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 as...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.展开更多
Many practical problems can be formulated as l0-minimization problems with nonnegativity constraints,which seek the sparsest nonnegative solutions to underdetermined linear systems.Recent study indicates that l1-minim...Many practical problems can be formulated as l0-minimization problems with nonnegativity constraints,which seek the sparsest nonnegative solutions to underdetermined linear systems.Recent study indicates that l1-minimization is efficient for solving l0-minimization problems.From a mathematical point of view,however,the understanding of the relationship between l0-and l1-minimization remains incomplete.In this paper,we further address several theoretical questions associated with these two problems.We 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 solution.This 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 l1-minimization.Motivated 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 vectors.This concept also enables us to develop a nonuniform recovery theory for sparse nonnegative vectors via the so-called weak range space property.展开更多
基金Supported by the National Natural Science Foundation of China(No.10871130)the Ph.D.Foundation Grant (0527003) of Chinese Education Ministry,the Shanghai Leading Academic Discipline Project(T0401)the Science Foundation Grant(05DZ11) of Shanghai Education Committee
文摘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.
基金supported by the Engineering and Physical Sciences Research Council(No.K00946X/1)was partially supported by the National Natural Science Foundation of China(No.11301016).
文摘Many practical problems can be formulated as l0-minimization problems with nonnegativity constraints,which seek the sparsest nonnegative solutions to underdetermined linear systems.Recent study indicates that l1-minimization is efficient for solving l0-minimization problems.From a mathematical point of view,however,the understanding of the relationship between l0-and l1-minimization remains incomplete.In this paper,we further address several theoretical questions associated with these two problems.We 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 solution.This 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 l1-minimization.Motivated 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 vectors.This concept also enables us to develop a nonuniform recovery theory for sparse nonnegative vectors via the so-called weak range space property.
