Willy Gochet

Multigroup Discriminant Analysis Using Linear Programming

In this paper we introduce a nonparametric linear programming formulation for the general multigroup classification problem. Previous research using linear programming formulations has either been limited to the two-group case, or required complicated constraints and many zero-one variables. We develop general properties of our multigroup formulation and illustrate its use with several small example problems and previously published real data sets. A comparative analysis on the real data sets shows that our formulation may offer an interesting robust alternative to parametric statistical formulations for the multigroup discriminant problem.

A branch-and-bound method for reversed geometric programming

A general or signomial geometric program is a nonlinear mathematical program involving general polynomials in several variables both in the objective function and the constraints. A branch-and-bound method is proposed for this extensive class of nonconvex optimization programs guaranteeing convergence to the global optimum. The subproblems to be solved are convex but the method can easily be combined with a cutting plane technique to generate subproblems which are linear. A simple example is given to illustrate the technique. The combined method involving linear subproblems has been coded and numerical experience with this code will be reported later.

Alternative formulations for a layout problem in the fashion industry

Abstract In this paper we propose alternative IP models for solving the layout problem in the fashion industry as described in Degraeve and Vandebroek [Management Science 44 (1998) 301–310]. Before cutting, several layers of cloth are put on a cutting table and several templates, indicating how to cut all material for a specific size, are fixed on top of the stack. The problem consists of finding good combinations of templates and the associated height of the stack of cloth to satisfy demand while minimizing total excess production. Computational results indicate that our alternative models generally outperform the originally proposed model. We also discuss how the new models can be adapted for the multicolor version of the problem and how we can include a total cost approach.

Mathematical programming based heuristics for improving LP-generated classifiers for the multiclass supervised classification problem

Mathematical programming is used as a nonparametric approach to supervised classification. However, mathematical programming formulations that minimize the number of misclassifications on the design dataset suffer from computational difficulties. We present mathematical programming based heuristics for finding classifiers with a small number of misclassifications on the design dataset. The basic idea is to improve an LPgenerated classifier with respect to the number of misclassifications on the design dataset. The heuristics are evaluated computationally on both simulated and real world datasets.

A dynamic-programming based heuristic for industrial buying of cardboard

Abstract A heuristic method is proposed for a deterministic two-dimensional assortment problem with a large number of different sizes. The method is applied using real-world data for cardboard buying. It is found to give satisfactory results in comparison with both the current company policy and results obtained by an existing heuristic.

Aggregating classifiers with mathematical programming

Bagging and boosting are popular and often successful ways to improve the performance of a classifier by means of aggregation. Classifiers can also be aggregated by means of an efficient and flexible mathematical programming model. This data-based approach guarantees that the aggregated classifier will be at least as good as the best predictor on the design data set for a user-defined criterion function. The mathematical programming approach is evaluated on real-world data sets from different contexts such as medical diagnosis, image segmentation and handwritten digit recognition. The real-world examples show that the approach can outperform both bagging and boosting.

A modified reduced gradient method for dual posynomial programming

In this paper, we present a method for solving the dual of a posynomial geometric program based on modifications of the reduced gradient method. The modifications are necessary because of the numerical difficulties associated with the nondifferentiability of the dual objective function. Some preliminary numerical results are included that compare the proposed method with the modified concave simplex method of Beck and Ecker of Ref. 1.

A modified reduced gradient method for a class of nondifferentiable problems

The class of nondifferentiable problems treated in this paper constitutes the dual of a class of convex differentiable problems. The primal problem involves faithfully convex functions of linear mappings of the independent variables in the objective function and in the constraints. The points of the dual problem where the objective function is nondifferentiable are known: the method presented here takes advantage of this fact to propose modifications necessary in the reduced gradient method to guarantee convergence.

Constraint sets of geometric programs characterized by auxiliary problems

This paper refines a previous classification scheme by using auxiliary problems of the dual geometric program. This new scheme provides a one-to-one correspondence between the possible states of primal and dual geometric programs. Moreover, a characterization of superconsistency, strong inconsistency and subconsistency without consistency is obtained in terms of properties of the dual problem.

COMPUTATIONAL ASPECTS OF GEOMETRIC PROGRAMMING 3. SOME PRIMAL AND DUAL ALGORITHMS FOR POSYNOMIAL AND SIGNOMIAL GEOMETRIC PROGRAMS

In this paper we consider algorithmic and computational aspects of selected methods for posynomial and signomial programming problems. In particular, we consider the modified concave simplex method for dual posynomial programs and discuss some recent developments suggesting a new dual algorithm. We also consider linear programming based algorithms for primal posynomial programs. Finally, we discuss a recently developed branch and bound method for signomial programming.