本文關鍵詞： 二階錐規(guī)劃 二階錐互補問題 低階罰函數(shù)算法 指數(shù)收斂速度　出處：《北方民族大學》2016年碩士論文　論文類型：學位論文

【摘要】：二階錐規(guī)劃隸屬于凸優(yōu)化,它的目標函數(shù)是線性函數(shù)的極小化或極大化問題,而約束域為一個仿射空間和若干個二階錐的笛卡爾乘積的交集.線性規(guī)劃、凸二次規(guī)劃、凸二次約束二次規(guī)劃等問題都可統(tǒng)一轉化為二階錐規(guī)劃問題,且它們又都是半定規(guī)劃的特例.二階錐規(guī)劃已經成為數(shù)學規(guī)劃領域一個重要的研究方向.二階錐互補問題是一類均衡優(yōu)化問題,是指在二階錐約束條件下,兩組決策變量之間滿足一種互補關系,它隸屬于對稱錐互補問題.近幾年來,人們在歐幾里得約當代數(shù)的基礎上,對二階錐互補問題的研究已取得許多成果,并且使之逐漸受到了重視.由于二階錐規(guī)劃的KKT條件是二階錐互補問題的一種特殊情況,因此可以運用求解二階錐互補問題的算法來解決二階錐規(guī)劃問題.低階罰函數(shù)算法是求解對稱錐互補問題的有效方法,其主要思路即將互補問題轉化為低階罰函數(shù)方程組,此算法的突出之處在于低階罰函數(shù)方程組的解序列在特定條件下以指數(shù)速度收斂于二階錐互補問題的解.由于低階罰函數(shù)算法有很多良好的性質,比如解的精確性等,所以將低階罰函數(shù)算法擴展到求解二階錐互補問題上是一個非常有意義的研究工作.本文主要研究求解二階錐線性互補問題的低階罰函數(shù)算法.其主要內容如下:1.針對二階錐線性互補問題,利用低階罰函數(shù)算法的思想及二階錐投影的冪的表達式將二階錐線性互補問題轉化為低階罰函數(shù)方程組.證明了矩陣正定(不一定對稱)的條件下低階罰函數(shù)方程組的解序列以指數(shù)速度收斂于二階錐線性互補問題的最優(yōu)解.數(shù)值實驗的結果進一步驗證了有關理論的結果.并將低階罰函數(shù)算法的數(shù)值結果與著名的光滑F(xiàn)ischer-Burmeister(F-B)函數(shù)算法進行比較,結果表明提出的算法是有效的,并且占有一定優(yōu)勢.2.針對一類廣義的二階錐線性互補問題,利用低階罰函數(shù)算法的思想也將其轉化為低階罰函數(shù)方程組.在矩陣正定(不一定對稱)的前提下,證明了在罰參數(shù)趨向于正無窮時,低階罰函數(shù)方程組的解序列以指數(shù)速度收斂于原廣義二階線性錐互補問題的解.最后,對本文的工作作了總結,并提出了有待進一步研究的工作.
[Abstract]:The second order cone programming belongs to convex optimization. Its objective function is the minimization or maximization of linear function, while the constraint domain is the intersection of the Cartesian product of an affine space and several second order cones. Problems such as convex quadratic programming and convex quadratic constrained quadratic programming can be unified into second-order cone programming problems. And they are special cases of semidefinite programming. Second order cone programming has become an important research direction in the field of mathematical programming. Second order cone complementarity problem is a kind of equilibrium optimization problem, which refers to the second order cone constraint problem. The two sets of decision variables satisfy a kind of complementary relation, which belongs to the symmetric cone complementarity problem. In recent years, on the basis of Euclidean approximate contemporary numbers, many achievements have been made in the study of the second-order cone complementarity problem. The KKT condition of the second-order cone programming is a special case of the second-order cone complementarity problem. Therefore, we can use the algorithm to solve the second-order cone complementarity problem, and the low-order penalty function algorithm is an effective method to solve the symmetric cone complementarity problem. The main idea is to transform the complementarity problem into a system of low-order penalty function equations. The outstanding point of this algorithm is that the sequence of solutions of the system of low-order penalty function converges to the solution of the second-order cone complementarity problem at exponential speed under certain conditions, because the low-order penalty function algorithm has many good properties. Such as the accuracy of the solution. Therefore, it is very meaningful to extend the low order penalty function algorithm to solve the second order cone complementarity problem. In this paper, we mainly study the low order penalty function algorithm for solving the second order cone linear complementarity problem. The main contents of this paper are as follows. :. 1. For the second order conical linear complementarity problem. By using the idea of the low-order penalty function algorithm and the expression of the power of the second-order cone projection, the linear complementarity problem of the second-order cone is transformed into the low-order penalty function equations. It is proved that the matrix is positive definite (not necessarily symmetric). The sequence of solutions of the system of penalty function of low order converges to the optimal solution of the linear complementarity problem of second order cone at exponential speed. The results of numerical experiments further verify the results of the theory. The number of the algorithm of low order penalty function is also given. Value results and the famous smooth Fischer-Burmeister (. F-B) function algorithm is compared. The results show that the proposed algorithm is effective and has a certain advantage .2. for a class of generalized second-order conical linear complementarity problems. By using the idea of low-order penalty function algorithm, it is also transformed into a set of low-order penalty function equations. On the premise of positive definite matrix (not necessarily symmetric), it is proved that the penalty parameters tend to be positive infinity when the penalty parameters tend to be positive infinity. The sequence of solutions of the system of penalty functions of low order converges to the solution of the original generalized second order linear cone complementarity problem at exponential speed. Finally, the work of this paper is summarized, and the work to be further studied is put forward.
【學位授予單位】：北方民族大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：O221

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1 趙雯宇;兩類二階錐線性互補問題的低階罰函數(shù)算法研究[D];北方民族大學;2016年

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