Computation of Minimum Volume Covering Ellipsoids

We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points a_1, …, a_m \in R^n. This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it … Read more

Unifying Condition Numbers for Linear Programming

In recent years, several condition numbers were defined for a variety of linear programming problems based upon relative distances to ill-posedness. In this paper we provide a unifying view of these condition numbers. To do so, we introduce yet another linear programming problem and show that its distance to ill-posedness naturally captures the most commonly … Read more

Interior-Point Methods for Nonconvex Nonlinear Programming: Complementarity Constraints

In this paper, we present the formulation and solution of optimization problems with complementarity constraints using an interior-point method for nonconvex nonlinear programming. We identify possible difficulties that could arise, such as unbounded faces of dual variables, linear dependence of constraint gradients and initialization issues. We suggest remedies. We include encouraging numerical results on the … Read more

Distance Weighted Discrimination

High Dimension Low Sample Size statistical analysis is becoming increasingly important in a wide range of applied contexts. In such situations, it is seen that the popular Support Vector Machine suffers from “data piling” at the margin, which can diminish generalizability. This leads naturally to the development of Distance Weighted Discrimination, which is based on … Read more

A Class of Hybrid Methods for Revenue Management

We consider a Markov decision process model of a network revenue management problem. Working within this formal framework, we study policies that combine aspects of mathematical programming approaches and pure Markov decision process methods. The policies employ heuristics early in the booking horizon, and switch to a more-detailed decision rule closer to the time of … Read more

TfMin: Short Reference Manual

This is a short guide to use the Fortran and Matlab package TfMin designed for the numerical solution of continuous 3D minimum-time orbit transfer around the Earth (with free final longitude), especially for low thrust engines. The underlying method is single shooting. The Matlab interface with the solver allows the user to define the problem … Read more

The Inverse Optimal Value Problem

This paper considers the following inverse optimization problem: given a linear program, a desired optimal objective value, and a set of feasible cost coefficients, determine a cost-coefficient vector such that the corresponding optimal objective value of the linear program is closest to the given value. The above problem, referred here as the inverse optimal value … Read more

A Primal-Dual Interior-Point Method for Nonlinear Programming with Strong Global and Local Convergence Properties.

An exact-penalty-function-based scheme—inspired from an old idea due to Mayne and Polak (Math. Prog., vol.~11, 1976, pp.~67–80)—is proposed for extending to general smooth constrained optimization problems any given feasible interior-point method for inequality constrained problems. It is shown that the primal-dual interior-point framework allows for a simpler penalty parameter update rule than that discussed and … Read more

Minimizing nonconvex nonsmooth functions via cutting planes and proximity control

We describe an extension of the classical cutting plane algorithm to tackle the unconstrained minimization of a nonconvex, not necessarily differentiable function of several variables. The method is based on the construction of both a lower and an upper polyhedral approximation to the objective function and it is related to the use of the concept … Read more

Parallel Interval Continuous Global Optimization Algorithms

We theorically study, on a distributed memory architecture, the parallelization of Hansen’s algorithm for the continuous global optimization with inequality constraints, using interval arithmetic. We propose a parallel algorithm based on a dynamic redistribution of the working list among the processors. On the other hand, we exploit the reduction technique, developped by Hansen, for computing … Read more