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研究生： 阮黎簪
Nguyen Le Tram
論文名稱： Interval valued optimization problems on Hadamard manifolds
Interval valued optimization problems on Hadamard manifolds
指導教授： 陳界山
Chen, Jein-Shan
口試委員： 陳界山
Chen, Jein-Shan
張毓麟
Chang, Yu-Lin
杜威仕
Du, Wei-Shih
許瑞麟
Sheu, Ruey-Lin
林仁彥
Lin, Jen-Yen
口試日期： 2024/01/05
學位類別： 博士
Doctor
系所名稱： 數學系
Department of Mathematics
論文出版年： 2024
畢業學年度： 112
語文別： 英文
論文頁數： 74
英文關鍵詞： Hadamard manifolds, interval variational inequalities, gH-diffirentiable, optimality condition, penalized, interval valued function, set valued function on manifolds
研究方法: 實驗設計法 、 紮根理論法
DOI URL： http://doi.org/10.6345/NTNU202400044
論文種類： 學術論文
相關次數： 點閱：89  下載：1 
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    • In this thesis, we study the interval valued optimization problems (IOPs)
    on Hadamard manifolds, including unconstrained and constrained problems. To
    achieve the theoretical results, we build up some new concepts about gH-directional derivative, gH-Gâteaux and gH-Fréchet differentiability of interval valued functions with their properties on Hadamard manifolds. More specifically, we characterize the optimality conditions for the IOPs on the Hadamard manifolds. For unconstrained problems, the existence of efficient points and the steepest descent algorithm are investigated. To the contrast, the optimality conditions, exact penalty, and duality approach are explored in the ones involving inequality constraints. The obtained results pave a way to further study on Riemannian interval optimization problems (RIOPs).

    Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii List of Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1 Introduction and Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.1 Background and Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.2 Contribution and Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 2.1 Interval analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 2.2 Notations on Riemannian manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 2.3 Interval valued functions on Riemannian manifolds. . . . . . . . . . . . . . . .14 2.3.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 2.3.2 gH-continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18 2.3.3 gH-directional differentiability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 2.3.4 gH-G^ateaux differentiability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.5 gH-Fr´echet differentiability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 3 Unconstrainted interval valued optimization problem on Hadamard manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 3.1 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 3.2 Existence of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 3.3 Steepest descent method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 3.4 Riemannian interval inequality problems . . . . . . . . . . . . . . . . . . . . . . . . . .50 4 Constrained interval valued optimization problem on Hadamard manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1 Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 53 4.2 Exact penalty approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70

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