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Solution uniqueness of convex optimization problems via the radial cone

Jalal Fadili, Tran T. A. Nghia, Duy Nhat Phan

Abstract

In this paper, we mainly study solution uniqueness of some convex optimization problems. Our characterizations of solution uniqueness are in terms of the radial cone. This approach allows us to know when a unique solution is a strong solution or even a tilt-stable one without checking second-order information. Consequently, we apply our theory to low-rank optimization problems. The radial cone is fully calculated in this case and numerical experiments show that our characterizations are sharp.

Solution uniqueness of convex optimization problems via the radial cone

Abstract

In this paper, we mainly study solution uniqueness of some convex optimization problems. Our characterizations of solution uniqueness are in terms of the radial cone. This approach allows us to know when a unique solution is a strong solution or even a tilt-stable one without checking second-order information. Consequently, we apply our theory to low-rank optimization problems. The radial cone is fully calculated in this case and numerical experiments show that our characterizations are sharp.
Paper Structure (10 sections, 16 theorems, 126 equations, 1 figure)

This paper contains 10 sections, 16 theorems, 126 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Solution uniqueness of convex composite optimization problems
  5. General regularizer
  6. Nuclear norm case
  7. Solution uniqueness of convex optimization problems with linear constraints
  8. General regularizer
  9. Nuclear norm case
  10. Numerical Experiments

Key Result

Theorem 3.1

Let $\bar{x}\in \Phi^{-1}\left({\rm int}\, (\hbox{\rm dom}\, f)\right)\cap \hbox{\rm dom}\, g$ be an optimal solution of problem CP and $\bar{y}\stackrel{\text{\rm\tiny def}}{=}-\Phi^*\nabla f(\Phi\bar{x})\in \partial g(\bar{x})$. Then $\bar{x}$ is the unique solution if and only if

Figures (1)

  • Figure 1: Proportions of cases for which $X_0$ is the unique solution with respect to the number of measurements.

Theorems & Definitions (34)

  • Definition 2.1: Quadratic growth condition
  • Theorem 3.1: Solution uniqueness via the radial cone
  • proof
  • Corollary 3.2: Strong minima and solution uniqueness
  • Remark 3.3: Closedness of the radial cone
  • Lemma 3.4: Subdifferentials of the nuclear norm and its conjugate
  • Lemma 3.5
  • Corollary 3.6: Solution uniqueness of low-rank optimization problems
  • Remark 3.7: Closedness of the radial cone for the nuclear norm
  • Remark 3.8: Checking condition \ref{['NonEq']} via optimization
  • ...and 24 more