Mathematics 4-1-2009 Weak Sharp Minima on Riemannian Manifolds

This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and establish their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and Hadamard manifolds. A number of the results obtained in this paper are also new for the case of conventional problems in finite-dimensional Euclidean spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper.

[1]  Rolf Walter,et al.  On the metric projection onto convex sets in riemannian spaces , 1974 .

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

[3]  Daniel Azagra Rueda,et al.  Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds , 2005 .

[4]  Jane J. Ye,et al.  Optimality conditions for bilevel programming problems , 1995 .

[5]  Chong Li,et al.  Stable and Total Fenchel Duality for Convex Optimization Problems in Locally Convex Spaces , 2009, SIAM J. Optim..

[6]  P. Priouret,et al.  Newton's method on Riemannian manifolds: covariant alpha theory , 2002, math/0209096.

[7]  Chong Li,et al.  Monotone vector fields and the proximal point algorithm on Hadamard manifolds , 2009 .

[8]  M. Ferris.  Iterative linear programming solution of convex programs , 1990 .

[9]  I. Ekeland.  Nonconvex minimization problems , 1979 .

[10]  Jane J. Ye,et al.  Exact Penalization and Necessary Optimality Conditions for Generalized Bilevel Programming Problems , 1997, SIAM J. Optim..

[11]  Chong Li,et al.  Existence of solutions for variational inequalities on Riemannian manifolds , 2009 .

[12]  M. Seethharama Gowda,et al.  An Analysis of Zero Set and Global Error Bound Properties of a Piecewise Affine Function via Its Recession Function , 1996, SIAM J. Matrix Anal. Appl..

[13]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[14]  John M. Lee.  Riemannian Manifolds: An Introduction to Curvature , 1997 .

[15]  Chong Li,et al.  Newton's method on Riemannian manifolds: Smale's point estimate theory under the γ-condition , 2006 .

[16]  Michael C. Ferris,et al.  Finite termination of the proximal point algorithm , 1991, Math. Program..

[17]  Orizon Pereira Ferreira,et al.  Singularities of Monotone Vector Fields and an Extragradient-type Algorithm , 2005, J. Glob. Optim..

[18]  L. Cromme.  Strong uniqueness , 1978 .

[19]  M. R. Pouryayevali,et al.  Invariant monotone vector fields on Riemannian manifolds , 2009 .

[20]  Robert E. Mahony,et al.  Optimization Algorithms on Matrix Manifolds , 2007 .

[21]  Tamás Rapcsák,et al.  Smooth Nonlinear Optimization in Rn , 1997 .

[22]  J. H. Wang,et al.  Monotone and Accretive Vector Fields on Riemannian Manifolds , 2010 .

[23]  R. Adler,et al.  Newton's method on Riemannian manifolds and a geometric model for the human spine , 2002 .

[24]  Heinz H. Bauschke,et al.  Projection algorithms and monotone operators , 1996 .

[25]  N. D. Yen,et al.  Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming , 2006 .

[26]  Sien Deng,et al.  Weak sharp minima revisited, part II: application to linear regularity and error bounds , 2005, Math. Program..

[27]  Detlef Gromoll,et al.  On the Structure of Complete Manifolds of Nonnegative Curvature , 1972 .

[28]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[29]  S. Helgason.  Differential Geometry, Lie Groups, and Symmetric Spaces , 1978 .

[30]  Michael C. Ferris,et al.  A Gauss—Newton method for convex composite optimization , 1995, Math. Program..

[31]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[32]  S. Yau.  Non-existence of continuous convex functions on certain Riemannian manifolds , 1974 .

[33]  Jane J. Ye,et al.  New Uniform Parametric Error Bounds , 1998 .

[34]  Boris S. Mordukhovich,et al.  Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs , 2010, Math. Program..

[35]  René Henrion,et al.  On constraint qualifications , 1992 .

[36]  Adrian S. Lewis,et al.  Optimal Stability and Eigenvalue Multiplicity , 2001, Found. Comput. Math..

[37]  B. F. Svaiter,et al.  Kantorovich's Theorem on Newton's Method , 2012 .

[38]  Michael McAsey,et al.  A multiplier rule on a metric space , 2008 .

[39]  J. J. Moré,et al.  On the identification of active constraints , 1988 .

[40]  I. Ekeland.  On the variational principle , 1974 .

[41]  Wu Li,et al.  Asymptotic constraint qualifications and global error bounds for convex inequalities , 1999, Math. Program..

[42]  Pierre-Antoine Absil,et al.  Trust-Region Methods on Riemannian Manifolds , 2007, Found. Comput. Math..

[43]  R. Mahony.  The constrained newton method on a Lie group and the symmetric eigenvalue problem , 1996 .

[44]  C. Udriste,et al.  Convex Functions and Optimization Methods on Riemannian Manifolds , 1994 .

[45]  S. Deng.  Global Error Bounds for Convex Inequality Systems in Banach Spaces , 1998 .

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

[47]  J. Burke,et al.  Weak sharp minima revisited Part I: basic theory , 2002 .

[48]  A. Lewis,et al.  Optimizing Matrix Stability , 1999 .

[49]  O. P. Ferreira,et al.  Proximal Point Algorithm On Riemannian Manifolds , 2002 .

[50]  Jérôme Malick,et al.  Newton methods for nonsmooth convex minimization: connections among -Lagrangian, Riemannian Newton and SQP methods , 2005, Math. Program..

[51]  Michael McAsey,et al.  A proof of a general maximum principle for optimal controls via a multiplier rule on metric space , 2008 .

[52]  Xi Yin Zheng,et al.  Error Bound Moduli for Conic Convex Systems on Banach Spaces , 2004, Math. Oper. Res..

[53]  R. Rockafellar.  Conjugate Duality and Optimization , 1987 .

[54]  Abderrahim Jourani,et al.  Erratum: Hoffman's Error Bound, Local Controllability, and Sensitivity Analysis , 2000, SIAM J. Control. Optim..

[55]  Alfred Auslender,et al.  Global Regularity Theorems , 1988, Math. Oper. Res..

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

[57]  Yu. S. Ledyaev,et al.  Nonsmooth analysis on smooth manifolds , 2007 .

[58]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[59]  O. P. Ferreira,et al.  Contributions to the Study of Monotone Vector Fields , 2002 .

[60]  Michael C. Ferris,et al.  Weak sharp minima and penalty functions in mathematical programming , 1988 .

[61]  F. Giannessi.  Variational Analysis and Generalized Differentiation , 2006 .

[62]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

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

[64]  Xi Yin Zheng,et al.  Metric Regularity and Constraint Qualifications for Convex Inequalities on Banach Spaces , 2003, SIAM J. Optim..

[65]  Boris S. Mordukhovich,et al.  Necessary Conditions for Nonsmooth Optimization Problems with Operator Constraints in Metric Spaces , 2008 .

[66]  D. Gabay.  Minimizing a differentiable function over a differential manifold , 1982 .

[67]  H. Karcher.  Riemannian center of mass and mollifier smoothing , 1977 .

[68]  Adrian S. Lewis,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[69]  Chong Li,et al.  Constraint Qualification, the Strong CHIP, and Best Approximation with Convex Constraints in Banach Spaces , 2003, SIAM J. Optim..

[70]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[71]  Heinz H. Bauschke,et al.  Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization , 1999, Math. Program..

[72]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .