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研究生： 蔡函儒
Tsai, Han-Ru
論文名稱： 使用統計學習方法訓練線性規劃機
Statistical Learning Method for training Linear Programming Machine
指導教授： 李雨青
Lee, Yu-Ching
口試委員: 冼芻蕘
翁偉泰
學位類別： 碩士
Master
系所名稱： 工學院 - 工業工程與工程管理學系
Department of Industrial Engineering and Engineering Management
論文出版年： 2018
畢業學年度： 106
語文別： 英文
論文頁數： 46
中文關鍵詞： 統計學習 、線性規劃 、LP機 、KKT條件 、LP機的複雜度
外文關鍵詞： Statistical learning, Linear programming, LP machine, KKT condition, Complexity of LP machine
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    • 統計學習是一門結合統計、最佳化、泛函分析使用資料來預測未來的學問。它已被應用在工業工程及電腦科學等領域。線性規劃是有著線性目標式及線性等式或不等式之數學規劃。但是鮮少有研究估計線性規劃的參數。因此，我們基於統計學習的框架建立了LP機的學習機器且證明解存在及得到KKT條件。進一步，我們將複雜度的概念融入線性規劃機當中並得到線性規劃機的學習邊界。

    Statistical learning is a subject that integrates statistics, optimization and functional analysis techniques for using data to predict the future. This subject can be applied in many fields such as industrial engineering and computer science. Linear programming (LP) is a mathematical program design with linear objective function subject to linear equality and inequality constraints. However, studies rarely estimate the LP parameter. We develop a learning machine, namely, LP machine, based on the framework of statistical learning to estimate the unknown LP parameter. We verify the existence of a solution, and derive the Karush-Kuhn-Tucker condition. Furthermore, we incorporate the concept of complexity into the LP machine and investigate its learning bound.

    1 Introduction 1 1.1 Background..........................................1 1.2 Motivation..........................................5 1.3 Organization of thesis..............................5 2 Literature review 6 2.1 LP..................................................6 2.1.1 Introduction..................................6 2.1.2 Duality.......................................7 2.1.3 Optimality condition..........................8 2.1.4 Sensitivity analysis..........................9 2.1.5 Parametric programming.......................10 2.2 Parameter estimation...............................12 2.2.1 Inverse problem..............................12 2.2.2 Ill-posed problem............................13 2.2.3 Modeling of physical system..................13 2.2.4 Inverse optimization.........................14 2.2.5 Optimal learning.............................14 2.3 Statistical learning...............................16 2.3.1 Classical model..............................16 2.3.2 Learnability.................................18 2.4 Summary............................................19 3 LP Machine 21 3.1 Problem statement..................................21 3.1.1 Framework of LP machine......................21 3.1.2 Assumptions of the LP machine................21 3.2 Modeling...........................................22 3.2.1 Notations....................................22 3.2.2 Modeling of LP machine.......................23 3.3 Training model.....................................24 3.3.1 Training model with parameter assumption.....25 3.3.2 Training model as quadratic programming......26 3.4 Existence of solutions.............................27 3.4.1 Learnability issue...........................27 3.4.2 Existence of optimal solution................29 3.4.3 KKT condition................................31 3.4.4 Geometric interpretation.....................33 4 Theoretical results of the LP machine 35 4.1 Issue of hypothesis set35 4.1.1 Growth function of LP machine................35 4.1.2 Minimum requirement of observations..........36 4.2 Complexity of LP machine...........................37 4.2.1 Leaning bound of LP machine..................37 4.3 Rademacher complexity bound of the LP machine......39 4.4 Sensitivity analysis of the LP machine.............40 5 Conclusions 42 5.1 Contributions......................................42 5.2 Limitation of research.............................43 5.3 Future direction...................................43 Reference 44

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