廣義FB函數與其merit函數的幾何觀點

Abstract

在這篇論文,我們主要研究廣義FB函數與其merit函數的一些幾何性質.非線性互補問題可以化成等價的約束最小化問題. 利用曲線與曲面的觀點,我們能得到直觀的想法來分析descent演算法的收斂行為. 幾何觀點更進一步指出在merit函數的方法下如何設定參數以改良演算法.

In this paper, we study some geometric properties of generalized Fischer-Burmeister function, ϕp(a, b) = ∥(a, b)∥p − (a + b) where p ∈ (1,+∞), and the merit function ψp(a, b) induced from ϕp(a, b). It is well known that the nonlinear complemen-tarity problem (NCP) can be reformulated as an equivalent unconstrained minimization by means of merit functions involving NCP-functions. From the geometric view of curve and surface, we have more intuitive ideas about convergent behaviors of the descent algo-rithms that we use. Furthermore, geometric view indicates how to improve the algorithm to achieve our goal by setting proper value of the parameter in merit function approach.