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SEMISMOOTH SQP METHOD FOR EQUALITYCONSTRAINED OPTIMIZATION PROBLEMS WITH AN APPLICATION TO THE LIFTED REFORMULATION OF MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS
, 2010
"... We consider the sequential quadratic programming algorithm (SQP) applied to equalityconstrained optimization problems, where the problem data is differentiable with Lipschitzcontinuous first derivatives. For this setting, DennisMoré type analysis of primal superlinear convergence is presented. Our ..."
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We consider the sequential quadratic programming algorithm (SQP) applied to equalityconstrained optimization problems, where the problem data is differentiable with Lipschitzcontinuous first derivatives. For this setting, DennisMoré type analysis of primal superlinear convergence is presented. Our main motivation is a special modification of SQP tailored to the structure of the lifted reformulation of mathematical programs with complementarity constraints (MPCC). For this problem, we propose a special positive definite modification of the matrices in the generalized Hessian, which is suitable for globalization of SQP based on the penalty function, and at the same time can be expected to satisfy our general DennisMoré type conditions, thus preserving local superlinear convergence. (Standard quasiNewton updates in the SQP framework require twice differentiability of the problem data at the solution for superlinear convergence.) Preliminary numerical results comparing a number of quasiNewton versions of semismooth SQP applied to MPCC are also reported. Key words: sequential quadratic programming, semismoothness, Bdifferential, BDregularity, semismooth Newton method, secondorder sufficiency, mathematical programs with complementarity constraints.
Copyright © 2005 SBMAC
"... www.scielo.br/cam Numerical results for a globalized activeset Newton method for mixed complementarity problems ..."
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www.scielo.br/cam Numerical results for a globalized activeset Newton method for mixed complementarity problems
Mathematical Programming manuscript No. (will be inserted by the editor)
, 2006
"... On attraction of Newtontype iterates to multipliers violating secondorder sufficiency conditions Dedicated to Professor Stephen Robinson on the occasion of his 65th birthday. The second author remembers, with a sense of privilege, the courses and advice he received from Professor Robinson during h ..."
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On attraction of Newtontype iterates to multipliers violating secondorder sufficiency conditions Dedicated to Professor Stephen Robinson on the occasion of his 65th birthday. The second author remembers, with a sense of privilege, the courses and advice he received from Professor Robinson during his stay at UWMadison.
Keywords Nonlinear Equations · Semismooth Functions · Newton’s Method · Nonlinear Complementarity Problems
"... Abstract We discuss local convergence of Newton’s method to a singular solution x ∗ of the nonlinear equations F (x) = 0, for F: IR n → IR n. It is shown that an existing proof of Griewank, concerning linear convergence to a singular solution x ∗ from a starlike domain around x ∗ for F twice Lipsch ..."
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Abstract We discuss local convergence of Newton’s method to a singular solution x ∗ of the nonlinear equations F (x) = 0, for F: IR n → IR n. It is shown that an existing proof of Griewank, concerning linear convergence to a singular solution x ∗ from a starlike domain around x ∗ for F twice Lipschitz continuously differentiable and x ∗ satisfying a particular regularity condition, can be adapted to the case in which F ′ is only strongly semismooth at the solution. Further, under this regularity assumption, Newton’s method can be accelerated to produce fast linear convergence to a singular solution by overrelaxing every second Newton step. These results are applied to a nonlinearequations reformulation of the nonlinear complementarity problem (NCP) whose derivative is strongly semismooth when the function f arising in the NCP is sufficiently smooth. Conditions on f are derived that ensure that the appropriate regularity conditions are satisfied for the nonlinearequations reformulation of the NCP at x ∗.
Equations with Semismooth Jacobians and Nonlinear Complementarity Problems
"... We dedicate this paper to Steve Robinson on the occasion of his 65th birthday, in recognition of his remarkable scholarly accomplishments and in appreciation for his collegiality, guidance, and kindness. Received: date / Accepted: date Abstract We discuss local convergence of Newton’s method to a si ..."
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We dedicate this paper to Steve Robinson on the occasion of his 65th birthday, in recognition of his remarkable scholarly accomplishments and in appreciation for his collegiality, guidance, and kindness. Received: date / Accepted: date Abstract We discuss local convergence of Newton’s method to a singular solution x ∗ of the nonlinear equations F(x) = 0, for F: IR n → IR n. It is shown that an existing proof of Griewank, concerning linear convergence to a singular solution x ∗ from a starlike domain around x ∗ for F twice Lipschitz continuously differentiable and x ∗ satisfying a particular regularity condition, can be adapted to the case in which F ′ is only strongly semismooth at the solution. Further, Newton’s method can be accelerated to produce fast linear convergence to a singular solution by overrelaxing every second Newton step. These results are applied to a nonlinearequations reformulation of the nonlinear complementarity problem (NCP) whose derivative is strongly semismooth when the function f arising in the NCP is sufficiently smooth. Conditions on f are derived that ensure that the appropriate regularity conditions are satisfied for the nonlinearequations reformulation of the NCP at x ∗.
PRONEX–Optimization, and by FAPERJ.
, 2013
"... Local convergence of the method of multipliers for variational and optimization problems under the noncriticality assumption ..."
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Local convergence of the method of multipliers for variational and optimization problems under the noncriticality assumption