Showing papers in "Mathematics of Operations Research in 1993"

Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations

Abstract: This paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally Lipschitzian functions. For the directional derivative-based and the generalized Jacobian-based Newton methods, both the iterates and the corresponding function values are locally, superlinearly convergent. Globally, a limiting point of the iterate sequence generated by the damped, directional derivative-based Newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one, provided that the system is strongly BD-regular and semismooth at this point. In this case, the convergence is superlinear. A general attraction theorem is presented, which can be applied to two algorithms proposed by Han, Pang and Rangaraj. A hybrid method, which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent, is also presented.

The Traveling Salesman Problem with Distances One and Two

Abstract: We present a polynomial-time approximation algorithm with worst-case ratio 7/6 for the special case of the traveling salesman problem in which all distances are either one or two. We also show that this special case of the traveling salesman problem is MAX SNP-hard, and therefore it is unlikely that it has a polynomial-time approximation scheme.

On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming

Abstract: We describe several adaptive-step primal-dual interior point algorithms for linear programming. All have polynomial time complexity while some allow very long steps in favorable circumstances. We provide heuristic reasoning for expecting that the algorithms will perform much better in practice than guaranteed by the worst-case estimates, based on an analysis using a nonrigorous probabilistic assumption.

Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming

Abstract: A Bregman function is a strictly convex, differentiable function that induces a well-behaved distance measure or D-function on Euclidean space. This paper shows that, for every Bregman function, there exists a "nonlinear" version of the proximal point algorithm, and presents an accompanying convergence theory. Applying this generalization of the proximal point algorithm to convex programming, one obtains the D-function proximal minimization algorithm of Censor and Zenios, and a wide variety of new multiplier methods. These multiplier methods are different from those studied by Kort and Bertsekas, and include nonquadratic variations on the proximal method of multipliers.

Hit-and-run algorithms for generating multivariate distributions

Abstract: We introduce a general class of Hit-and-Run algorithms for generating essentially arbitrary absolutely continuous distributions on Rd. They include the Hypersphere Directions algorithm and the Coordinate Directions algorithm that have been proposed for identifying nonredundant linear constraints and for generating uniform distributions over subsets of Rd. Given a bounded open set S in Rd, an absolutely continuous probability distribution π on S the target distribution and an arbitrary probability distribution I? on the boundary of the d-dimensional unit sphere centered at the origin the direction distribution, the I?, π-Hit-and-Run algorithm produces a sequence of iteration points as follows. Given the nth iteration point x, choose a direction I? according to the distribution I? and then choose the n + 1st iteration point according to the conditionalization of the distribution π along the line defined by x and x + I?. Under mild conditions, we show that this sequence of points is a Harris recurrent reversible Markov chain converging in total variation to the target distribution π.

Asymptotic Theory for Solutions in Statistical Estimation and Stochastic Programming

Abstract: New techniques of local sensitivity analysis for nonsmooth generalized equations are applied to the study of sequences of statistical estimates and empirical approximations to solutions of stochastic programs. Consistency is shown to follow from a certain local invertibility property, and asymptotic distributions are derived from a generalized implicit function theorem that characterizes asymptotic behavior in situations where estimates are subjected to constraints and estimation functionals are nonsmooth.

Average Optimality in Dynamic Programming with General State Space

Abstract: A Markovian decision model with general state space, compact action space, and the average cost as criterion is considered. The existence of an optimal policy is shown via an optimality inequality in terms of the minimal average cost g and a relative value function w. The existence of some w is usually shown via relative compactness in a space of real-valued functions on the state space. Here it shall be shown that one can instead do with pointwise relative compactness in the set of real numbers if one makes use of a generalized lower limit of functions. An application to an inventory model is given.

Stable Matchings, Optimal Assignments, and Linear Programming

Abstract: Vande Vate 1989 described the polytope whose extreme points are the stable core matchings in the Marriage Problem. Rothblum 1989 simplified and extended this result. This paper explores a corresponding linear program, its dual and consequences of the fact that the dual solutions have an unusually direct relation to the primal solutions. This close relationship allows us to provide simple proofs both of Vande Vate and Rothblum's results and of other important results about the core of marriage markets. These proofs help explain the structure shared by the marriage problem without sidepayments and the assignment game with sidepayments. The paper further explores "fractional" matchings, which may be interpreted as lotteries over possible matches or as time-sharing arrangements. We show that those fractional matchings in the Stable Marriage Polytope form a lattice with respect to a partial ordering that involves stochastic dominance. Thus, all expected utility functions corresponding to the same ordinal preferences will agree on the relevant comparisons. Finally, we provide linear programming proofs of slightly stronger versions of known incentive compatibility results.

