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A QQPMinimization Method for Semidefinite and Smooth Nonconvex Programs
 Working Paper, Abteilung Mathematik, Universitat
, 1998
"... . In several applications, semidefinite programs with nonlinear equality constraints arise. We give two such examples to emphasize the importance of this class of problems. We then propose a new solution method that also applies to smooth nonconvex programs. The method combines ideas of a predictor ..."
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. In several applications, semidefinite programs with nonlinear equality constraints arise. We give two such examples to emphasize the importance of this class of problems. We then propose a new solution method that also applies to smooth nonconvex programs. The method combines ideas of a predictor corrector interiorpoint method, of the SQP method, and of trust region methods. In particular, we believe that the new method combines the advantagesgenerality and robustness of trust region methods, local convergence of the SQPmethod and dataindependence of interiorpoint methods. Some convergence results are given, and some very preliminary numerical experiments suggest a high robustness of the proposed method. AMS 1991 subject classification. Primary: 90C. Key words. Predictor corrector method, SQP method, trust region method, semidefinite program. 1. Introduction This work was motivated by two applications from semidefinite programming with nonlinear equality constraints as outlin...
Affine Scaling Algorithm Fails For Semidefinite Programming
 Mathematical Programming 83
, 1997
"... In this paper, we introduce an affine scaling algorithm for semidefinite programming, and give an example of a semidefinite program such that the affine scaling algorithm converges to a nonoptimal point. Both our program and its dual have interior feasible solutions, and unique optimal solutions wh ..."
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In this paper, we introduce an affine scaling algorithm for semidefinite programming, and give an example of a semidefinite program such that the affine scaling algorithm converges to a nonoptimal point. Both our program and its dual have interior feasible solutions, and unique optimal solutions which satisfy strict complementarity, and they are nondegenerate everywhere. Abbreviated Title: Affine scaling fails for SDP Key Words: Semidefinite Programming, Affine Scaling Algorithm, Global Convergence Analysis. Affiliation: Department of Mechanical Engineering, Sophia University Address: 71, Kioicho, Chiyodaku, Tokyo 102 Japan Email address: muramatu@me.sophia.ac.jp URL: http://www.or.me.sophia.ac.jp/~muramatu 1 Introduction Semidefinite programming (SDP) has remarkable resemblance with linear programming (LP), and it is known that several interior point methods for LP and their polynomial convergence analysis can be naturally extended to SDP (Alizadeh [1], Alizadeh, Haeberly and...
Convergence properties of Dikinâ€™s affine scaling algorithm for nonconvex quadratic minimization
 J. Global Optim
, 2001
"... Abstract We study convergence properties of Dikin's affine scaling algorithm applied to nonconvex quadratic minimization. First, we show that the objective function value either diverges or converges Qlinearly to a limit. Using this result, we show that, in the case of box constraints, the ite ..."
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Abstract We study convergence properties of Dikin's affine scaling algorithm applied to nonconvex quadratic minimization. First, we show that the objective function value either diverges or converges Qlinearly to a limit. Using this result, we show that, in the case of box constraints, the iterates converge to a unique point satisfying 1storder and weak 2ndorder optimality conditions, assuming the objective function Hessian Q is rank dominant with respect to the principal submatrices that are maximally positive semidefinite. Such Q include matrices that are positive semidefinite or negative semidefinite or nondegenerate or have negative diagonals. Preliminary numerical experience is reported. Key words. Nonconvex quadratic minimization, affinescaling algorithm, trust region subproblem, Hoffman's error bound, linear convergence. 1 Introduction We consider the nonconvex quadratic program (QP):