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Intro; Preface; Contents; A Collection of Nonsmooth Riemannian Optimization Problems; 1 Introduction; 2 Sparse PCA; 3 Secant-Based Dimensionality Reduction; 4 Economic Load Dispatch; 5 Range-Based Independent Component Analysis; 6 Sphere Packing on Manifolds; 7 Robust Low-Rank Matrix Completion; 8 Finding the Sparsest Vector in a Subspace; 9 Restoring Manifold-Valued Images; 10 Oriented Bounding Box; 11 Conclusion; References; An Approximate ADMM for Solving Linearly Constrained Nonsmooth Optimization Problems with Two Blocks of Variables; 1 Introduction; 2 Spherical Subdifferentials
1 Introduction2 A Fast Overview of VU-Theory; 2.1 The U-Lagrangian; 2.2 Fast Tracks; 2.3 Smooth Manifolds and Partial Smoothness; 3 Approximating the V-Subspace; 3.1 Semicontinuity Notions; 3.2 Subdifferential Enlargements; 4 Impact of the Chosen Enlargement; 4.1 The -Subdifferential Enlargement; 4.2 A Separable Enlargement; 4.3 A Not-So-Large Enlargement; 4.4 The Enlargement as a Stabilization Device of the VU-Scheme; 4.4.1 Exact VU-Approach; 4.4.2 Exact V-Step with U-Step; 4.4.3 Approximating Both Steps; 4.4.4 Implicit V Step; 4.4.5 Explicit V Step; 4.4.6 Iteration Update for All the Cases
5 Maximum of Convex Functions5.1 -Activity Sets; 5.2 Polyhedral Functions: Enlargement and VU-Subspaces; 5.3 Polyhedral Functions: Manifold and Manifold Relaxation; 5.3.1 Illustration on a Simple Example; 5.4 General Max-Functions: Enlargement and VU-Subspaces; 6 Sublinear Functions; 6.1 Enlargement and VU-Subspaces; 6.2 A Variety of Sublinear Functions; 6.2.1 The Absolute Value Function; 6.2.2 Finite Valued Sublinear Polyhedral Functions; 6.2.3 The Maximum Eigenvalue Function; 7 Concluding Remarks; References
Proximal Mappings and Moreau Envelopes of Single-Variable Convex Piecewise Cubic Functions and Multivariable GaugeFunctions1 Introduction; 2 Preliminaries; 2.1 Notation; 2.2 Definitions and Facts; 3 Properties of the Moreau Envelope of Convex Functions; 3.1 The Set of Convex Moreau Envelopes; 3.2 Characterizations of the Moreau Envelope; 3.3 Differentiability of the Moreau Envelope; 3.4 Moreau Envelopes of Piecewise Differentiable Functions; 4 The Moreau Envelope of Piecewise Cubic Functions; 4.1 Motivation; 4.2 Convexity; 4.3 Examples; 4.4 Main Result
1 Introduction2 A Fast Overview of VU-Theory; 2.1 The U-Lagrangian; 2.2 Fast Tracks; 2.3 Smooth Manifolds and Partial Smoothness; 3 Approximating the V-Subspace; 3.1 Semicontinuity Notions; 3.2 Subdifferential Enlargements; 4 Impact of the Chosen Enlargement; 4.1 The -Subdifferential Enlargement; 4.2 A Separable Enlargement; 4.3 A Not-So-Large Enlargement; 4.4 The Enlargement as a Stabilization Device of the VU-Scheme; 4.4.1 Exact VU-Approach; 4.4.2 Exact V-Step with U-Step; 4.4.3 Approximating Both Steps; 4.4.4 Implicit V Step; 4.4.5 Explicit V Step; 4.4.6 Iteration Update for All the Cases
5 Maximum of Convex Functions5.1 -Activity Sets; 5.2 Polyhedral Functions: Enlargement and VU-Subspaces; 5.3 Polyhedral Functions: Manifold and Manifold Relaxation; 5.3.1 Illustration on a Simple Example; 5.4 General Max-Functions: Enlargement and VU-Subspaces; 6 Sublinear Functions; 6.1 Enlargement and VU-Subspaces; 6.2 A Variety of Sublinear Functions; 6.2.1 The Absolute Value Function; 6.2.2 Finite Valued Sublinear Polyhedral Functions; 6.2.3 The Maximum Eigenvalue Function; 7 Concluding Remarks; References
Proximal Mappings and Moreau Envelopes of Single-Variable Convex Piecewise Cubic Functions and Multivariable GaugeFunctions1 Introduction; 2 Preliminaries; 2.1 Notation; 2.2 Definitions and Facts; 3 Properties of the Moreau Envelope of Convex Functions; 3.1 The Set of Convex Moreau Envelopes; 3.2 Characterizations of the Moreau Envelope; 3.3 Differentiability of the Moreau Envelope; 3.4 Moreau Envelopes of Piecewise Differentiable Functions; 4 The Moreau Envelope of Piecewise Cubic Functions; 4.1 Motivation; 4.2 Convexity; 4.3 Examples; 4.4 Main Result