Solving second order cone programming via a reduced augmented system approach

The standard Schur complement equation based implementation of interior-point methods for second order cone programming may encounter stability problems in the computation of search directions, and as a consequence, accurate approximate optimal solutions are sometimes not attainable. Based on the eigenvalue decomposition of the $(1,1)$ block of the augmented equation, a reduced augmented equation approach … Read more

Implementation of Interior Point Methods for Mixed Semidefinite and Second Order Cone Optimization Problems

There is a large number of implementational choices to be made for the primal-dual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different research groups. This is also the first paper to provide … Read more

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