Constraint selection in a build-up interior-point cutting-plane method for solving relaxations of the stable-set problem

The stable-set problem is an NP-hard problem that arises in numerous areas such as social networking, electrical engineering, environmental forest planning, bioinformatics clustering and prediction, and computational chemistry. While some relaxations provide high-quality bounds, they result in very large and expensive conic optimization problems. We describe and test an integrated interior-point cutting-plane method that efficiently handles the large number of nonnegativity constraints in the popular doubly-nonnegative relaxation. This algorithm identifies relevant inequalities dynamically and selectively adds new constraints in a build-up fashion. We present computational results showing the significant benefits of this approach in comparison to a standard interior-point cutting-plane method.     

Enhanced intersection cutting-plane approach for linear complementarity problems

In this paper, we develop an enhanced intersection cutting-plane algorithm for solving a mixed integer 0–1 bilinear programming formulation of the linear complementarity problem (LCP). The matrixM associated with the LCP is not assumed to possess any special structure, except that the corresponding feasible region is assumed to be bounded. A procedure is described to generate cuts that are deeper versions of the Tuy intersection cuts, based on a relaxation of the usual polar set. The proposed algorithm then attempts to find an LCP solution in the process of generating either a single or a pair of such strengthened intersection cuts. The process of generating these cuts involves a vertexranking scheme that either finds an LCP solution, or else these cuts eliminate the entire feasible region leading to the conclusion that no LCP solution exists. Computational experience on various test problems is provided.This material is based upon work supported by the National Science Foundation under Grant No. DMII-9121419 to the first author and Grant No. DMII-9114489 to the third author. The authors gratefully acknowledge the constructive suggestions of a referee that helped focus the approach and its presentation.     

On the complexity of cutting-plane proofs using split cuts

We prove a monotone interpolation property for split cuts which, together with results from Pudlák (1997) [20], implies that cutting-plane proofs which use split cuts (or, equivalently, mixed-integer rounding cuts or Gomory mixed-integer cuts) have exponential length in the worst case.     

Processing second-order stochastic dominance models using cutting-plane representations

Second-order stochastic dominance (SSD) is widely recognised as an important decision criterion in portfolio selection. Unfortunately, stochastic dominance models are known to be very demanding from a computational point of view. In this paper we consider two classes of models which use SSD as a choice criterion. The first, proposed by Dentcheva and Ruszczyński (J Bank Finance 30:433–451, 2006), uses a SSD constraint, which can be expressed as integrated chance constraints (ICCs). The second, proposed by Roman et al. (Math Program, Ser B 108:541–569, 2006) uses SSD through a multi-objective formulation with CVaR objectives. Cutting plane representations and algorithms were proposed by Klein Haneveld and Van der Vlerk (Comput Manage Sci 3:245–269, 2006) for ICCs, and by Künzi-Bay and Mayer (Comput Manage Sci 3:3–27, 2006) for CVaR minimization. These concepts are taken into consideration to propose representations and solution methods for the above class of SSD based models. We describe a cutting plane based solution algorithm and outline implementation details. A computational study is presented, which demonstrates the effectiveness and the scale-up properties of the solution algorithm, as applied to the SSD model of Roman et al. (Math Program, Ser B 108:541–569, 2006).     

A cutting-plane approach for the two-dimensional orthogonal non-guillotine cutting problem

The two-dimensional orthogonal non-guillotine cutting problem (NGCP) appears in many industries (like wood and steel industries) and consists in cutting a rectangular master surface into a number of rectangular pieces, each with a given size and value. The pieces must be cut with their edges always parallel or orthogonal to the edges of the master surface (orthogonal cuts). The objective is to maximize the total value of the pieces cut.In this paper, we propose a two-level approach for solving the NGCP, where, at the first level, we select the subset of pieces to be packed into the master surface without specifying the layout, while at a second level we check only if a feasible packing layout exists. This approach has been already proposed by Fekete and Schepers [S.P. Fekete, J. Schepers, A new exact algorithm for general orthogonal d-dimensional knapsack problems, ESA 97, Springer Lecture Notes in Computer Science 1284 (1997) 144–156; S.P. Fekete, J. Schepers, On more-dimensional packing III: Exact algorithms, Tech. Rep. 97.290, Universität zu Köln, Germany, 2000; S.P. Fekete, J. Schepers, J.C. van der Veen, An exact algorithm for higher-dimensional orthogonal packing, Tech. Rep. Under revision on Operations Research, Braunschweig University of Technology, Germany, 2004] and Caprara and Monaci [A. Caprara, M. Monaci, On the two-dimensional knapsack problem, Operations Research Letters 32 (2004) 2–14]. We propose improved reduction tests for the NGCP and a cutting-plane approach to be used in the first level of the tree search to compute effective upper bounds.Computational tests on problems derived from the literature show the effectiveness of the proposed approach, that is able to reduce the number of nodes generated at the first level of the tree search and the number of times the existence of a feasible packing layout is tested.     

A cutting-plane method for quadratic semi infinite programming problems

A cutting plane algorithm for solving convex quadratic semi-infinite programming problems is presented. Nonbinding constraints can be dropped. Its arithmetic convergence rate is proved by taking into consideration the error of the approximate solution of the auxiliary problem to calculate the most violate constraint. An implementable variant of this method is described which is due to the adaptive discretization of the index set and its stability is shown. Computational experiments show the behaviour of the method.     