Lot-Sizing with Constant Batches: Formulation and Valid Inequalities

Abstract: We consider the classical lot-sizing problem with constant production capacities LCC and a variant in which the capacity in each period is an integer multiple of some basic batch size LCB. We first show that the classical dynamic program for LCC simplifies for LCB leading to an On2 min{n, C} algorithm where n is the number of periods and C the batch size. Using this new algorithm, we show how to formulate both problems as linear programs with On3 variables and constraints. A class of so-called k, l, S, I inequalities are described for LCB which capture both the dynamic nature of the problem as well as the capacity aspects. For LCB, we prove that these inequalities are the only facet-defining inequalities of a certain form. For LCC, we show that these inequalities include all the known classes of valid inequalities. Finally, we discuss several open questions and possible extensions.

Asymptotic behavior of optimal solutions in stochastic programming

Abstract: Asymptotic behavior of optimal solutions xI‚n of a sequence of stochastic programming problems is studied. Variational and generalized equations approaches are discussed. An expansion of xI‚n in terms of a parametrized mathematical programming problem, depending on a single random vector, is given. When optimal solutions of the parametrized program are directionally differentiable, this expansion leads to a close form expression for the asymptotic distribution of xI‚n. Applicability of the involved regularity conditions to nondifferentiable cases, and in particular to stochastic programming with recourse, is discussed.

On the Convergence Rate of Dual Ascent Methods for Linearly Constrained Convex Minimization

Abstract: We analyze the rate of convergence of certain dual ascent methods for the problem of minimizing a strictly convex essentially smooth function subject to linear constraints. Included in our study are dual coordinate ascent methods and dual gradient methods. We show that, under mild assumptions on the problem, these methods attain a linear rate of convergence. Our proof is based on estimating the distance from a feasible dual solution to the optimal dual solution set by the norm of a certain residual.

Unbounded utility for Savage's “Foundations of statistics,” and other models

Abstract: A general procedure for extending finite-dimensional "additive-like" representations for binary relations to infinite-dimensional "integral-like" representations is developed by means of a condition called truncation-continuity. The restriction of boundedness of utility, met throughout the literature, can now be dispensed with, and for instance normal distributions, or any other distribution with finite first moment, can be incorporated. Classical representation results of expected utility, such as Savage 1954, von Neumann and Morgenstern 1944, Anscombe and Aumann 1963, de Finetti 1937, and many others, can now be extended. The results are generalized to Schmeidler's 1989 approach with nonadditive measures and Choquet integrals, and Quiggin's 1982 rank-dependent utility. The different approaches have been brought together in this paper to bring to the fore the unity in the extension process.

Approximations and Metric Regularity in Mathematical Programming in Banach Space

Abstract: This paper establishes verifiable conditions ensuring the important notion of metric regularity for general nondifferentiable programming problems in Banach spaces. These conditions are used to obtain Lagrange-Kuhn-Tucker multipliers for minimization problems with infinitely many inequality and equality constraints.

Continuity properties of expectation functions in stochastic integer programming

Abstract: Sufficient conditions for the (Lipschitz) continuity of the expectation of second-stage costs are given for two-stage stochastic programs, where the optimization problem in the second stage is a mixed-integer linear program. We also present counterexamples to show that, in general, the results can no longer be maintained when relaxing assumptions as well as multivariate probability distributions for which the theory works.

Rearrangement, majorization and stochastic scheduling

Abstract: Rearrangement inequalities, such as the classical Hardy-Littlewood-Polya inequality and the more general Day's inequality, and related majorization results are often useful in solving scheduling problems. Among other things, they are essential for pairwise interchange arguments. Motivated by solving stochastic scheduling problems, we develop stochastic versions of Day's Inequality, over both unrestricted and restricted (specifically, one-cycle) permutations. These lead to a general and unified approach, which we apply to solve the stochastic versions of several classical deterministic scheduling problems. In most cases, the approach leads to new or stronger results; in other cases it recovers known results with new insight. The approach is built upon recent developments in stochastic majorization and multivariate characterization of stochastic order relations.

