) space. The ideas behind this algorithm reveal rather strong connections between graph decomposition, homomorphisms of graphs and posets, and comparability graph recognition. Furthermore, the well-known series-parallel decomposition for graphs and posets is contained as a special case.
TL;DR: Several lifting procedures for facets of the uncapacitated plant location problem are discussed and necessary and sufficient conditions for nontrivial facets with 0-1 integer coefficients are derived.
Abstract: Several lifting procedures for facets of the uncapacitated plant location problem are discussed. Also, necessary and sufficient conditions for nontrivial facets with 0-1 integer coefficients are derived. In addition, all of the facets for the case of three or more plants and three destinations are identified.
TL;DR: Applications to inverse balayage problems, measures with prescribed moments or values of Laplace transforms, measuresWith prescribed marginal distributions and the queueing system GI/M/1 are presented.
Abstract: Let K be the set of probability measures on a metric space having prescribed values for the integrals of a finite number of prescribed functions fi. Extreme points of K are characterized in general. Under a restriction on the fi extreme points and more general faces of K are characterized. Applications to inverse balayage problems, measures with prescribed moments or values of Laplace transforms, measures with prescribed marginal distributions and the queueing system GI/M/1 are presented.
TL;DR: A computational method is described which finds the global minimum of a smooth concave function over a polyhedron in Rn which is partitioned into a rectangular domain, and r ≤ 2n subdomains, at least one of which contains the globalminimum.
Abstract: The problem of minimizing a concave function subject to linear inequality constraints may have many local solutions. Therefore, finding the global constrained minimum is a computationally difficult problem. A computational method is described which finds the global minimum of a smooth concave function over a polyhedron in Rn. The feasible domain is partitioned into a rectangular domain, which can be excluded from further consideration, and r ≤ 2n subdomains, at least one of which contains the global minimum. A known algorithm can be applied sequentially (or in parallel) to each of these r subdomains to compute the global minimum. A method is also presented (Appendix B) for the construction of nontrivial test problems for which the global minimum point is known. Given an arbitrary polyhedron and a selected vertex, it is shown how to determine a concave quadratic function (generally with many local minima) with its global minimum at the selected vertex.
TL;DR: This study studies convex processes between topological vector spaces with particular emphasis on their adjoints and applies this study to produce general duality results for classes of convex programs involving processes.
Abstract: We study convex processes between topological vector spaces with particular emphasis on their adjoints. This study is then applied to produce general duality results for classes of convex programs involving processes. The use of processes allows one to exploit the symmetry of linear programming and to obtain significantly broader and stronger results.
TL;DR: This paper establishes the existence of a solution to the optimality equations in undis-counted semi-Markov decision models with countable state space, under conditions generalizing the hitherto obtained results.
Abstract: This paper establishes the existence of a solution to the optimality equations in undis-counted semi-Markov decision models with countable state space, under conditions generalizing the hitherto obtained results. In particular, we merely require the existence of a finite set of states in which every pair of states can reach each other via some stationary policy, instead of the traditional and restrictive assumption that every stationary policy has a single irreducible set of states. A replacement model and an inventory model illustrate why this extension is essential. Our approach differs fundamentally from classical approaches; we convert the optimality equations into a form suitable for the application of a fixed point theorem.
TL;DR: This paper presents a parametric scheme that follows from the characterization of proper efficiency by an extended form of the generalized Tchebycheff norm for nonconvex vector maximum problems.
Abstract: Proper efficient solutions of nonconvex vector maximum problems can be generated by solving a parametric family of ordinary nonlinear programs. This parametric scheme follows from the characterization of proper efficiency by an extended form of the generalized Tchebycheff norm.
TL;DR: Under relatively weak conditions on the information structure of a game it is shown that any mixed strategy of a player can be replaced by a certain pure strategy without affecting any player’s expected payoff appreciably, no matter what strategies the other players use.
Abstract: Under relatively weak conditions on the information structure of a game it is shown that any mixed strategy of a player can be replaced by a certain pure strategy without affecting any player’s expected payoff appreciably, no matter what strategies the other players use. Under even weaker conditions the existence of a Nash equilibrium in mixed strategies is shown to guarantee the existence of a pure-strategy combination which is an approximate equilibrium.
TL;DR: The paper gives a complete treatment of the existence and nonexistence of optimal policies and new theorems for the optimal control of degenerate diffusions are given, as well as explicit formulas for the value function.
Abstract: An agent can distribute his wealth between two investments, one with a fixed rate of return r and the other with a random rate of return (modeled as a diffusion) with mean r. The agent seeks to maximize total discounted utility from consumption over an infinite horizon. Consumption may be constrained from below. Various models for bankruptcy, including welfare, are considered. The agent has a strictly concave utility function for consumption; however, it is shown that the utility function for wealth may have convex portions, thus the agent may be risk seeking. The paper gives a complete treatment of the existence and nonexistence of optimal policies. New theorems for the optimal control of degenerate diffusions are given, as well as explicit formulas for the value function.
TL;DR: It is shown that as the traffic goes to zero, the probability of delay depends only on the mean of the service-time distributions and that the delay when positive converges in distribution to the minimum of c independent equilibrium-excess service-times is zero.
