Connectivity of Quadratic Hypersurfaces and Its Applications in Optimization, Part I: General Theory

In this paper we establish a basic theory on the connectivity of quadratic hypersurfaces. In addition, we examine how both the connectivity and the hidden convexity of quadratic functions are related to each other and prove several connectivity theorems on quadratic functions that combine and generalize Dines's and Brickman's results on the hidden convexity of quadratic mappings in real field. Many classical results in quadratic optimization are reproved by the unified approach, and more properties involving the matrix pencils are given.

[1]  K. Veselic,et al.  Trace minimization and definiteness of symmetric pencils , 1995 .

[2]  E. Feron.  Nonconvex Quadratic Programming, Semidefinite Relaxations and Randomization Algorithms in Information and Decision Systems , 2000 .

[3]  H. Wolkowicz,et al.  Trust Region Problems and Nonsymmetric Eigenvalue Perturbations , 1994 .

[4]  L. Brickman.  ON THE FIELD OF VALUES OF A MATRIX , 1961 .

[5]  Jean-Baptiste Hiriart-Urruty,et al.  Potpourri of Conjectures and Open Questions in Nonlinear Analysis and Optimization , 2007, SIAM Rev..

[6]  Yan Zi-zong,et al.  Some Equivalent Results with Yakubovich's S-Lemma , 2010, SIAM J. Control. Optim..

[7]  Über eine KlasseJ-selbstadjungierter Operatoren , 1964 .

[8]  Xin Chen,et al.  A note on quadratic forms , 1999, Math. Program..

[9]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[10]  Dick den Hertog,et al.  Hidden conic quadratic representation of some nonconvex quadratic optimization problems , 2014, Math. Program..

[11]  Henry Wolkowicz,et al.  The generalized trust region subproblem , 2014, Comput. Optim. Appl..

[12]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[13]  Yik-Hoi Au-Yeung.  On the semi-definiteness of the real pencil of two Hermitian matrices , 1975 .

[14]  Samuel Burer,et al.  The trust region subproblem with non-intersecting linear constraints , 2015, Math. Program..

[15]  L. L. Dines.  On the mapping of quadratic forms , 1941 .

[16]  E. Calabi.  Linear systems of real quadratic forms. II , 1964 .

[17]  Laurent El Ghaoui,et al.  Advances in linear matrix inequality methods in control: advances in design and control , 1999 .

[18]  J. Stoer,et al.  Convexity and Optimization in Finite Dimensions I , 1970 .

[19]  Ronald I. Becker.  Necessary and sufficient conditions for the simultaneous diagonability of two quadratic forms , 1980 .

[20]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[21]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[22]  Chi-Kwong Li,et al.  Joint Ranges of Hermitian Matrices and Simultaneous Diagonalization , 1991 .

[23]  Amir Beck,et al.  On the convexity of a class of quadratic mappings and its application to the problem of finding the smallest ball enclosing a given intersection of balls , 2007, J. Glob. Optim..

[24]  C. Hamburger,et al.  Two Extensions to Finsler's Recurring Theorem , 1999 .

[25]  Paul Pinsler.  Über das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen , 1936 .

[26]  F. Uhlig.  A recurring theorem about pairs of quadratic forms and extensions: a survey , 1979 .

[27]  Ya-Xiang Yuan,et al.  On a subproblem of trust region algorithms for constrained optimization , 1990, Math. Program..

[28]  Tamás Terlaky,et al.  A Survey of the S-Lemma , 2007, SIAM Rev..

[29]  Henry Wolkowicz,et al.  Indefinite Trust Region Subproblems and Nonsymmetric Eigenvalue Perturbations , 1995, SIAM J. Optim..

[30]  M. Levin.  Estimation of a system pulse transfer function in the presence of noise , 1964 .

[31]  Henry Wolkowicz,et al.  The trust region subproblem and semidefinite programming , 2004, Optim. Methods Softw..

[32]  M. Hestenes.  Pairs of quadratic forms , 1968 .

[33]  Boris Polyak.  Convexity of Quadratic Transformations and Its Use in Control and Optimization , 1998 .

[34]  J. J. Moré.  Generalizations of the trust region problem , 1993 .

[35]  An Extension of Polyak’s Theorem in a Hilbert Space , 2009 .

[36]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[37]  Mustafa Ç. Pinar,et al.  On the S-procedure and Some Variants , 2006, Math. Methods Oper. Res..

[38]  Eric Schechter,et al.  Handbook of Analysis and Its Foundations , 1996 .