Showing papers in "Mathematical Programming in 1998"

A branch and bound method for stochastic global optimization

Abstract: A stochastic branch and bound method for solving stochastic global optimization problems is proposed. As in the deterministic case, the feasible set is partitioned into compact subsets. To guide the partitioning process the method uses stochastic upper and lower estimates of the optimal value of the objective function in each subset. Convergence of the method is proved and random accuracy estimates derived. Methods for constructing stochastic upper and lower bounds are discussed. The theoretical considerations are illustrated with an example of a facility location problem.

A simulation-based approach to two-stage stochastic programming with recourse

Abstract: In this paper we consider stochastic programming problems where the objective function is given as an expected value function. We discuss Monte Carlo simulation based approaches to a numerical solution of such problems. In particular, we discuss in detail and present numerical results for two-stage stochastic programming with recourse where the random data have a continuous (multivariate normal) distribution. We think that the novelty of the numerical approach developed in this paper is twofold. First, various variance reduction techniques are applied in order to enhance the rate of convergence. Successful application of those techniques is what makes the whole approach numerically feasible. Second, a statistical inference is developed and applied to estimation of the error, validation of optimality of a calculated solution and statistically based stopping criteria for an iterative alogrithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

An SQP method for general nonlinear programs using only equality constrained subproblems

Abstract: In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [P. Spellucci, Han's method without solving QP, in: A. Auslender, W. Oettli, J. Stoer (Eds), Optimization and Optimal Control, Lecture Notes in Control and Information Sciences, vol. 30, Springer, Berlin, 1981, pp. 123–141.] it is much simpler to implement and allows any kind of changes of the working set in every step. Our method relies on a strong regularity condition. As far as it is applicable the new approach is superior to conventional SQP-methods, as demonstrated by extensive numcrical tests. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Solving quadratic (0,1)-problems by semidefinite programs and cutting planes

Abstract: We present computational experiments for solving quadratic (0, 1) problems. Our approach combines a semidefinite relaxation with a cutting plane technique, and is applied in a Branch and Bound setting. Our experiments indicate that this type of approach is very robust, and allows to solve many moderately sized problems, having say, less than 100 binary variables, in a routine manner. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

L -shaped decomposition of two-stage stochastic programs with integer recourse

Abstract: We consider two-stage stochastic programming problems with integer recourse. The L-shaped method of stochastic linear programming is generalized to these problems by using generalized Benders decomposition. Nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques. Finite convergence of the method is established when Gomory’s fractional cutting plane algorithm or a branch-and-bound algorithm is applied.

Minimum cost capacity installation for multicommodity network flows

Abstract: Consider a directed graphG = (V,A), and a set of traffic demands to be shipped between pairs of nodes inV. Capacity has to be installed on the edges of this graph (in integer multiples of a base unit) so that traffic can be routed. In this paper we consider the problem of minimum cost installation of capacity on the arcs to ensure that the required demands can be shipped simultaneously between node pairs. We study two different approaches for solving problems of this type. The first one is based on the idea of metric inequalities (see Onaga and Kakusho, On feasibility conditions of multicommodity flows in networks, IEEE Transactions on Circuit Theory, CT-18 (4) (1971) 425---429.), and uses a formulation with only |A| variables. The second uses an aggregated multicommodity flow formulation and has |V||A| variables. We first describe two classes of strong valid inequalities and use them to obtain a complete polyhedral description of the associated polyhedron for the complete graph on three nodes. Next we explain our solution methods for both of the approaches in detail and present computational results. Our computational experience shows that the two formulations are comparable and yield effective algorithms for solving real-life problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Minimizing symmetric submodular functions.