Abstract: Several approximations for the expected delay in an M/G/c queue depend on both its light-and heavy-traffic behavior. Although the required heavy-traffic result has been proved, the light-traffic result has only been conjectured by Boxma, Cohen, and Huffels. We prove this conjecture when the service is of phase-type; intuitively, any M/G/c queue can be arbitrarily closely approximated by such a system. In particular, as the traffic goes to zero, we show that the probability of delay depends only on the mean of the service-time distributions and that the delay when positive converges in distribution to the minimum of c independent equilibrium-excess service-times. This result justifies an efficient computational approach to obtain numerical results for M/G/c queues and provides useful methodology for the approximation of other complicated stochastic systems.
TL;DR: An algorithm is given, which solves the open-shop problem in polynomial time, whenever the sum of execution times for one processor is large enough with respect to the maximal execution time.
Abstract: The open-shop problem is known to be NP-complete. However we give an algorithm, which solves the problem in polynomial time, whenever the sum of execution times for one processor is large enough with respect to the maximal execution time. According to the schedule given by our algorithm one of the processors works without idle time. Construction of this schedule is based on a suitable generalization of several “integer-making” techniques.
TL;DR: In this article, two simple versions of this two-stage hierarchical scheduling problem are presented, and heuristic solution methods are described and their performance is asymptotically optimal both in expectation and in probability.
Abstract: Certain multistage decision problems that arise frequently in operations management planning and control allow a natural formulation as multistage stochastic programs. In job shop scheduling, for example, the first stage could correspond to the acquisition of resources subject to probabilistic information about the jobs to be processed, and the second stage to the actual allocation of the resources to the jobs given deterministic information about their processing requirements. For two simple versions of this two-stage hierarchical scheduling problem, we describe heuristic solution methods and show that their performance is asymptotically optimal both in expectation and in probability.
TL;DR: A homotopy is presented, having a variable dimension feature, whose implementation in a PL algorithm effectively exploits the sparsity of the Jacobian and separability of the mapping.
Abstract: We consider here the problem of solving a system of n nonlinear equations in n variables, when n is large, but the underlying mapping has a sparse Jacobian, and is also structured. We present a homotopy, having a variable dimension feature, whose implementation in a PL algorithm effectively exploits the sparsity of the Jacobian and separability of the mapping. The implementation given here uses the Cholesky factorization and is thus stable. An application to a large system is also discussed.
TL;DR: This paper investigates the properties of a particular bang-bang control (sigma) 0, and finds necessary and sufficient conditions on R for (s Sigma) 0 to be optimal.
Abstract: We consider the optimal control of a one-dimensional diffusion process over a finite time interval The process may be controlled by varying the diffusion coefficient The objective is to maximize the expected value of some function of the state, R, at final time In this paper we investigate the properties of a particular bang-bang control σ0, and find necessary and sufficient conditions on R for σ0 to be optimal
TL;DR: The main result states that the supremum of the expected total rewards under all stationary policies is equal to the supremus under all (possibly randomized and non-Markovian) policies.
Abstract: The present work deals with the usual stationary decision model of dynamic programming The imposed convergence condition on the expected total rewards is so general that both the negative (unbounded) case and the positive (unbounded) case are included However, the gambling model studied by Dubins and Savage is not covered by the present model In addition to the convergence condition, a continuity and compactness condition is imposed The main result states that the supremum of the expected total rewards under all stationary policies is equal to the supremum under all (possibly randomized and non-Markovian) policies
TL;DR: This work considers the problem of finding a global extremum for a nonconvex function by constructing another smoothed function, which has a better “behavior” than the original one, and finds the global extremu for the original function.
Abstract: We consider the problem of finding a global extremum for a nonconvex function by constructing another smoothed function, which has a better “behavior” than the original one. Then, operating only with the smoothed function, we find the global extremum for the original function.
TL;DR: The ellipsoid method is applied to the unconstrained minimization of a general convex function, and equating the Steiner polynomial associated to the optimal set, and the volume of the ellipseid at a given iteration, will give an upper bound on the minimum recorded function value.
Abstract: The ellipsoid method is applied to the unconstrained minimization of a general convex function. The method converges at a geometric rate, which depends only upon the dimension of the space but not on the actual function. This rate can be improved somewhat if the function satisfies some Lipschitz-type condition, or if the minimum set has dimension greater than zero. If the ellipsoid entirely contains the optimal set, equating the Steiner polynomial associated to the optimal set, and the volume of the ellipsoid at a given iteration, will give an upper bound on the minimum recorded function value.
TL;DR: A unanimity order is constructed based upon characteristics of preference convexity in conjunction with finitely many elicited preference responses and the corresponding efficient frontier may be smaller.
Abstract: The approach which results in the definition of stochastic dominance orders for risky decision making can also be applied to the nonrisky case, that is, one may postulate that alternative x dominates alternative y if x is preferred to y by every possible decision maker with specified preference characteristics. One such “unanimity” order is the standard componentwise vector order of multiple objective optimization. In this paper, a unanimity order is constructed based upon characteristics of preference convexity in conjunction with finitely many elicited preference responses. Since the new order properly includes the vector order, the corresponding efficient frontier may be smaller.