Cutting planes and a biased Newton direction for minimizing quasiconvex functions

Abstract

A biased Newton direction is introduced for minimizing quasiconver functions with bounded level sets. It is a generalization of the usual Newton’s direction for strictly convex quadratic functions. This new direction can be derived from the intersection of approzimating hyperplanes to the epigraph at points on the boundary of the same level set. Based on that direction, an unconstrained minimization algorithm is presented. It is proved to have global and local-quadratic convergence under standard hypotheses. These theoretical results may lead to different methods based on computing search directions using only first order information at points on the level sets. Most of all if the computational cost can be reduced by relaxing some of the conditions according for instance to the results presented in the Appendix. Some tests are presented to show the qualitative behavior of the new direction and with the purpose to stimulate further research on these kind of algorithms.