Abstract: We describe a purely combinatorial algorithm which, given a submodular set functionf on a finite setV, finds a nontrivial subsetA ofV minimizingf[A] + f[V ∖ A]. This algorithm, an extension of the Nagamochi—Ibaraki minimum cut algorithm as simplified by Stoer and Wagner [M. Stoer, F. Wagner, A simple min cut algorithm, Proceedings of the European Symposium on Algorithms ESA '94, LNCS 855, Springer, Berlin, 1994, pp. 141–147] and by Frank [A. Frank, On the edge-connectivity algorithm of Nagamochi and Ibaraki, Laboratoire Artemis, IMAG, Universite J. Fourier, Grenbole, 1994], minimizes any symmetric submodular function using O(|V|3) calls to a function value oracle. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

On the projected subgradient method for nonsmooth convex optimization in a Hilbert space

Abstract: We consider the method for constrained convex optimization in a Hilbert space, consisting of a step in the direction opposite to anek-subgradient of the objective at a current iterate, followed by an orthogonal projection onto the feasible set. The normalized stepsizesek are exogenously given, satisfyingΣk=0∞ αk = ∞, Σk=0∞ αk2 0. We prove that the sequence generated in this way is weakly convergent to a minimizer if the problem has solutions, and is unbounded otherwise. Among the features of our convergence analysis, we mention that it covers the nonsmooth case, in the sense that we make no assumption of differentiability off, and much less of Lipschitz continuity of its gradient. Also, we prove weak convergence of the whole sequence, rather than just boundedness of the sequence and optimality of its weak accumulation points, thus improving over all previously known convergence results. We present also convergence rate results. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Minimizing average completion time in the presence of release dates

Abstract: A natural and basic problem in scheduling theory is to provide good average quality of service to a stream of jobs that arrive over time. In this paper we consider the problem of schedulingn jobs that are released over time in order to minimize the average completion time of the set of jobs. In contrast to the problem of minimizing average completion time when all jobs are available at time 0, all the problems that we consider are NP-hard, and essentially nothing was known about constructing good approximations in polynomial time. We give the first constant-factor approximation algorithms for several variants of the single and parallel machine models. Many of the algorithms are based on interesting algorithmic and structural relationships between preemptive and nonpreemptive schedules and linear programming relaxations of both. Many of the algorithms generalize to the minimization of averageweighted completion time as well. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

A Lagrangian-based heuristic for large-scale set covering problems

Abstract: We present a new Lagrangian-based heuristic for solving large-scale set-covering problems arising from crew-scheduling at the Italian Railways (Ferrovie dello Stato). Our heuristic obtained impressive results when compared to state-of-the-art codes on a test-bed provided by the company, which includes instances with sizes ranging from 50,000 variables and 500 constraints to 1,000,000 variables and 5000 constraints. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Approximating the independence number via the j -function

Abstract: We describe an approximation algorithm for the independence number of a graph. If a graph onn vertices has an independence numbern/k + m for some fixed integerk ⩾ 3 and somem > 0, the algorithm finds, in random polynomial time, an independent set of size\(\tilde \Omega (m^{{3 \mathord{\left/ {\vphantom {3 {(k + 1)}}} \right. \kern- ulldelimiterspace} {(k + 1)}}} )\), improving the best known previous algorithm of Boppana and Halldorsson that finds an independent set of size Ω(m1/(k−1)) in such a graph. The algorithm is based on semi-definite programming, some properties of the Lovaszϑ-function of a graph and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph. If theϑ-function of ann vertex graph is at leastMn1−2/k for some absolute constantM, we describe another, related, efficient algorithm that finds an independent set of sizek. Several examples show the limitations of the approach and the analysis together with some related arguments supply new results on the problem of estimating the largest possible ratio between theϑ-function and the independence number of a graph onn vertices. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Approximate iterations in Bregman-function-based proximal algorithms

Abstract: This paper establishes convergence of generalized Bregman-function-based proximal point algorithms when the iterates are computed only approximately. The problem being solved is modeled as a general maximal monotone operator, and need not reduce to minimization of a function. The accuracy conditions on the iterates resemble those required for the classical “linear” proximal point algorithm, but are slightly stronger; they should be easier to verify or enforce in practice than conditions given in earlier analyses of approximate generalized proximal methods. Subjects to these practically enforceable accuracy restrictions, convergence is obtained under the same conditions currently established for exact Bregman-function-based proximal methods.