A fully polynomial-time approximation algorithm for computing a stationary point of the general linear complementarity problem

Abstract: We apply a potential reduction algorithm to solve the general linear complementarity problem GLCP minimize xTy subject to Ax + By + Cz = q and x, y, z ≥ 0. We show that the algorithm is a fully polynomial-time approximation scheme FPTAS for computing an e-approximate stationary point of the GLCP. Note that there are some GLCPs in which every stationary point is a solution xTy = 0. These include the LCPs with row sufficient matrices. We also show that the algorithm is a polynomial-time algorithm for a special class of GLCPs.

Stability results for Ekeland's e-variational principle and cone extremal solutions

Abstract: Given X a Banach space and f: X → ℝ âˆa {+∞} a proper lower semicontinuous function which is bounded from below, the Ekeland's e-variational principle asserts the existence of a point xI„ in X, which we call e-extremal with respect to f, which satisfies fu >fxI„-e‖u-xI„‖ for all u ∈ X, u ≠xI„. By using set convergence notions Kuratowski-Painleve, Mosco, bounded Hausdorff and their epigraphical versions we study the semi continuity properties of the mapping which to f associates e-ext f the set of such e-extremal points. The key for the geometrical understanding of such properties is to consider the equivalent Phelps extremization principle which, given a closed set D in X and a partial ordering with respect to a pointed cone, associates the set of elements of D maximal with respect to this order. Direct or potential applications are given in various fields multicriteria optimization, numerical algorithmic, calculus of variations.

A bound on the proportion of pure strategy equilibria in generic games

Abstract: In a generic finite normal form game with 2α + 1 Nash equilibria, at least α of the equilibria are nondegenerate mixed strategy equilibria (that is, they involve randomization by some players).

Quadratic convergence in a primal-dual method

Abstract: We show that the Mizuno-Todd-Ye O(√nL) iteration predictor-corrector primal-dual interior-point algorithm for linear programming is quadratically convergent. Our proof does not assume that the problems be nondegenerate. We do not assume that the iterate generated by the algorithm be convergent, an assumption common to all previous asymptotic convergence analysis.

Stability of solutions for stochastic programs with complete recourse

Abstract: Quantitative continuity of optimal solution sets to convex stochastic programs with (linear) complete recourse and random right-hand sides is investigated when the underlying probability measure varies in a metric space. The central result asserts that, under a strong-convexity condition for the expected recourse in the unperturbed problem, optimal tenders behave Holder-continuous with respect to a Wasserstein metric. For linear stochastic programs this carries over to the Hausdorff distance of optimal solution sets A general sufficient condition for the crucial strong-convexity assumption is given and verified for recourse problems with separable and nonseparable objectives.

The complexity of eliminating dominated strategies

Abstract: This paper deals with the computational complexity of some yes/no problems associated with sequential elimination of strategies using three domination relations: strong domination (strict inequalities), weak domination (weak inequalities), and domination (the asymmetric part of weak domination). Classification of various problems as polynomial or NP-complete seems to suggest that strong domination is a simple notion, whereas weak domination and domination are complicated ones.

Existence of Interior Points and Interior Paths in Nonlinear Monotone Complementarity Problems

Abstract: This paper establishes basic results on the existence of interior points and interior paths in a nonlinear monotone complementarity problem in ℝn under very weak interior conditions. We show that the interior paths are bounded, continuous, and all the limit points of the paths are solutions to the complementarity problem. We prove that certain sets, including the solution set to the complementary problem, form a compact convex set. We also prove the existence of generalized interior points and interior paths. These generalized paths are also continuous and contain readily available starting points from which we can follow the paths to locate the solutions to the complementarity problem. We prove our results in the context of maximal monotone operators. The result presented here can be used to develop polynomial time interior point algorithms for general monotone complementarity problems.

A general framework of continuation methods for complementarity problems

Abstract: A general class of continuation methods is presented which, in particular, solve linear complementarity problems with compositive-plus and L*-matrices. Let a, b ∈ Rn be nonnegative vectors. We embed the complementarity problem with a continuously differentiable mapping f: Rn → Rn in an artificial system of equations *   Fx, y = µa, I¶b and x, y ≥ 0, where F: R2n → R2n is defined by Fx, y = x1y1, ', xnyn, y-fx and µ ≥ 0 and I¶ ≥ 0 are parameters. A pair x, y is a solution of the complementarity problem if and only if it solves * for µ = 0 and I¶ = 0. A general idea of continuation methods founded on the system * is as follows: 1 Choose n-dimensional vectors a ≥ 0 and b > 0 such that the system * has a trivial solution x1, y1 for some µ1, I¶1 ≥ 0. 2 Trace solutions of * from x1, y1 with µ = µ1 and I¶ = I¶1 as the parameters µ and I¶ are decreased to zero. This idea provides a theoretical basis for various methods such as Lemke's method and a method of tracing the central trajectory of linear complementarity problems.

