The stochastic linear programming problem with recourse has a dual block angular structure. It can thus be handled by Benders decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block angular structure and can be handled by Dantzig-Wolfe decomposition--- the two approaches are in fact identical by duality. Here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations. The only significant difference form the aforementioned methods is that new cutting planes (or columns, by duality) will be generated not from the optimum of the linear programming relaxation, but from the analytic center of the set of localization. 1 Introduction The study of optimization problems in the presence of uncertainty still taxes the limits of methodology and software. One of the most approachable settings is that of two-staged planning under uncertainty, in which a first stage decision has to be taken bef...

We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a self-contained convergence analysis, that uses the formalism of the theory of self-concordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.

Introduction We are concerned in this note with the Goffin Haurie and Vial's [7] Analytic Center Cutting Plane Method (ACCPM for short) for large-scale convex optimization. Its state-of-the-art implementation [10] is now available upon request for academic research use. Cutting plane methods for convex optimization have a long history that goes back at least to a fundamental paper of Kelley [14]. There exist numerous strategies that can be applied to "solve" subsequent relaxed master problems in the cutting planes optimization scheme. In the Analytic Center Cutting Plane Method, subsequent relaxed master problems are not solved to optimality. Instead of it, an approximate analytic center of the current localization set is looked for. The theoretical development of ACCPM started from Goffin and Vial [9]. It was later continued in [7, 8] and led to a development of the prototype implementation of the method due to du Merle [15] that was successfully applied to solve several nont

A new local search heuristic, called J-Means, is proposed for solving the minimum sum-of-squares clustering problem. The neighborhood of the current solution is defined by all possible centroid-to-entity relocations followed by corresponding changes of assignments. Moves are made in such neighborhoods until a local optimum is reached. The new heuristic is compared with two other well-known local search heuristics, KMeans and H-Means as well as with H-Means+, an improved version of the latter in which degeneracy is removed. Moreover, another heuristic, which fits into the Variable Neighborhood Search metaheuristic framework and uses J-Means in its local search step, is proposed too. Results on standard test problems from the literature are reported. It appears that J-Means outperforms the other local search methods, quite substantially when many entities and clusters are considered. 1 Introduction Consider a set X = fx 1 ; : : : ; xN g, x j = (x 1j ; : : : ; x qj ) 2 R q of N entiti...

The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with the Dijkstra's d-heap algorithm. An implementation is described that that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on well-known nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities). This research has been supported by the Fonds National de la Recherche Scientifique Suisse, grant #12 \Gamma 34002:92, NSERC-Canada and ...

We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables within the trust regions defined by Dikin's primal and dual ellipsoids. The new primal and dual directions use the variance--covariance matrix of the normals to the new cuts in the metric given by Dikin's ellipsoid. We prove that the recovery of a new analytic center from the optimal restoration direction can be done in O(p log(p + 1)) damped Newton steps, where p is the number of new cuts added by the oracle, which may vary with the iteration. The results and the proofs are independent of the specific scaling matrix ---primal, dual or primal-dual--- that is used in the computations. The computation of the optimal direction uses Newton's method applied to a self-concordant function of p variab...

A practical warm-start procedure is described for the infeasible primal-dual interior-point method 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. Key words. Warm start, primal-dual algorithm, cutting-plane methods. Supported by the Fonds National de la Recherche Scientifique Su...

In this paper we consider a new analytic center cutting plane method in a projective space. We prove the efficiency estimates for the general scheme and show that these results can be used in the analysis of a feasibility problem, the variational inequality problem and the problem of constrained minimization. Our analysis is valid even for the problems whose solution belongs to the boundary of the domain. Keywords: Cutting plane, analytic centers. This research is partially supported by the Fonds National Suisse (grant # 12-42503.94) 1 Introduction Cutting plane methods are designed to solve convex problems with the following property. A so-called oracle provides a first order information in the form of cutting planes that separate the query point from the set of solutions. Given a sequence of query points, the oracle answers a set of cutting planes that generates a polyhedral relaxation of the solution set. As the sequence of query points increases, the relaxation gets increasin...

Abstract. An exact algorithm is proposed for minimum sum-of-squares nonhierarchical clustering, i.e., for partitioning a given set of points from a Euclidean m-space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong. This problem is expressed as a constrained hyperbolic program in 0-1 variables. The resolution method combines an interior point algorithm, i.e., a weighted analytic center column generation method, with branch-and-bound. The auxiliary problem of determining the entering column (i.e., the oracle) is an unconstrained hyperbolic program in 0-1 variables with a quadratic numerator and linear denominator. It is solved through a sequence of unconstrained quadratic programs in 0-1 variables. To accelerate resolution, variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached. Estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region. Proved minimum sum-of-squares partitions are determined for the first time for several fairly large data sets from the literature, including Fisher’s 150 iris. Key words. classification and discrimination, cluster analysis, interior-point methods, combinatorial optimization

Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.