混合性互補問題和無窮維二階錐非光滑函數

Abstract

在本篇論文中, 首先我們將範數從2 放寬到p (p > 1) 的廣義Fischer-Burmeister (GFB)函數應用在由Kanzow 等人發展的兩種解混合性互補問題的解法上。這兩種方法是將混合性互補問題視為一種帶條件的極小值問題或是非線性系統方程。同時我們也經由MCPLIB 問題庫中不同的p 值計算疊代次數與函數值的performance profiles 來探討改變p 值所帶來的影響與效能改善狀況。 接下來, 我們處理希爾伯空間(H) 中的互補問題。為此, 我們先介紹無窮維度二階錐 K 的向量值函數f^H(x)。詳細來說, 對任意 x 在 H 中, 引進x 的譜分解式。然後對任意實值函數 f : R -> R, 定義在 H 上相對應的向量值函數 f^H(x) 為由 x 在 H 中的譜分解值所生成。我們證明了由 f 引申的這個向量值函數 f^H(x) 具有連續、Lipschitz 連續、可微分、光滑與s-半光滑等性質。這些結果, 在設計與分析無窮維度上二階錐規劃和互補問題的解法上是非常有用的。In this thesis, we first employ the generalized Fischer-Burmeister (GFB) function where the 2-norm in the Fischer-Burmeister function is relaxed to a general p-norm (p > 1) for the two methods which is proposed by Kanzow et al. to recast the mixed complementarity problem (MCP) as a constrained minimization problem and a nonlinear system of equations, we also investigate how much the improvement is by changing the parameter p as well as which method is influenced more when we do so, by the performance profiles of iterations and functions evaluations for the two methods with different p on MCPLIB collection. Next, we deal with complementarity problems in Hilbert space. To this end, we introduce vector-valued function f^H(x) associated with the infinite-dimensional second order cone K. More specifically, for any x ∈ H, a spectral decomposition is introduced, and for any function f : R → R, we define a corresponding vector-valued function f^H(x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x ∈ H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results are useful for designing and analyzing solutions methods for solving infinite-dimensional second-order cone programs and complementarity problems.