Sylvain Perron

Exact and heuristic solutions of the global supply chain problem with transfer pricing

We examine the example of a multinational corporation that attempts to maximize its global after tax profits by determining the flow of goods, the transfer prices, and the transportation cost allocation between each of its subsidiaries. Vidal and Goetschalckx [Vidal, C.J., Goetschalckx, M., 2001. A global supply chain model with transfer pricing and transportation cost allocation. European Journal of Operational Research 129 (1), 134-158] proposed a bilinear model of this problem and solved it by an Alternate heuristic. We propose a reformulation of this model reducing the number of bilinear terms and accelerating considerably the exact solution. We also present three other solution methods: an implementation of Variable Neighborhood Search (VNS) designed for any bilinear model, an implementation of VNS specifically designed for the problem considered here and an exact method based on a branch and cut algorithm. The solution methods are tested on artificial instances. These results show that our implementation of VNS outperforms the two other heuristics. The exact method found the optimal solution of all small instances and of 26% of medium instances.

Merging the local and global approaches to probabilistic satisfiability

The probabilistic satisfiability problem is to verify the consistency of a set of probability values or intervals for logical propositions. The (tight) probabilistic entailment problem is to find best bounds on the probability of an additional proposition. The local approach to these problems applies rules on small sets of logical sentences and probabilities to tighten given probability intervals. The global approach uses linear programming to find best bounds. We show that merging these approaches is profitable to both: local solutions can be used to find global solutions more quickly through stabilized column generation, and global solutions can be used to confirm or refute the optimality of the local solutions found. As a result, best bounds are found, together with their step-by-step justification.

Probabilistic satisfiability with imprecise probabilities

Abstract Treatment of imprecise probabilities within the probabilistic satisfiability approach to uncertainty in knowledge-based systems is surveyed and discussed. Both probability intervals and qualitative probabilities are considered. Analytical and numerical methods to test coherence and bound the probability of a conclusion are reviewed. They use polyhedral combinatorics and advanced methods of linear programming.

The minimum diameter octagon with unit-length sides: Vincze's wife's octagon is suboptimal

This paper answers a query of Vincze (Acta Univ. Szeged, Sect. Sci. Math. 12 A (1950) 136-142): find the convex octagon with unit-length sides and minimum diameter. It also shows that the solution is e-unique. The proof uses geometric arguments and a global optimization algorithm to solve a nonconvex quadratic program.

Exact L 2 -norm plane separation

We consider the problem of separating two sets of points in an n-dimensional real space with a (hyper)plane that minimizes the sum of Lp-norm distances to the plane of points lying on the wrong side of it. Despite recent progress, practical techniques for the exact solution of cases other than the L1 and L∞-norm were unavailable. We propose and implement a new approach, based on non-convex quadratic programming, for the exact solution of the L2-norm case. We solve in reasonable computing times artificial problems of up to 20000 points (in 6 dimensions) and 13 dimensions (with 2000 points). We also observe that, for difficult real-life instances from the UCI Repository, computation times are substantially reduced by incorporating heuristic results in the exact solution process. Finally, we compare the classification performance of the planes obtained for the L1, L2 and L∞ formulations. It appears that, despite the fact that L2 formulation is computationally more expensive, it does not give significantly better results than the L1 and L∞ formulations.

MaxMinMin p-dispersion problem

In this work we have developed a Variable Neighborhood Search (VNS) method in order to solve the MaxMinMin p-dispersion problem, which adds a new type of plateau search mechanism to the classical VNSmetaheuristic framework. Besides, several other contributions have been made to the basic VNSheuristic in terms of the ascent and perturbation functions. To the best of our knowledge this is the first application of the VNSto the MaxMinMin problem and our approach, compared to previous methods, finds or improves the results for all of the large-sized benchmarks with low computational efforts. Finding most of the proven optimal solutions in a fraction of a second, the robustness and quality of the solutions and the low complexity of the methods demonstrate the strength of the proposed heuristic solution procedures.

Algorithms for l 1 -Embeddability and Related Problems

Assouad has shown that a real-valued distance d = (dij)1 ≤ i < j ≤ n is isometrically embeddable in ℓ1space if and only if it belongs to the cut cone on n points. Determining if this condition holds is NP-complete. We use Assouads result in a constructive column generation algorithm for ℓ1-embeddability. The subproblem is an unconstrained 0-1 quadratic program, solved by Tabu Search and Variable Neighborhood Search heuristics as well as by an exact enumerative algorithm. Computational results are reported. Several ways to approximate a distance which is not ℓ1-embeddable by another one which is are also studied.

The small hexagon and heptagon with maximum sum of distances between vertices

The hexagon and heptagon with unit diameter and maximum sum of Euclidean distances between vertices are determined by enumerating diameter configurations, and by using a branch and cut algorithm for nonconvex quadratic programming. Lower bounds on the value on this sum are presented for polygon with a larger number of vertices.

A Column Generation Heuristic for Districting the Price of a Financial Product

This paper studies a districting problem that arises in the context of financial product pricing. The challenge lies in partitioning a set of small geographical regions into a set of larger territories. In each territory, the customers will share a common price. These territories need to be contiguous, contain enough customers and be as homogeneous as possible in terms of customer value. To address this problem, we present a column generation-based heuristic where the subproblem generates contiguous territories taken into account a nonlinear objective function. Computational results indicate that the territories produced by this heuristic are about 35% more homogeneous than those previously used in practice. The developed algorithm has been transferred to a financial firm and is now used to help craft more competitive financial products.

DGR-ELM–Distributed Generalized Regularized ELM for classification

Abstract Extreme Learning Machine (ELM) has recently increased popularity and has been successfully applied to a wide range of applications. Variants using regularization are now a common practice in the state of the art in ELM field. The most commonly used regularization is the l2 norm, which improves generalization but result in a dense network. Regularization based on the elastic net has also been proposed but mainly applied to regression and binary classification problems. In this paper, we propose a generalization of regularized ELM (R-ELM) for multiclass classification problems, termed GR-ELM. We achieve such generalization using the l2,1 and Frobenius norm. Traditional R-ELM is a particular case of our method when binary classification tasks are considered. We also propose an alternative algorithm for GR-ELM when training data is distributed, namely DGR-ELM. We use Alternating Direction Method of Multipliers (ADMM) for solving the resulting optimization problems. Message Passing Interface (MPI) in a Single Program, Multiple Data (SPMD) programming style is chosen for implementing DGR-ELM. Extensive experiments are conducted to evaluate the proposed method. Our experiments show that GR-ELM and DGR-ELM have similar training and testing accuracy when compared to R-ELM, although usually faster testing time is obtained with our method due to the compactness of the resulting network.