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DC Semidefinite Programming and Cone Constrained DC Optimization: Theory and Local Search Methods

M. V. Dolgopolik

Abstract

In this paper, we study possible extensions of the main ideas and methods of constrained DC optimization to the case of nonlinear semidefinite programming problems and more general nonlinear and nonsmooth cone constrained optimization problems. In the first part of the paper, we analyse two different approaches to the definition of DC matrix-valued functions (namely, order-theoretic and componentwise), study some properties of convex and DC matrix-valued mappings and demonstrate how to compute DC decompositions of some nonlinear semidefinite constraints appearing in applications. We also compute a DC decomposition of the maximal eigenvalue of a DC matrix-valued function. This DC decomposition can be used to reformulate DC semidefinite constraints as DC inequality constrains. Finally, we study local optimality conditions for general cone constrained DC optimization problems. The second part of the paper is devoted to a detailed convergence analysis of two extensions of the well-known DCA method for solving DC (Difference of Convex functions) optimization problems to the case of general cone constrained DC optimization problems. We study the global convergence of the DCA for cone constrained problems and present a comprehensive analysis of a version of the DCA utilizing exact penalty functions. In particular, we study the exactness property of the penalized convex subproblems and provide two types of sufficient conditions for the convergence of the exact penalty method to a feasible and critical point of a cone constrained DC optimization problem from an infeasible starting point. In the numerical section of this work, the exact penalty DCA is applied to the problem of computing compressed modes for variational problems and the sphere packing problem on Grassmannian.

DC Semidefinite Programming and Cone Constrained DC Optimization: Theory and Local Search Methods

Abstract

In this paper, we study possible extensions of the main ideas and methods of constrained DC optimization to the case of nonlinear semidefinite programming problems and more general nonlinear and nonsmooth cone constrained optimization problems. In the first part of the paper, we analyse two different approaches to the definition of DC matrix-valued functions (namely, order-theoretic and componentwise), study some properties of convex and DC matrix-valued mappings and demonstrate how to compute DC decompositions of some nonlinear semidefinite constraints appearing in applications. We also compute a DC decomposition of the maximal eigenvalue of a DC matrix-valued function. This DC decomposition can be used to reformulate DC semidefinite constraints as DC inequality constrains. Finally, we study local optimality conditions for general cone constrained DC optimization problems. The second part of the paper is devoted to a detailed convergence analysis of two extensions of the well-known DCA method for solving DC (Difference of Convex functions) optimization problems to the case of general cone constrained DC optimization problems. We study the global convergence of the DCA for cone constrained problems and present a comprehensive analysis of a version of the DCA utilizing exact penalty functions. In particular, we study the exactness property of the penalized convex subproblems and provide two types of sufficient conditions for the convergence of the exact penalty method to a feasible and critical point of a cone constrained DC optimization problem from an infeasible starting point. In the numerical section of this work, the exact penalty DCA is applied to the problem of computing compressed modes for variational problems and the sphere packing problem on Grassmannian.

Paper Structure

This paper contains 14 sections, 21 theorems, 175 equations, 1 figure, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Two Approaches to DC Matrix-Valued Functions
  3. DC Structure of the Maximal Eigenvalue Function
  4. Cone Constrained DC Optimization
  5. Some Properties of Convex Mappings
  6. Optimality Conditions
  7. DCA for Cone Constrained DC Optimization
  8. DCA2/The Penalty Convex-Concave Procedure
  9. Exactness of the Convex Subproblem
  10. Two Approaches to Convergence Analysis
  11. Numerical experiments
  12. Compressed modes for variational problems
  13. Sphere packing on Grassmannian
  14. Conclusions

Key Result

Theorem \oldthetheorem

Let a map $F \colon \mathbb{R}^d \to \mathbb{S}^{\ell}$ be twice continuously differentiable and suppose that there exists $M > 0$ such that $\| \nabla^2 F_{ij}(x) \|_F \le M$ for all $i, j \in \{ 1, \ldots, \ell \}$. Then the mapping $F$ is DC and for any $\mu \ge \ell M$ both pairs $(G_k, H_k)$, $ and are DC decompositions of $F$.

Figures (1)

  • Figure 1: The solutions of problem \ref{['prob:CompressedModes']} (the columns of matrix $\Psi$) with $\nu = 0$ (left figure) and $\nu = 0.2$ (right figure) corresponding to the best value of the objective function computed in our numerical experiments.

Theorems & Definitions (62)

  • Example 1
  • Theorem \oldthetheorem
  • proof
  • Definition 1
  • Example 2
  • Theorem \oldthetheorem
  • proof
  • Corollary 1
  • Corollary 2: Busemann-Feller-Aleksandrov theorem for matrix-va-lued functions
  • Remark 1
  • ...and 52 more