Quasiconvex Programming

Summary

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.

1. Introduction

Quasiconvex programming is a form of geometric optimization, introduced in [Amenta et al. 1999] in the context of mesh improvement techniques and since applied to other problems in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics [Bern and Eppstein 2001; 2003; Chan 2004; Eppstein 2004]. If a problem can be formulated as a quasiconvex program of bounded dimension, it can be solved algorithmically in a linear number of constant-complexity primitive operations by generalized linear programming techniques, or numerically by generalized gradient descent techniques. In this paper we survey quasiconvex programming algorithms and applications.

1.1. Quasiconvex functions. Let Y be a totally ordered set, for instance the real numbers ℝ or integers ℤ ordered numerically. For any function f : X ⟼ Y, and any value ƛ ∈ Y, we define the lower level set A function q : X ⟼ Y, where X is a convex subset of R^d, is called quasiconvex [Dharmadhikari and Joag-Dev 1988] when its lower level sets are all convex.