Nonsmooth optimization and its applications

Preface......Page 6
Contents......Page 8
 1 Introduction......Page 9
 2 Sparse PCA......Page 12
 3 Secant-Based Dimensionality Reduction......Page 13
 5 Range-Based Independent Component Analysis......Page 14
 6 Sphere Packing on Manifolds......Page 15
 7 Robust Low-Rank Matrix Completion......Page 16
 8 Finding the Sparsest Vector in a Subspace......Page 17
 9 Restoring Manifold-Valued Images......Page 18
 11 Conclusion......Page 19
 References......Page 20
 1 Introduction......Page 24
 2 Spherical Subdifferentials......Page 26
 3 An Approximate Alternating Direction of Multipliers Method......Page 28
 4 An Algorithm for Finding (τ,δ)-Stationary Points......Page 31
 5 Convergence of SGBUN Method......Page 35
 6 Computational Results......Page 42
 7 Conclusions......Page 45
 Appendix......Page 46
 References......Page 50
 1 Introduction......Page 52
 2 Preliminaries......Page 54
  2.2 Bouligand Tangent and Normal Cone to M≤k......Page 55
 3 Clarke and Mordukhovich Normal Cones......Page 56
 4 Implications to Necessary Optimality Conditions......Page 58
 References......Page 60
Subdifferential Enlargements and Continuity Properties of the VU-Decomposition in Convex Optimization......Page 61
 1 Introduction......Page 62
 2 A Fast Overview of VU-Theory......Page 63
  2.2 Fast Tracks......Page 64
  2.3 Smooth Manifolds and Partial Smoothness......Page 65
 3 Approximating the V-Subspace......Page 66
  3.1 Semicontinuity Notions......Page 67
  3.2 Subdifferential Enlargements......Page 69
  4.1 The -Subdifferential Enlargement......Page 71
  4.2 A Separable Enlargement......Page 72
  4.3 A Not-So-Large Enlargement......Page 73
   4.4.1 Exact VU-Approach......Page 74
   4.4.2 Exact V-Step with U-Step......Page 76
   4.4.4 Implicit V Step......Page 77
   4.4.6 Iteration Update for All the Cases......Page 78
  5.1 -Activity Sets......Page 79
  5.2 Polyhedral Functions: Enlargement and VU-Subspaces......Page 80
  5.3 Polyhedral Functions: Manifold and Manifold Relaxation......Page 82
   5.3.1 Illustration on a Simple Example......Page 84
  5.4 General Max-Functions: Enlargement and VU-Subspaces......Page 85
  6.1 Enlargement and VU-Subspaces......Page 86
   6.2.2 Finite Valued Sublinear Polyhedral Functions......Page 88
   6.2.3 The Maximum Eigenvalue Function......Page 90
 7 Concluding Remarks......Page 91
 References......Page 92
 1 Introduction......Page 94
  2.2 Definitions and Facts......Page 96
  3.1 The Set of Convex Moreau Envelopes......Page 98
  3.2 Characterizations of the Moreau Envelope......Page 101
  3.3 Differentiability of the Moreau Envelope......Page 102
  3.4 Moreau Envelopes of Piecewise Differentiable Functions......Page 103
  4.1 Motivation......Page 109
  4.2 Convexity......Page 110
  4.3 Examples......Page 111
  4.4 Main Result......Page 116
  5.1 Definitions......Page 120
  5.2 Main Result and Illustrations......Page 121
 6 Conclusion......Page 133
 References......Page 134
Newton-Like Dynamics Associated to Nonconvex OptimizationProblems......Page 136
 1 Introduction and Preliminaries......Page 137
 2 Asymptotic Analysis......Page 140
 3 Convergence of the Trajectory When the Objective Function Satisfies the Kurdyka-Łojasiewicz Property......Page 144
 4 Convergence Rates......Page 149
 References......Page 152