Computational aspects of cutting-plane algorithms for geometric programming problems

This paper presents the results of computational studies of the properties of cutting plane algorithms as applied to posynomial geometric programs. The four cutting planes studied represent the gradient method of Kelley and an extension to develop tangential cuts; the geometric inequality of Duffin and an extension to generate several cuts at each iteration. As a result of over 200 problem solutions, we will draw conclusions regarding the effectiveness of acceleration procedures, feasible and infeasible starting point, and the effect of the initial bounds on the variables. As a result of these experiments, certain cutting plane methods are seen to be attractive means of solving large scale geometric programs.This author's research was supported in part by the Center for the Study of Environmental Policy, The Pennsylvania State University.     

A cutting-plane algorithm for solving a weighted influence interdiction problem

We consider a two-stage defender-attacker game that takes place on a network, in which the attacker seeks to take control over (or “influence”) as many nodes as possible. The defender acts first in this game by protecting a subset of nodes that cannot be influenced by the attacker. With full knowledge of the defender’s action, the attacker can then influence an initial subset of unprotected nodes. The influence then spreads over a finite number of time stages, where an uninfluenced node becomes influenced at time t if a threshold number of its neighbors are influenced at time t?1. The attacker’s objective is to maximize the weighted number of nodes that are influenced over the time horizon, where the weights depend both on the node and on the time at which that is influenced. This defender-attacker game is especially difficult to optimize, because the attacker’s problem itself is NP-hard, which precludes a standard inner-dualization approach that is common in many interdiction studies. We provide three models for solving the attacker’s problem, and develop a tailored cutting-plane algorithm for solving the defender’s problem. We then demonstrate the computational efficacy of our proposed algorithms on a set of randomly generated instances.     

A cutting-plane algorithm with linear and geometric rates of convergence

This paper presents a cutting-plane algorithm for nonlinear programming which, under suitable conditions, exhibits a linear or geometric global rate of convergence. Other known rates of convergence for cutting-plane algorithms are no better than arithmetic for problems not satisfying a Haar condition. The feature responsible for this improved rate of convergence is the addition at each iteration of a new cut for each constraint, rather than adding only one new cut corresponding to the most violated constraint as is typically the case. Certain cuts can be dropped at each iteration, and there is a uniform upper bound on the number of old cuts retained. Geometric convergence is maintained if the subproblems at each iteration are approximated, rather than solved exactly, so the algorithm is implementable. The algorithm is flexible with respect to the point used to generate new cuts.The author is grateful to W. Oettli for bringing to his attention the linearly convergent cutting-plane algorithm of Ref. 15 and to the referee for a comment that stimulated an extension of the convergence rate results from an earlier version where _k depended on certain parameters of the problem.     

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

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. Supported by the Fonds National de la Recherche Scientifique Suisse, grant #12-42503.94.     

Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities

Let VIP(F,C) denote the variational inequality problem associated with the mapping F and the closed convex set C. In this paper we introduce weak conditions on the mapping F that allow the development of a convergent cutting-plane framework for solving VIP(F,C). In the process we introduce, in a natural way, new and useful notions of generalized monotonicity for which first order characterizations are presented. Received: September 25, 1997 / Accepted: March 2, 1999?Published online July 20, 2000     

A cutting-plane approach for large-scale capacitated multi-period facility location using a specialized interior-point method

We propose a cutting-plane approach (namely, Benders decomposition) for a class of capacitated multi-period facility location problems. The novelty of this approach lies on the use of a specialized interior-point method for solving the Benders subproblems. The primal block-angular structure of the resulting linear optimization problems is exploited by the interior-point method, allowing the (either exact or inexact) efficient solution of large instances. The consequences of different modeling conditions and problem specifications on the computational performance are also investigated both theoretically and empirically, providing a deeper understanding of the significant factors influencing the overall efficiency of the cutting-plane method. The methodology proposed allowed the solution of instances of up to 200 potential locations, one million customers and three periods, resulting in mixed integer linear optimization problems of up to 600 binary and 600 millions of continuous variables. Those problems were solved by the specialized approach in less than one hour and a half, outperforming other state-of-the-art methods, which exhausted the (144 GB of) available memory in the largest instances.     

Implementing the simplex method as a cutting-plane method, with a view to regularization

We show that the simplex method can be interpreted as a cutting-plane method, assuming that a special pricing rule is used. This approach is motivated by the recent success of the cutting-plane method in the solution of special stochastic programming problems. We focus on the special linear programming problem of finding the largest ball that fits into a given polyhedron. In a computational study we demonstrate that ball-fitting problems have such special characteristics which indicate their utility in regularization schemes.     

Optimal challenges in tennis

The use of technology in sport to assist umpires has been gradually introduced into several sports. This has now been extended to allow players to call upon technology to arbitrate when they disagree with the umpire's decision. Both tennis and cricket now allow the players to challenge a doubtful decision, which is reversed if the evidence shows it to be incorrect. However, the number of challenges is limited, and players must balance any possible immediate gain with the loss of a future right to challenge. With similar challenge rules expected to be introduced in other sports, this situation has been a motivation to consider challenges more widely. We use Dynamic Programming to investigate the optimal challenge strategy and obtain some general rules. In a traditional set of tennis, players should be more aggressive in challenging in the latter stages of the games and sets, and when their opponent is ahead. Optimal challenge strategy can increase a player's chance of winning an otherwise even five-set match to 59%.     

Passive walking towards running

Passive walking emerges autonomously on a slight slope without an external input of energy. It is known that the walking motion on a steep slope evolves into a chaotic motion. In this paper a biped model for walking and running is presented, and a strategy is proposed to expand the range of stable passive walking by using a chaos-control technique based on the Ott?–?Grebogi?–?Yorke method. The resultant controller is a discrete type so that the input value changes at every step, and the generated walking motion is kept non-chaotic. Fast walking on a steep slope is achieved, and pseudorunning has also been realized in simulations. By adding an input to the biped model, in which the input corresponds to the effect of the artificial gravity field, it has been verified that pseudorunning can be realized on level ground.