Graph partitioning using linear and semidefinite programming

 Graph partition is used in the telecommunication industry to subdivide a transmission network into small clusters. We consider both linear and semidefinite relaxations for the equipartition problem and present numerical results on real data from France Telecom networks with up 900 nodes, and also on randomly generated problems. Received: August 8, 2001 / Accepted: November 9, 2001 Published online: December 9, 2002 RID="★★" ID="★★" This research was carried out while this author was working at France Telecom R & D, 38–40 rue du Général Leclerc, F-92794 Issy-Les-Moulineaux Cedex 9, France. RID="★" ID="★" This author greatfully acknowledges financial support from the Austrian Science Foundation FWF Project P12660-MAT. Key words. graph partitioning – semidefinite programming     

Conditioning of convex piecewise linear stochastic programs

 In this paper we consider stochastic programming problems where the objective function is given as an expected value of a convex piecewise linear random function. With an optimal solution of such a problem we associate a condition number which characterizes well or ill conditioning of the problem. Using theory of Large Deviations we show that the sample size needed to calculate the optimal solution of such problem with a given probability is approximately proportional to the condition number. Received: May 2000 / Accepted: May 2002-07-16 Published online: September 5, 2002 RID="★" The research of this author was supported, in part, by grant DMS-0073770 from the National Science Foundation Key Words. stochastic programming – Monte Carlo simulation – large deviations theory – ill-conditioned problems     

Maximum stable set formulations and heuristics based on continuous optimization

 The stability number α(G) for a given graph G is the size of a maximum stable set in G. The Lovász theta number provides an upper bound on α(G) and can be computed in polynomial time as the optimal value of the Lovász semidefinite program. In this paper, we show that restricting the matrix variable in the Lovász semidefinite program to be rank-one and rank-two, respectively, yields a pair of continuous, nonlinear optimization problems each having the global optimal value α(G). We propose heuristics for obtaining large stable sets in G based on these new formulations and present computational results indicating the effectiveness of the heuristics. Received: December 13, 2000 / Accepted: September 3, 2002 Published online: December 19, 2002 RID="★" ID="★" Computational results reported in this paper were obtained on an SGI Origin2000 computer at Rice University acquired in part with support from NSF Grant DMS-9872009. Key Words. maximum stable set – maximum clique – minimum vertex cover – semidefinite program – semidefinite relaxation – continuous optimization heuristics – nonlinear programming Mathematics Subject Classification (2000): 90C06, 90C27, 90C30     

A feasible semismooth asymptotically Newton method for mixed complementarity problems

 Semismooth Newton methods constitute a major research area for solving mixed complementarity problems (MCPs). Early research on semismooth Newton methods is mainly on infeasible methods. However, some MCPs are not well defined outside the feasible region or the equivalent unconstrained reformulations of other MCPs contain local minimizers outside the feasible region. As both these problems could make the corresponding infeasible methods fail, more recent attention is on feasible methods. In this paper we propose a new feasible semismooth method for MCPs, in which the search direction asymptotically converges to the Newton direction. The new method overcomes the possible non-convergence of the projected semismooth Newton method, which is widely used in various numerical implementations, by minimizing a one-dimensional quadratic convex problem prior to doing (curved) line searches. As with other semismooth Newton methods, the proposed method only solves one linear system of equations at each iteration. The sparsity of the Jacobian of the reformulated system can be exploited, often reducing the size of the system that must be solved. The reason for this is that the projection onto the feasible set increases the likelihood of components of iterates being active. The global and superlinear/quadratic convergence of the proposed method is proved under mild conditions. Numerical results are reported on all problems from the MCPLIB collection [8]. Received: December 1999 / Accepted: March 2002 Published online: September 5, 2002 RID="★" ID="★" This work was supported in part by the Australian Research Council. Key Words. mixed complementarity problems – semismooth equations – projected Newton method – convergence AMS subject classifications. 90C33, 90C30, 65H10     

A computational study of a gradient-based log-barrier algorithm for a class of large-scale SDPs

 The authors of this paper recently introduced a transformation [4] that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based on the transformation, we proposed a globally convergent, first-order (i.e., gradient-based) log-barrier algorithm for solving a class of linear SDPs. In this paper, we discuss an efficient implementation of the proposed algorithm and report computational results on semidefinite relaxations of three types of combinatorial optimization problems. Our results demonstrate that the proposed algorithm is indeed capable of solving large-scale SDPs and is particularly effective for problems with a large number of constraints. Received: June 22, 2001 / Accepted: January 20, 2002 Published online: December 9, 2002 RID="†" ID="†"Computational results reported in this paper were obtained on an SGI Origin2000 computer at Rice University acquired in part with support from NSF Grant DMS-9872009. RID="⋆" ID="⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203426 RID="⋆⋆" ID="⋆⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203113 RID="⋆⋆⋆" ID="⋆⋆⋆"This author was supported in part by DOE Grant DE-FG03-97ER25331, DOE/LANL Contract 03891-99-23 and NSF Grant DMS-9973339. Key Words. semidefinite program – semidefinite relaxation – nonlinear programming – interior-point methods – limited memory quasi-Newton methods. Mathematics Subject Classification (1991): 90C06, 90C27, 90C30.     

Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function

 We propose and analyze a class of penalty-function-free nonmonotone trust-region methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint violation and the value of the Lagrangian function. Similar to the Byrd–Omojokun class of algorithms, each step is composed of a quasi-normal and a tangential step. Both steps are required to satisfy a decrease condition for their respective trust-region subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior comparable to filter-based methods. We establish the global convergence of the method. Furthermore, transition to quadratic local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach. Received: December 14, 2000 / Accepted: August 30, 2001 Published online: September 27, 2002 Key words. nonmonotone trust-region methods – sequential quadratic programming – penalty function – global convergence – equality constraints – local convergence – large-scale optimization Mathematics Subject Classification (2000): 65K05, 90C30     

A multi-item production planning model with setup times: algorithms, reformulations, and polyhedral characterizations for a special case

 We study a special case of a structured mixed integer programming model that arises in production planning. For the most general case of the model, called PI, we have earlier identified families of facet–defining valid inequalities: (l, S) inequalities (introduced for the uncapacitated lot–sizing problem by Barany, Van Roy, and Wolsey), cover inequalities, and reverse cover inequalities. PI is 𝒩𝒫–hard; in this paper we focus on a special case, called PIC. We describe a polynomial algorithm for PIC, and we use this algorithm to derive an extended formulation of polynomial size for PIC. Projecting from this extended formulation onto the original space of variables, we show that (l, S) inequalities, cover inequalities, and reverse cover inequalities suffice to solve the special case PIC by linear programming. We also describe fast combinatorial separation algorithms for cover and reverse cover inequalities for PIC. Finally, we discuss the relationship between our results for PIC and a model studied previously by Goemans. Received: December 13, 2000 / Accepted: December 13, 2001 Published online: October 9, 2002 RID="★" ID="★" Some of the results in this paper have appeared in condensed form in Miller et al. (2001). Key words. mixed integer programming – polyhedral combinatorics – production planning – capacitated lot–sizing – fixed charge network flow – setup times This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors. This research was also supported by NSF Grant No. DMI-9700285 and by Philips Electronics North America.     

Dynamic knapsack sets and capacitated lot-sizing

 A dynamic knapsack set is a natural generalization of the 0-1 knapsack set with a continuous variable studied recently. For dynamic knapsack sets a large family of facet-defining inequalities, called dynamic knapsack inequalities, are derived by fixing variables to one and then lifting. Surprisingly such inequalities have the simultaneous lifting property, and for small instances provide a significant proportion of all the facet-defining inequalities. We then consider single-item capacitated lot-sizing problems, and propose the joint study of three related sets. The first models the discrete lot-sizing problem, the second the continuous lot-sizing problem with Wagner-Whitin costs, and the third the continuous lot-sizing problem with arbitrary costs. The first set that arises is precisely a dynamic knapsack set, the second an intersection of dynamic knapsack sets, and the unrestricted problem can be viewed as both a relaxation and a restriction of the second. It follows that the dynamic knapsack inequalities and their generalizations provide strong valid inequalities for all three sets. Some limited computation results are reported as an initial test of the effectiveness of these inequalities on capacitated lot-sizing problems. Received: March 28, 2001 / Accepted: November 9, 2001 Published online: September 27, 2002 RID="★" ID="★" Research carried out with financial support of the project TMR-DONET nr. ERB FMRX–CT98–0202 of the European Union. Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium. Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium. Key words. knapsack sets – valid inequalities – simultaneous lifting – lot-sizing – Wagner-Whitin costs     

Karush-Kuhn-Tucker systems: regularity conditions,error bounds and a class of Newton-type methods

 We consider optimality systems of Karush-Kuhn-Tucker (KKT) type, which arise, for example, as primal-dual conditions characterizing solutions of optimization problems or variational inequalities. In particular, we discuss error bounds and Newton-type methods for such systems. An exhaustive comparison of various regularity conditions which arise in this context is given. We obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for KKT systems, such as quasi-regularity or semistability (equivalently, the R ₀-property). Error bounds are useful, among other things, for identifying active constraints and developing efficient local algorithms. We propose a family of local Newton-type algorithms. This family contains some known active-set Newton methods, as well as some new methods. Regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth Newton methods and sequential quadratic programming methods. Received: December 10, 2001 / Accepted: July 28, 2002 Published online: February 14, 2003 Key words. KKT system – regularity – error bound – active constraints – Newton method Mathematics Subject Classification (1991): 90C30, 65K05     

Multiobjective optimization problem with variational inequality constraints

 We study a general multiobjective optimization problem with variational inequality, equality, inequality and abstract constraints. Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived. They lead to Kuhn-Tucker type necessary optimality conditions under additional constraint qualifications including the calmness condition, the error bound constraint qualification, the no nonzero abnormal multiplier constraint qualification, the generalized Mangasarian-Fromovitz constraint qualification, the strong regularity constraint qualification and the linear constraint qualification. We then apply these results to the multiobjective optimization problem with complementarity constraints and the multiobjective bilevel programming problem. Received: November 2000 / Accepted: October 2001 Published online: December 19, 2002 Key Words. Multiobjective optimization – Variational inequality – Complementarity constraint – Constraint qualification – Bilevel programming problem – Preference – Utility function – Subdifferential calculus – Variational principle Research of this paper was supported by NSERC and a University of Victoria Internal Research Grant Research was supported by the National Science Foundation under grants DMS-9704203 and DMS-0102496 Mathematics Subject Classification (2000): Sub49K24, 90C29     

Solving Problems with Semidefinite and Related Constraints Using Interior-Point Methods for Nonlinear Programming

 In this paper, we describe how to reformulate a problem that has second-order cone and/or semidefiniteness constraints in order to solve it using a general-purpose interior-point algorithm for nonlinear programming. The resulting problems are smooth and convex, and numerical results from the DIMACS Implementation Challenge problems and SDPLib are provided. Received: March 10, 2001 / Accepted: January 18, 2002 Published online: September 27, 2002 Key Words. semidefinite programming – second-order cone programming – interior-point methods – nonlinear programming Mathematics Subject Classification (2000): 20E28, 20G40, 20C20     

Optimal labeling of point features in rectangular labeling models

 We investigate the NP–hard label number maximization problem (LNM): Given a set of rectangular labels Λ, each of which belongs to a point feature in the plane, the task is to find a labeling for a largest subset Λ _P of Λ. A labeling is a placement such that none of the labels overlap and each λΛ _P is placed according to a labeling model: In the discrete models, the label must be placed so that the labeled point coincides with an element of a predefined subset of corners of the rectangular label, in the continuous or slider models, the point must lie on a subset of boundaries of the label. Obviously, the slider models allow a continuous movement of a label around its point feature, leading to a significantly higher number of labels that can be placed. We present exact algorithms for this problem that are based on a pair of so-called constraint graphs that code horizontal and vertical positioning relations. The key idea is to link the two graphs by a set of additional constraints, thus characterizing all feasible solutions of LNM. This enables us to formulate a zero-one integer linear program whose solution leads to an optimal labeling. We can express LNM in both the discrete and the slider labeling models. To our knowledge, we present the first algorithm that computes provably optimal solutions in the slider models. We find it remarkable that our approach is independent of the labeling model and results in a discrete algorithm even if the problem is of continuous nature as in the slider models. Extensive experimental results on both real-world instances and point sets created by a widely used benchmark generator show that the new approach - although being an exponential time algorithm - is applicable in practice. Received: November 20, 2000 / Accepted: April 9, 2002 Published online: September 5, 2002 RID="★" ID="★" This work is partially supported by the German Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (No. 03-MU7MP1-4). Key words. map labeling – point feature map labeling – constraint graphs – combinatorial optimization – integer programming     

Index information algorithm with local tuning for solving multidimensional global optimization problems with multiextremal constraints

 Multidimensional optimization problems where the objective function and the constraints are multiextremal non-differentiable Lipschitz functions (with unknown Lipschitz constants) and the feasible region is a finite collection of robust nonconvex subregions are considered. Both the objective function and the constraints may be partially defined. To solve such problems an algorithm is proposed, that uses Peano space-filling curves and the index scheme to reduce the original problem to a H?lder one-dimensional one. Local tuning on the behaviour of the objective function and constraints is used during the work of the global optimization procedure in order to accelerate the search. The method neither uses penalty coefficients nor additional variables. Convergence conditions are established. Numerical experiments confirm the good performance of the technique. Received: April 2002 / Accepted: December 2002 Published online: March 21, 2003 RID="⋆" ID="⋆" This research was supported by the following grants: FIRB RBNE01WBBB, FIRB RBAU01JYPN, and RFBR 01–01–00587. Key Words. global optimization – multiextremal constraints – local tuning – index approach     

