跳到主要內容

臺灣博碩士論文加值系統

(3.236.225.157) 您好!臺灣時間:2022/08/16 00:44
字體大小: 字級放大   字級縮小   預設字形  
回查詢結果 :::

詳目顯示

: 
twitterline
研究生:蔡佩旻
研究生(外文):Pei-Min Tsai
論文名稱:關於變分不等式的輔助問題原理
論文名稱(外文):Auxiliary Problem Principle On Variational Inequalities
指導教授:朱亮儒朱亮儒引用關係
指導教授(外文):Liang-Ju Chu
學位類別:碩士
校院名稱:國立臺灣師範大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:24
中文關鍵詞:變分不等式近似點方法輔助問題原理偽單調算子強偽單調算子(lw)-上半連續(ws)-上半連續
外文關鍵詞:variational inequalityproximal point algorithmauxiliary principle problempseudomonotonestrongly pseudomonotonepseudo-Dunn property(lw)-u.s.c.(ws)-u.s.c.
相關次數:
  • 被引用被引用:0
  • 點閱點閱:161
  • 評分評分:
  • 下載下載:15
  • 收藏至我的研究室書目清單書目收藏:0
輔助問題原理允許我們藉由解決輔助問題的一個數列去尋找最佳化的問題
(例如:最小化問題,鞍點問題,變分不等式問題,...等)的解。
根據 Cohen 的輔助問題原理,我們介紹並分析一個演算法來解決一般性的變分不等式 VI(T,C)問題。
為了解決關於一般的非單調算子在自反的巴那赫空間中多值的變分不等式問題,所以在這篇文章裡,近似方法的觀念被介紹而且一個收斂的演算法也被提出。而我們文章的目標就是為了輔助問題原理去建立類似的連結。事實上,這篇論文的要旨有兩層:
(1)一般化單調算子的條件之下,以輔助問題原理為基礎,
我們處理演算法的收斂性,例如: pseudo-Dunn property,強偽單
調性,$\alpha$-強偽單調性,...等。
(2)我們提出一個修改的演算法,在一個缺乏強單調性質的輔助函數條
件之下,來解決變分不等式的解之收斂性。

The auxiliary problem principle allows us to find the solution
of an optimization problem (minimization problem, saddle-point
problem, variational inequality problem, etc.) by solving a
sequence of auxiliary problem. Following the auxiliary problem
principle of Cohen, we introduce and analyze an algorithm to
solve the usual variational inequality VI(T,C). In this
paper, the concept of proximal method is introduced and a
convergent algorithm is proposed for solving set-valued variational
inequalities involving nonmonotone operators in reflexive
Banach spaces. The aim of our work is to establish similar
links for the auxiliary problem principle. In fact, the purpose
of this paper has two folds :
(1) We first deal with the convergence of algorithm based on
the auxiliary problem principle under generalized
monotonicity, such as, pseudo-Dunn property, strong
pseudomonotonicity, $\alpha$-strong pseudomonotonicity, etc.
(2) We present a modified algorithm for solving our variational
inequalities under a weaker condition on the auxiliary
function without strong monotonicity.

1. Introduction and Preliminaries
2. A Generalized Proximal Point Algorithm
3. Convergence Results With Strong Convexity
4. A Modified Algorithm Without Strong Convexity

