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Superlinear Convergence of Affine Scaling Interior Point Newton Method for Linear Inequality Constrained Minimization without strict complementarity

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Acta Mathematicae Applicatae Sinica 2009年第2期25卷 183-194页

作者： De-tong Zhu Business College Shanghai Normal University Shanghai 200234 China

In this paper we extend and improve the classical affine scaling interior-point Newton method for solving nonlinear optimization subject to linear inequality constraints in the absence of the strict complementarity assumption. Introducing a computationally efficient technique and employing an identification function for the definition of the new affine scaling matrix, we propose and analyze a new affine scaling interior-point Newton method which improves the Coleman and Li affine sealing matrix in [2] for solving the linear inequlity constrained optimization. Local superlinear and quadratical convergence of the proposed algorithm is established under the strong second order sufficiency condition without assuming strict complementarity of the solution.

关键词： Interior method affine scaling strict complementarity

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A CONSTRAINED OPTIMIZATION APPROACH FOR LCP

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Journal of Computational Mathematics 2004年第4期22卷 509-522页

作者： Ju-liangZhang JianChen Xin-jianZhuo DepartmentofManagementScienceandEngineering SchoolofEconomicsandManagementTsinghuaUniversityBeijing100084China SchoolofInformationEngineering BeijingUniversityofpostsandTelecommunicationBeijing100876China

In this paper, LCP is converted to an equivalent nonsmooth nonlinear equation system H(x,y) = 0 by using the famous NCP function-Fischer-Burmeister function. Note that some equations in H(x, y) = 0 are nonsmooth and nonlinear hence difficult to solve while the others are linear hence easy to solve. Then we further convert the nonlinear equation system H(x, y) = 0 to an optimization problem with linear equality constraints. After that we study the conditions under which the K-T points of the optimization problem are the solutions of the original LCP and propose a method to solve the optimization problem. In this algorithm, the search direction is obtained by solving a strict convex programming at each iterative point, However, our algorithm is essentially different from traditional SQP method. The global convergence of the method is proved under mild conditions. In addition, we can prove that the algorithm is convergent superlinearly under the conditions: M is P0 matrix and the limit point is a strict complementarity solution of LCP. Preliminary numerical experiments are reported with this method.

关键词： LCP strict complementarity Nonsmooth equation system PQ matrix Super-linear convergence

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Equivalence and Strong Equivalence Between the Sparsest and Least l1-Norm Nonnegative Solutions of Linear Systems and Their Applications

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Journal of the Operations Research Society of China 2014年第2期2卷 171-193页

作者： Yun-Bin Zhao School of Mathematics University of BirminghamEdgbastonBirmingham B152TTUK

Many practical problems can be formulated as l0-minimization problems with nonnegativity constraints,which seek the sparsest nonnegative solutions to underdetermined linear *** study indicates that l1-minimization is efficient for solving l0-minimization *** a mathematical point of view,however,the understanding of the relationship between l0-and l1-minimization remains *** this paper,we further address several theoretical questions associated with these two *** prove that the fundamental strict complementarity theorem of linear programming can yield a necessary and sufficient condition for a linear system to admit a unique least l1-norm nonnegative *** condition leads naturally to the so-called range space property(RSP)and the “full-column-rank”property,which altogether provide a new and broad understanding of the equivalence and the strong equivalence between l0-and *** by these results,we introduce the concept of “RSP of order K”that turns out to be a full characterization of uniform recovery of all K-sparse nonnegative *** concept also enables us to develop a nonuniform recovery theory for sparse nonnegative vectors via the so-called weak range space property.

关键词： strict complementarity Linear programming Underdetermined linear system Sparsest nonnegative solution Range space property Uniform recovery Nonuniform recovery

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