A bundle-Newton method for nonsmooth unconstrained minimization

Abstract: An algorithm based on a combination of the polyhedral and quadratic approximation is given for finding stationary points for unconstrained minimization problems with locally Lips-chitz problem functions that are not necessarily convex or differentiable. Global convergence of the algorithm is established. Under additional assumptions, it is shown that the algorithm generates Newton iterations and that the convergence is superlinear. Some encouraging numerical experience is reported.

A variable-penalty alternating directions method for convex optimization

Abstract: We study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints. The method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both Lagrangian and proximal terms. The minimizers determine the new proximal terms and a simple update of the Lagrangian terms follows. We prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers. Another novelty is that we allow penalty matrices, and these may vary per iteration. This can be beneficial in applications, since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties. As an application, we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems.

An improved approximation ratio for the minimum latency problem

Abstract: Given a tour visitingn points in a metric space, thelatency of one of these pointsp is the distance traveled in the tour before reachingp. Theminimum latency problem (MLP) asks for a tour passing throughn given points for which the total latency of then points is minimum; in effect, we are seeking the tour with minimum average “arrival time”. This problem has been studied in the operations research literature, where it has also been termed the “delivery-man problem” and the “traveling repairman problem”. The approximability of the MLP was first considered by Sahni and Gonzalez in 1976; however, unlike the classical traveling salesman problem (TSP), it is not easy to give any constant-factor approximation algorithm for the MLP. Recently, Blum et al. (A. Blum, P. Chalasani, D. Coppersimith, W. Pulleyblank, P. Raghavan, M. Sudan, Proceedings of the 26th ACM Symposium on the Theory of Computing, 1994, pp. 163–171) gave the first such algorithm, obtaining an approximation ratio of 144. In this work, we develop an algorithm which improves this ratio to 21.55; moreover, combining our algorithm with a recent result of Garg (N. Garg, Proceedings of the 37th IEEE Symposium on Foundations of Computer Science, 1996, pp. 302–309) provides an approximation ratio of 10.78. The development of our algorithm involves a number of techniques that seem to be of interest from the perspective of the TSP and its variants more generally. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

An interior point method with Bregman functions for the variational inequality problem with paramonotone operators

Abstract: We present an algorithm for the variational inequality problem on convex sets with nonempty interior. The use of Bregman functions whose zone is the convex set allows for the generation of a sequence contained in the interior, without taking explicitly into account the constraints which define the convex set. We establish full convergence to a solution with minimal conditions upon the monotone operatorF, weaker than strong monotonicity or Lipschitz continuity, for instance, and including cases where the solution needs not be unique. We apply our algorithm to several relevant classes of convex sets, including orthants, boxes, polyhedra and balls, for which Bregman functions are presented which give rise to explicit iteration formulae, up to the determination of two scalar stepsizes, which can be found through finite search procedures. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Plant location with minimum inventory

Abstract: We present an integer programming model for plant location with inventory costs. The linear programming relaxation has been solved by Dantzig-Wolfe decomposition. In this case the subproblems reduce to the minimum cut problem. We have used subgradient optimization to accelerate the convergence of the D-W algorithm. We present our experience with problems arising in the design of a distribution network for computer spare parts. In most cases, from a fractional solution we were able to derive integer solutions within 4% of optimality.

