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Nonsingularity Study of the Parametric FB System for Nonlinear Semidefinite Programming

Author: QuJiaJin
Tutor: PanShaoHua
School: South China University of Technology
Course: Operational Research and Cybernetics
Keywords: Nonlinear semidefinite programming Strong second-order sufficient condition Constrained non- degenerative Semidefinite cone complementary function of the parameter type FB Directional derivative B first derivative clarke generalized Jacobian (Strongly) semismooth Smoothing algorithm Quadratic convergence rate
CLC: O221.2
Type: Master's thesis
Year: 2011
Downloads: 6
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Abstract


Nonlinear semidefinite programming and a wide range of applications in the areas of engineering design, system control, finance, combinatorial optimization, robust optimization, and pattern recognition. Semidefinite cone complementary function of the parameter type FB (?) Where T ∈ (1,1) is a parameter) and its smooth function (?), This problem Karush-Kuhn-Tucker (KKT) optimality conditions into a non-smooth equations Er (.) = 0 or augmented equations Fr (.) = 0. When the non-smooth Newton method Er (.) = 0 and the classical Newton's method to solve a (.) 10, in order to ensure that the algorithm has fast convergence rate, in addition to the requirements of the mapping Er and Fr (strongly) semismooth most The key is to know under what conditions, Er and Fr the KKT point at clarke generalized Jacobian nonsingularity. In this paper, the nature of the study parameters FB the semidefinite cone complementary function directional derivative function and clarke generalized Jacobian, nonlinear semidefinite programming local optimal solution in Robinson constraint specification, it is proved that the strong second order sufficient conditions and constraints non-degenerative, Er strong in the the KKT point at clarke generalized Jacobian nonsingularity well KKT point regular sex is equivalent to each other. When the degradation of linear semidefinite programming, we get the original constraint nondegenerate dual of Er strong at the the KKT point at clarke generalized Jacobian nonsingularity KKT point between regular equivalence. Augmented equations Fr (.) = 0, we have also established a similar theoretical results. These results on the one hand for nonlinear semidefinite planning parameter type FB semismooth and smoothing algorithms guarantee fast convergence theory; On the other hand, provide a new characterization of the nature of the local optimal solution for nonlinear semidefinite programming . Parameter type FB semidefinite cone complementarity function contains FB semidefinite cone complementary function (when T = 0) and NR semidefinite cone complementary function (when T = -1), the results of this paper [1. This thesis consists of five chapters, each chapter as follows. The first chapter is an introduction, a brief nonlinear semidefinite programming algorithms, especially semi-smooth and smoothing algorithm Research; given in Chapter II of this article required prior knowledge and lemma, and the proof of (? ) is semidefinite cone complementary function and smooth function (?); the Chapter by studying the directional derivative function clarke generalized Jacobian nature, Robinson constraint specification, the establishment of a strong second-order sufficient condition constraint nondegeneracy of Er strong at the the KKT point at clarke generalized Jacobian nonsingular KKT point regular sex is equivalent to each other: Chapter smoothing function by studying the smoothing parameter 0 directional derivative function clarke generalized Jacobian of nature, given Fr nonsingular equivalent conditions in the KKT point clarke generalized Jacobi; Chapter literature \group Fr (.) = 0, the establishment of ultra-linear (quadratic convergence) the rate of convergence of the algorithm, and the effectiveness of the algorithm by calculating the standard exam test.

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CLC: > Mathematical sciences and chemical > Mathematics > Operations Research > Planning Theory ( mathematical programming) > Nonlinear Programming
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