本文選題：二階錐規(guī)劃 + 線性規(guī)劃��； 參考：《內(nèi)蒙古大學》2017年碩士論文

【摘要】：線性規(guī)劃問題是研究變量在仿射集和凸多面體交集上的一類凸優(yōu)化問題.作為線性規(guī)劃的推廣,二階錐規(guī)劃也是一類凸優(yōu)化問題,它是在一個仿射子空間和有限個二階錐的笛卡爾乘積的交集上極大化或極小化一個線性函數(shù).許多數(shù)學規(guī)劃問題,都可以轉化為二階錐問題求解.線性規(guī)劃和二階錐規(guī)劃在工程、控制與設計等諸多領域的廣泛應用,使其成為數(shù)學規(guī)劃的一個重要研究方向.本文主要研究線性規(guī)劃和二階錐規(guī)劃的光滑牛頓法.全文共分為四章.第一章,介紹線性規(guī)劃和二階錐規(guī)劃的研究背景及現(xiàn)狀.第二章,通過光滑逼近Fischer-Burmeister函數(shù),構造出一個新的光滑函數(shù),得出該函數(shù)的連續(xù)可微性.基此給出一個求解線性規(guī)劃問題的光滑牛頓法.此外,證明了算法的全局收斂性.在解點處雅可比矩陣可逆的條件下,得到算法的二次收斂速度.最后通過數(shù)值實驗證明了算法的有效性.第三章,通過對稱擾動Fischer-Burmeister函數(shù),提出一個新的互補函數(shù).基于該函數(shù),把二階錐規(guī)劃問題轉化為一個參數(shù)化的光滑方程組,并利用光滑牛頓法求解.此外,證明了算法的全局收斂性.在解點處雅可比矩陣可逆的條件下,得到算法的二次收斂速度.最后進行數(shù)值實驗,數(shù)值結果表明了算法的有效性.第四章是對本文的總結.
[Abstract]:Linear programming problem is a class of convex optimization problems for variables on affine sets and convex polyhedron intersection. As a generalization of linear programming, second-order cone programming is also a class of convex optimization problems. It is a linear function that is maximized or minimized on the intersection of the Cartesian product of an affine subspace and the finite second-order cone. Many mathematical programming problems can be transformed into second-order cone problems. Linear programming and second-order cone programming are widely used in many fields such as engineering, control and design, which make them become an important research direction of mathematical programming. In this paper, the smooth Newton method for linear programming and second order cone programming is studied. The full text is divided into four chapters. The first chapter introduces the research background and present situation of linear programming and second-order cone programming. In chapter 2, a new smooth function is constructed by smoothing the Fischer-Burmeister function, and the continuous differentiability of the function is obtained. Based on this, a smooth Newton method for solving linear programming problems is given. In addition, the global convergence of the algorithm is proved. The quadratic convergence rate of the algorithm is obtained under the condition that Jacobian matrix is reversible at the solution point. Finally, the effectiveness of the algorithm is proved by numerical experiments. In chapter 3, a new complementary function is proposed by symmetric perturbation Fischer-Burmeister function. Based on this function, the second order cone programming problem is transformed into a parameterized smooth equation system and solved by the smooth Newton method. In addition, the global convergence of the algorithm is proved. The quadratic convergence rate of the algorithm is obtained under the condition that Jacobian matrix is reversible at the solution point. Finally, numerical experiments are carried out, and the numerical results show the effectiveness of the algorithm. The fourth chapter is the summary of this paper.
【學位授予單位】：內(nèi)蒙古大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O221

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