Error Bounds for Eigenvalue and Semidefinite Matrix Inequality Systems

In this paper we give sufficient conditions for existence of error bounds for systems expressed in terms of eigenvalue functions (such as in eigenvalue optimization) or positive semidefiniteness (such as in semidefinite programming).

[1]  R. Bhatia.  Matrix Analysis , 1996 .

[2]  B. Mordukhovich.  Variational analysis and generalized differentiation , 2006 .

[3]  Henry Wolkowicz,et al.  Handbook of Semidefinite Programming , 2000 .

[4]  ABDERRAHIM JOURANI,et al.  Hoffman's Error Bound, Local Controllability, and Sensitivity Analysis , 2000, SIAM J. Control. Optim..

[5]  O. Cornejo,et al.  Conditioning and Upper-Lipschitz Inverse Subdifferentials in Nonsmooth Optimization Problems , 1997 .

[6]  Xi Yin Zheng,et al.  Error Bounds for Lower Semicontinuous Functions in Normed Spaces , 2001, SIAM J. Optim..

[7]  B. Mordukhovich.  Variational Analysis and Generalized Differentiation II: Applications , 2006 .

[8]  A. Lewis,et al.  Error Bounds for Convex Inequality Systems , 1998 .

[9]  Lionel Thibault,et al.  Approximate subdifferential and metric regularity: The finite-dimensional case , 1990, Math. Program..

[10]  Michael L. Overton,et al.  Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices , 2015, Math. Program..

[11]  R. Fletcher.  Semi-Definite Matrix Constraints in Optimization , 1985 .

[12]  B. Mordukhovich,et al.  Nonsmooth sequential analysis in Asplund spaces , 1996 .

[13]  A. Hoffman.  On approximate solutions of systems of linear inequalities , 1952 .

[14]  Sien Deng.  Computable Error Bounds For Convex Inequality Systems In Reflexive Banach Spaces , 1997, SIAM J. Optim..

[15]  Adrian S. Lewis,et al.  Nonsmooth analysis of eigenvalues , 1999, Math. Program..

[16]  Jean-Baptiste Hiriart-Urruty,et al.  Optimal Hoffman-Type Estimates in Eigenvalue and Semidefinite Inequality Constraints , 2002, J. Glob. Optim..

[17]  Jong-Shi Pang,et al.  Error bounds in mathematical programming , 1997, Math. Program..

[18]  Jane J. Ye,et al.  Sufficient Conditions for Error Bounds , 2002, SIAM J. Optim..

[19]  Hui Hu,et al.  Computable Error Bounds for Semidefinite Programming , 1999, J. Glob. Optim..

[20]  Jane J. Ye,et al.  On error bounds for lower semicontinuous functions , 2002, Math. Program..

[21]  M. Ferris,et al.  Weak sharp minima in mathematical programming , 1993 .

[22]  A. Ioffe.  Regular points of Lipschitz functions , 1979 .

[23]  René Henrion,et al.  Sufficient Conditions for Error Bounds and Applications , 2004 .

[24]  Alexander Shapiro,et al.  First and second order analysis of nonlinear semidefinite programs , 1997, Math. Program..

[25]  Jane J. Ye,et al.  First-Order and Second-Order Conditions for Error Bounds , 2003, SIAM J. Optim..

[26]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .