系統識別號 U0026-1606201516104400
論文名稱(中文) An SDP approach for non-convex quadratic fractional programming problems
論文名稱(英文) An SDP approach for non-convex quadratic fractional programming problems
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 103
學期 2
出版年 104
研究生(中文) 阮文蓬
研究生(英文) Van-Bong Nguyen
學號 L18007016
學位類別 博士
語文別 英文
論文頁數 98頁
口試委員 指導教授-許瑞麟
中文關鍵字 none 
英文關鍵字 Quadratic fractional programming  Dinkelbach algorithm  S-lemma  semidefinite relaxation  generalized trust region subproblem  sum-of-ratios  Rayleigh quotient  generalized Rayleigh quotient 
中文摘要 在本論文中,我們關心兩類的二次分式型規劃問題:其一是帶有雙邊二次限制式之二次分式型問題(P),其二是分式和規劃問題(Q)。雖然我們的問題(P)與(Q)是以二次函數的分式型態展現,透過參數化技巧,分式的結構可以被轉換成一系列特殊型的二次限制式二次規劃(QCQP)問題。QCQP問題是文獻中備受關注,存在已久且困難求解的非凸最佳化問題。因此,在處理(P)與(Q),我們不只面對傳統QCQP問題,另外還有從分式結構衍生出的參數要處理。由於我們是直接著手處理二次分式型規劃問題,顯然許多結果對QCQP來說是也會是新且有用的。儘管(P)與(Q)本身是非凸的, 我們主要的想法是探討(P)與(Q)的隱藏凸性。我們主要的工具包含一個強大的機制,即已知的S-引理,以及半正定鬆弛、秩一分解技術。為了引用上述的工具來求解(P)與(Q),我們將問題轉化或拆開成一些子問題,並研究S-引理的新版本、修改秩一分解程序。結果是,我們證明了問題(P)的計算複雜度屬於P類問題,而NP-困難的問題(Q)可以被簡化成一個一維度搜尋問題。我們的解法是創新的,而且容易在電腦上執行。我們的計算結果顯示,在求解全局最佳解方面我們的演算法比其他現有的求解方法更有效率。
英文摘要 In this dissertation, we are concerned with two types of quadratic fractional programming problems: one is a single ratio quadratic fractional problem with a twosided quadratic constraint (P) and the other is a sum-of-ratios problem (Q). Although we pose the problems (P) and (Q) in the form of ratios of quadratic functions, after some parameterization scheme, the fractional structure can be exchanged with a family of special types of quadratically constrained quadratic programming (QCQP) problems. The QCQP problem is a long-standing difficult non-convex optimization problem which has attracted much attention in literature. Therefore, in dealing with (P) and (Q), we not only face the traditional QCQP problem, but also an additional parameter coming from the ratio structure. Since we feel that we can address the two issues all together, we directly attack the quadratic fractional programming but apparently many results are new and useful to QCQP alone. Our main idea is to probe the hidden convexity of (P) and (Q), in spite that both (P) and (Q) are themselves non-convex. Our main tool is a powerful mechanism, known as the S-lemma, together with the semi-definite relaxation and the rank-one decomposition technique.
In order to apply the aforementioned tools for solving (P) and (Q), we transform or divide the problems into a few subcases; study new versions of S-lemma; and modify the rank-one decomposition procedure. As a result, we prove that (P) is indeed in the class of P, and the NP-hard problem (Q) can be reduced to a much simpler one-dimensional search problem. Our solution method is novel, and can be easily implemented on the computer. Our computational results show that the algorithms are very efficient in finding the global optimal solution compared with other existing methods.
論文目次 Cover......... i
Oral presentation document .....ii
Chinese version ......ii
English version ...... iii
Abstract (Chinese) ......iv
Abstract (English) .......v
Acknowledgments....... vii
Table of Contents .......viii
Chapter 1.Introduction ......1
1.1 A single-ratio quadratic fractional program ...1
1.1.1 Problem format and background ....1
1.1.2 Dinkelbach Algorithm ......3
1.1.3 A modern technique: Semi-definite relaxation ...4
1.1.4 Our approach .......6
1.2 A sum-of-ratios problem .....8
1.2.1 Problem format and existing methods ...8
1.2.2 Our approach .......10
1.3 Contributions of the dissertation .....12
1.4 Notation ......13
Chapter 2.Quadratically Constrained Quadratic Programming and Its
SDP Relaxation ........16
2.1 Semidefinite Programming ( SDP) ....16
2.2 Quadratically constrained quadratic programming and its SDP relaxation 19
2.2.1 The generalized Trust region subproblems (GTRS) .23
2.2.2 The generalized Trust region subproblems with equality constraint .28
2.2.3 The interval bounded generalized trust region subproblem .34
2.2.4 Homogeneous quadratic problems with two quadratic constraints
(QP2QC) .........36
Chapter 3.Single-ratio quadratic fractional programming problems .39
3.1 The issue of boundedness and attainment ....40
3.2 A two-sided quadratically constrained quadratic fractional problem .. 41
3.3 A quadratic fractional problem with one inequality quadratic constraint
(QF1QC) ........49
3.4 On well-definedness of (QF1QC) .....53
3.5 Chapter Conclusion .......55
Chapter 4.A sum-of-quadratic-ratios problem ....57
4.1 An equivalent reformulation of the problem ....57
4.2 Solving the reformulation problem via SDP ....60
4.2.1 Evaluating the optimal value . .....60
4.2.2 Solving a global solution of the reformulation problem ..63
4.3 Finding __ numerically ......66
4.4 Computational Experiments ......70
4.5 Chapter Conclusion ........74
Chapter 5.Concluding remarks and future researches ...76
5.1 Summary of the dissertation ......76
5.2 Future researches .......77
References .....80
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