[1] Chadli, O., Chbani, Z. & Riahi, H. (2000). Equilibrium Problems with Generalized
Monotone Bifunctions and Applications to Variational Inequalities, J. Optim.
Theory Appl. 105(2), 299-323.
[2] Cohen, G. & Zhu, R. U. (1983). Decomposition-Coordination Methods in Large
Scale Optimization Problems : The Nondierentiable Case and the Use of Augmented
Lagrangians, Advances in Large Scale Systems Theory and Applications, Edited by
J. B. Cruz, JAI Press, Greenwich, Connecticut, Vol. 1, 203-266.
[3] Cohen, G. (1988). Auxiliary Problem Principle Extended to Variational Inequality,
J. Optim. Theory Appl. 59, 325-333.
[4] Cohen, G. (1980). Auxiliary Problem Principle and Decomposition of Optimization
Problem, J. Optim. Theory Appl. 32, 277-305.
[5] Cohen, G. (1978). Optimization by Decomposition and Coordination : A Unified
Approach, IEEE Transactions on Automatic Control, Vol. AC-23, No. 2, 222-232.
[6] Crouzeix, J. P. (1997). Pseudomonotone Variational Inequality Problems : Existence
of Solutions, Mathematical Programming 78, 305-314.
[7] Dafermos, D. (1983). An Iterative Scheme for Variational Inequalities, Mathematical
Programming 26, 40-47.
[8] El Farouq, N. & Cohen, G. (1998). Progressive Regularization of Variational
Inequalities and Decomposition Algorithms, J. Optim. Theory Appl. 97, 407-433.
[9] El Farouq, N. (2001). Pseudomonotone Variational Inequalites : Convergence of
Proximal Method, J. Optim. Theory Appl. 109(2), 311-326.
[10] El Farouq, N. (2001). Pseudomonotone Variational Inequalites : Convergence of
the Auxiliary Problem Method, J. Optim. Theory Appl. 111(2), 305-326.
[11] Ekeland, I. & Temam, R. (1976). Convex Analysis and Variational Problems,
North-Holland, Amsteram, Holland.
[12] Harker, P. T. & Pang, J. S. (1990). Finite-Dimensional Variational Inequality
and Nonlinear Complementarity Problems : A Survey of Theory, Algorithms, and
Applications, Mathematical Programming 48, 161-220.
[13] Kanzow, C. (1996). Nonlinear Complementarity as Unconstrained Optimization,
J. Optim. Theory Appl. 88, 139-155.
[14] Karamardian, S. & Schaible, S. (1990). Seven Kinds of Monotone Maps, J.
Optim. Theory Appl. 66, 37-47.
[15] Karamardian, S. (1969). The Nonlinear Complementarity Problem with Applications,
Part 2, J. Optim. Theory Appl. 4, 167-181.
[16] Karamardian, S. (1976). Complementarity Problems over Cones with Monotone
and Pseudomonotone Maps, J. Optim. Theory Appl. 18, 445-455.
[17] Karamardian, S., Schaible, S. & Crouzexi, J. P. (1993). Characterizations of
Generalized Monotone Maps, J. Optim. Theory Appl. 76, 399-413.
[18] Komlosi, S. (1995). Generalized Monotonicity and Generalized Convexity, J. Optim.
Theory Appl. 84, 361-376.
[19] Nagurnty, A. (1993). Network Economics : A Variational Inequality Approach,
Kluwer Academic Publishers, Boston, Massachusetts.
[20] Ortega, J. M. & Rheinboldt, W. C. (1970). Iteractive Solutions of Nonlinear
Equations in Several Variables, Academic Press, New York, New York.
[21] Rockafellar, R. T. (1976). Monotone Operators and Proximal Point Algorithms,
SIAM. Journal on Control and Optimization 14, 877-898.
[22] Schaible, S. (1995). Generalized Monotonicity: Concepts and Uses, Variational
Inequalities and Network Equilibrium Problems, Edited by F. Giannessi and A.
Maugeri, Plenum Publishing Corporation, New York, NY, pp.289-299.
[23] Shih, M. H. & Tan, K. K. (1988). Browder-Hartman-Stampacchia Variational
Inequality for Multi-valued Monotone Operators, J. Math. Analysis and Applications
134, 431-440.
[24] Verma, R. U. (1998). Variational Inequality Involving Strongly Pseudomonotone
Hemicontinuous Mappings in Nonreflexive Banach Spaces, Appl. Math. Lett. 11(2),
41-43.
[25] Yao, J. C. (2001). Multi-valued Variational Inequalities with K-Pseudomonotone
Operators , J. Optim. Theory Appl. 83(2), 391-403.
[26] Yao, J. C. (1994). Variational Inequalities with Generalized Monotone Operators ,
Mathematics of Operations Research. 19, 691-705.
[27] Zhu, D. L. & Marcotte, P. (1996). Cocoercivity and Its Role in the Convergence of
Iterative Schemes for Solving Variational Inequalities, SIAM Journal on Optimization
6, 714-726.
[28] Zhu, D. L. & Marcotte, P. (1995). New Classes of Generalized Monotonicity, J.
Optim. Theory Appl. 87, 457-471.

QRCODE
 
 
 
 
 
                                                                                                                                                                                                                                                                                                                                                                                                               
第一頁 上一頁 下一頁 最後一頁 top