A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming

Abstract: We present a unified analysis for a class of long-step primal-dual path-following algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central path Hp(XS) -- [PXSP -~ + (PXSP 1)TI/2 = #I, introduced by Zhang. At an iterate (X, S), we choose a scaling matrix P from the class of nonsingular matrices P such that PXSP -~ is symmetric. This class of matrices includes the three well-known choices, namely: P = S ~/2 and P - X -~/2 proposed by Monteiro, and the matrix P corresponding to the Nesterov Todd direction. We show that within the class of algorithms studied in this paper, the one based on the Nesterov-Todd direction has the lowest possible iteration-complexity bound that can provably be derived from our analysis. More specifically, its iteration-complexity bound is of the same order as that of the corresponding long-step primal-dual path-following algorithm for linear programming introduced by Kojima, Mizuno and Yoshise. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Local convergence of predictor-corrector infeasible-interior-point algorithms for SDPs and SDLCPs

Abstract: An example of an SDP (semidefinite program) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the Mizuno--Todd--Ye type predictor--corrector primal-dual interior-point method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A Mizuno--Todd--Ye type predictor--corrector infeasible-interior-point algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Merit functions for semi-definite complementarity problems

Abstract: Merit functions such as the gap function, the regularized gap function, the implicit Lagrangian, and the norm squared of the Fischer-Burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors. We study the extension of these merit functions to complementarity problems defined over the cone of block-diagonal symmetric positive semi-definite real matrices. The extension suggests new solution methods for the latter problems.

The node capacitated graph partitioning problem: a computational study

Abstract: In this paper we consider the problem ofk-partitioning the nodes of a graph with capacity restrictions on the sum of the node weights in each subset of the partition, and the objective of minimizing the sum of the costs of the edges between the subsets of the partition. Based on a study of valid inequalities, we present a variety of separation heuristics for cycle, cycle with ears, knapsack tree and path-block cycle inequalities among others. The separation heuristics, plus primal heuristics, have been implemented in a branch-and-cut routine using a formulation including variables for the edges with nonzero costs and node partition variables. Results are presented for three classes of problems: equipartitioning problems arising in finite element methods and partitioning problems associated with electronic circuit layout and compiler design. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Generalized semi-infinite optimization: a first order optimality condition and examples

Abstract: We consider a generalized semi-infinite optimization problem (GSIP) of the form (GSIP) min{f(x)‖xeM}, where M={x∈ℝn|hi(x)=0i=l,...m, G(x,y)⩾0, y∈Y(x)} and all appearing functions are continuously differentiable. Furthermore, we assume that the setY(x) is compact for allx under consideration and the set-valued mappingY(.) is upper semi-continuous. The difference with a standard semi-infinite problem lies in thex-dependence of the index setY. We prove a first order necessary optimality condition of Fritz John type without assuming a constraint qualification or any kind of reduction approach. Moreover, we discuss some geometrical properties of the feasible setM.

Capacitated facility location: separation algorithms and computational experience

Abstract: We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack cover, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities. The separation problem based on the family of submodular inequalities is NP-hard in general. For the well known subclass of flow cover inequalities, however, we show that if the client set is fixed, and if all capacities are equal, then the separation problem can be solved in polynomial time. For the flow cover inequalities based on an arbitrary client set and general capacities, and for the effective capacity and single depot inequalities we develop separation heuristics. An important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining. The way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities. The family of combinatorial inequalities was originally developed for the uncapacitated facility location problem, but is also valid for the capacitated problem. No computational experience using the combinatorial inequalities has been reported so far. Here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities. We report on computational results from solving 60 medium size problems.

Makespan minimization in open shops : a polynomial time approximation scheme

Abstract: In this paper, we demonstrate the existence of a polynomial time approximation scheme for makespan minimization in the open shop scheduling problem with an arbitrary fixed numberm of machines. For the variant of the problem where the number of machines is part of the input, it is known that the existence of an approximation scheme would implyP = NP. Hence, our result draws a precise separating line between approximable cases (i.e., withm fixed) and non-approximable cases (i.e., withm part of the input) of this shop problem. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Solving stochastic programs with integer recourse by enumeration: a framework using Gro¨bner basis reductions

Abstract: In this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right-hand side vector, using Grobner basis methods from computational algebra to solve the numerous second-stage integer programs. Using structural properties of the expected integer recourse function, we prove that under mild conditions an optimal solution is contained in a finite set. Furthermore, we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed.