On linear independence of the values of generalized Heine series

 A linear independence measure is obtained for values of solutions of a certain system of functional equations. This result is then applied to a rather general class of q–hypergeometric series, for example to the values of q–analogues of exponential and Bessel functions at several algebraic points. Received: 18 October 2000 / Revised version: 2 August 2001 / Published online: 16 October 2002 RID="★" ID="★" The author is grateful to Alexander von Humboldt Foundation for support and to the Department of Mathematics of the University of Cologne for the kind hospitality. He also thanks Peter Bundschuh for many useful discussions.     

Convex optimization problems involving finite autocorrelation sequences

 We discuss convex optimization problems in which some of the variables are constrained to be finite autocorrelation sequences. Problems of this form arise in signal processing and communications, and we describe applications in filter design and system identification. Autocorrelation constraints in optimization problems are often approximated by sampling the corresponding power spectral density, which results in a set of linear inequalities. They can also be cast as linear matrix inequalities via the Kalman-Yakubovich-Popov lemma. The linear matrix inequality formulation is exact, and results in convex optimization problems that can be solved using interior-point methods for semidefinite programming. However, it has an important drawback: to represent an autocorrelation sequence of length $n$, it requires the introduction of a large number ($n(n+1)/2$) of auxiliary variables. This results in a high computational cost when general-purpose semidefinite programming solvers are used. We present a more efficient implementation based on duality and on interior-point methods for convex problems with generalized linear inequalities. Received: August 20, 2001 / Accepted: July 16, 2002 Published online: September 27, 2002 RID="★" ID="★" This material is based upon work supported by the National Science Foundation under Grant No. ECS-9733450.     

Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra

 We consider stochastic programming problems with probabilistic constraints involving random variables with discrete distributions. They can be reformulated as large scale mixed integer programming problems with knapsack constraints. Using specific properties of stochastic programming problems and bounds on the probability of the union of events we develop new valid inequalities for these mixed integer programming problems. We also develop methods for lifting these inequalities. These procedures are used in a general iterative algorithm for solving probabilistically constrained problems. The results are illustrated with a numerical example. Received: October 8, 2000 / Accepted: August 13, 2002 Published online: September 27, 2002 Key words. stochastic programming – integer programming – valid inequalities     

Easy and difficult objective functions for max cut

 This note investigates the boundary between polynomially-solvable Max Cut and NP Hard Max Cut instances when they are classified only on the basis of the sign pattern of the objective function coefficients, i.e., of the orthant containing the objective function vector. It turns out that the matching number of the subgraph induced by the positive edges is the key parameter that allows us to differentiate between polynomially-solvable and hard instances of the problem. We give some applications of the polynomially solvable cases. Received: November 29, 2000 / Accepted: August 17, 2001 Published online: December 9, 2002 RID="★" ID="★" The research of this author was partially supported by an NSERC Research Grant.     

Subharmonic functions on Carnot groups

 Many results of classical Potential Theory are extended to sub-Laplacians ▵_𝔾 on Carnot groups 𝔾. Some characterizations of ▵_𝔾-subharmonicity, representation formulas of Poisson-Jensen's kind and Nevanlinna-type theorems are proved. We also characterize the Riesz-measure related to bounded-above ▵_𝔾-subharmonic functions in ℝ ^N . Received: 21 June 2000 / Revised version: 12 March 2002 / Published online: 2 December 2002 RID="★" ID="★" Investigation supported by University of Bologna. Funds for selected research topics. Mathematics Subject Classification (2000): 31B05, 35J70, 35H20     

Non-Interior continuation methods for solving semidefinite complementarity problems

 There recently has been much interest in non-interior continuation/smoothing methods for solving linear/nonlinear complementarity problems. We describe extensions of such methods to complementarity problems defined over the cone of block-diagonal symmetric positive semidefinite real matrices. These extensions involve the Chen-Mangasarian class of smoothing functions and the smoothed Fischer-Burmeister function. Issues such as existence of Newton directions, boundedness of iterates, global convergence, and local superlinear convergence will be studied. Preliminary numerical experience on semidefinite linear programs is also reported. Received: October 1999 / Accepted: April 2002 Published online: December 19, 2002 RID="⋆" ID="⋆" This research is supported by National Science Foundation Grant CCR-9731273. Key words. semidefinite complementarity problem – smoothing function – non-interior continuation – global convergence – local superlinear convergence