Generalized quasi-variational-like inequality problem

Abstract: This paper gives some very general results on the generalized quasi-variational-like inequality problem. Since the problem includes all the existing extensions of the classical variational inequality problem as special cases, our existence theorems extend the previous results in the literature by relaxing both continuity and concavity of the functional. The approach adopted in this paper is based on continuous selection-type arguments and thus is quite different from the Berge Maximum Theorem or Hahn-Banach Theorem approach used in the literature.

Cut-polytopes, Boolean quadratic polytopes and nonnegative quadratic pseudo-Boolean functions

Abstract: In this paper we present a class of valid inequalities as well as a class of facets for the cut-polytope of the complete graph. It is shown that many of the known classes of valid inequalities, e.g., triangle, hypermetric and cycle inequalities, belong to this class. By specializing some of the parameters, large classes of facets can be defined. It is shown that each of these classes contains at least n/3n/3 facets, where n ≥ 10 denotes the number of vertices, and that the corresponding separation problems are polynomially solvable. These results are based on a one-to-one correspondence established between the set of valid inequalities facets for the cut-polytope of Kn+1, and the set of nonnegative extremal quadratic pseudo-Boolean functions in n variables.

Applications of Degree Theory to Linear Complementarity Problems

Abstract: In this paper, we consider two applications of degree theory to linear complementarity problems. In the first application, we study the stability of an LCP at a solution point. Specifically we prove the stability of an LCP corresponding to a P0-matrix at an isolated solution. Using a recent degree formula due to Stewart 1991, we strengthen a stability result of Gowda and Pang 1992. In the second application, we use the same degree formula of Stewart to describe the number of solutions of LCPM, q when M is a negative almost N-matrix. This analysis leads to a Lipschitzian characterization of the solution map I¦: q ↦ SOLM, q corresponding to a nondegenerate negative matrix.

Asymptotically optimal loss network control

Abstract: We consider a loss network which employs alternative routing and derive the asymptotically optimal call acceptance and routing policy, the limit being as the number of links becomes large. We show that a well known control policy, least busy alternative routing with trunk reservation, is asymptotically optimal. The approach of the paper is not specific to the loss network control problem that we consider but is applicable more generally. The results are obtained by combining weak convergence with linear programming techniques.

On line bin packing with items of random size

Abstract: Consider an iid sequence of random variables X1, ', Xn distributed according to a given distribution µ on [0, 1] Let cµ be the asymptotic optimal packing ratio, ie, cµ = limn→∞ETn/n, where Tn is the minimum number of unit size bins needed to pack X1, ', Xn We prove the existence of an on line algorithm, dependent on µ, that packs X1, ', Xn in ncµ + On1/2log n3/4 bins

Optimal stopping by means of point process observations with applications in reliability

Abstract: A problem in reliability is considered in which only partial information is available. Some technical system is assumed to work in one of N unobservable states. The changes of the states are driven by a Markov process with known characteristics. The system fails from time to time according to a point process with a failure rate (intensity) which depends on the unobservable state. After failure a minimal repair is carried out immediately which leaves the state of the system unchanged. It is investigated under which conditions there exists an optimal time to stop operating the system with respect to some reward functional. The only available information is given by the failure point process observations. An explicit solution to this optimal stopping problem with partial information is derived. The problem is solved in the martingale framework. Results for monotone stopping problems are used and a generalization of the so-called monotone case is considered. The well-known disruption or disorder or detection ...

Queues in series via interacting particle systems

Abstract: In this paper, we consider a serial network with very large infinite number of single server queues. Initially, the network is empty. The input process to the network is Poisson with rate less than one unsaturated case or one saturated case and the service times at each queue is exponential with mean one. We exploit results for interacting particle systems, in particular, the zero-range and simple-exclusion processes to investigate the approach to equilibrium of this network. We derive the hydrodynamic limit of this queueing network which dramatically describes the transient behavior of this network.