Warm start of the primal-dual method applied in the cutting-plane scheme

Abstract: A practical warm-start procedure is described for the infeasible primal-dual interior-point method (IPM) employed to solve the restricted master problem within the cutting-plane method. In contrast to the theoretical developments in this field, the approach presented in this paper does not make the unrealistic assumption that the new cuts are shallow. Moreover, it treats systematically the case when a large number of cuts are added at one time. The technique proposed in this paper has been implemented in the context of HOPDM, the state of the art, yet public domain, interior-point code. Numerical results confirm a high degree of efficiency of this approach: regardless of the number of cuts added at one time (can be thousands in the largest examples) and regardless of the depth of the new cuts, reoptimizations are usually done with a few additional iterations.

Global one-dimensional optimization using smooth auxiliary functions

Abstract: In this paper new global optimization algorithms are proposed for solving problems where the objective function is univariate and has Lipschitzean first derivatives. To solve this problem, smooth auxiliary functions, which are adaptively improved during the course of the search, are constructed. Three new algorithms are introduced: the first used the exact a priori known Lipschitz constant for derivatives; the second, when this constant is unknown, estimates it during the course of the search and finally, the last method uses neither the exact global Lipschitz constant nor its estimate but instead adaptively estimates the local Lipschitz constants in different sectors of the search region during the course of optimization. Convergence conditions of the methods are investigated from a general viewpoint and some numerical results are also given. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Approximability of flow shop scheduling

Abstract: Shop scheduling problems are notorious for their intractability, both in theory and practice. In this paper, we construct a polynomial approximation scheme for the flow shop scheduling problem with an arbitrary fixed number of machines. For the three common shop models (open, flow, and job), this result is the only known approximation scheme. Since none of the three models can be approximated arbitrarily closely in the general case (unless P = NP), the result demonstrates the approximability gap between the models in which the number of machines is fixed, and those in which it is part of the input of the instance. The result can be extended to flow shops with job release dates and delivery times and to flow shops with a fixed number of stages, where the number of machines at any stage is fixed. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

An efficient approximation algorithm for the survivable network design problem

Abstract: The survivable network design problem (SNDP) is to construct a minimum-cost subgraph satisfying certain given edge-connectivity requirements. The first polynomial-time approximation algorithm was given by Williamson et al. (Combinatorica 15 (1995) 435–454). This paper gives an improved version that is more efficient. Consider a graph ofn vertices and connectivity requirements that are at mostk. Both algorithms find a solution that is within a factor 2k − 1 of optimal fork ⩾ 2 and a factor 2 of optimal fork = 1. Our algorithm improves the time from O(k 3n4) to O $$(k^2 n^2 + kn^2 \sqrt {\log \log n} )$$ ). Our algorithm shares features with those of Williamson et al. (Combinatorica 15 (1995) 435–454) but also differs from it at a high level, necessitating a different analysis of correctness and accuracy; our analysis is based on a combinatorial characterization of the “redundant” edges. Several other ideas are introduced to gain efficiency. These include a generalization of Padberg and Rao's characterization of minimum odd cuts, use of a representation of all minimum (s, t) cuts in a network, and a new priority queue system. The latter also improves the efficiency of the approximation algorithm of Goemans and Williamson (SIAM Journal on Computing 24 (1995) 296–317) for constrained forest problems such as minimum-weight matching, generalized Steiner trees and others. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Partitioning mathematical programs for parallel solution

Abstract: This paper describes heuristics for partitioning a generalM × N matrix into arrowhead form. Such heuristics are useful for decomposing large, constrained, optimization problems into forms that are amenable to parallel processing. The heuristics presented can be easily implemented using publicly available graph partitioning algorithms. The application of such techniques for solving large linear programs is described. Extensive computational results on the effectiveness of our partitioning procedures and their usefulness for parallel optimization are presented. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.