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Optimality Conditions for Sparse Bilinear Least Squares Problems

Zixin Deng, Zheng-Hai Huang, Yun-Bin Zhao

TL;DR

The paper addresses first-order optimality for the sparse bilinear least squares problem $\min f(x,y)=\tfrac{1}{2}\|\mathscr{A}xy-b\|^2$ subject to $e_{\gamma_x}^T x=1$ and sparsity constraints $\|x\|_0\le s$, $\|y\|_0\le t$. It develops a comprehensive variational framework based on Bouligand and Clarke tangent/normal cones to define N-type, T-type, CW, L-like, and M-stationarity, and proves that optimal solutions satisfy these necessary conditions. The work provides exact cone expressions for the feasible set and establishes formal relationships among the stationary notions, including equivalences between Bouligand and Clarke stationarity in key cases. These results lay a rigorous foundation for designing and analyzing algorithms for sparse bilinear problems arising in blind deconvolution, Hammerstein identification, self-calibration, and matrix sensing, by enabling precise first-order checks of optimality and guiding numerical strategies.

Abstract

The first-order optimality conditions of sparse bilinear least squares problems are studied. The so-called T-type and N-type stationary points for this problem are characterized in terms of tangent cone and normal cone in Bouligand and Clarke senses, and another stationarity concept called the coordinate-wise minima is introduced and discussed. Moreover, the L-like stationary point for this problem is introduced and analyzed through the newly introduced concept of like-projection, and the M-stationary point is also investigated via a complementarity-type reformulation of the problem. The relationship between these stationary points is discussed as well. It turns out that all stationary points discussed in this work satisfy the necessary optimality conditions for the sparse bilinear least squares problem.

Optimality Conditions for Sparse Bilinear Least Squares Problems

TL;DR

The paper addresses first-order optimality for the sparse bilinear least squares problem subject to and sparsity constraints , . It develops a comprehensive variational framework based on Bouligand and Clarke tangent/normal cones to define N-type, T-type, CW, L-like, and M-stationarity, and proves that optimal solutions satisfy these necessary conditions. The work provides exact cone expressions for the feasible set and establishes formal relationships among the stationary notions, including equivalences between Bouligand and Clarke stationarity in key cases. These results lay a rigorous foundation for designing and analyzing algorithms for sparse bilinear problems arising in blind deconvolution, Hammerstein identification, self-calibration, and matrix sensing, by enabling precise first-order checks of optimality and guiding numerical strategies.

Abstract

The first-order optimality conditions of sparse bilinear least squares problems are studied. The so-called T-type and N-type stationary points for this problem are characterized in terms of tangent cone and normal cone in Bouligand and Clarke senses, and another stationarity concept called the coordinate-wise minima is introduced and discussed. Moreover, the L-like stationary point for this problem is introduced and analyzed through the newly introduced concept of like-projection, and the M-stationary point is also investigated via a complementarity-type reformulation of the problem. The relationship between these stationary points is discussed as well. It turns out that all stationary points discussed in this work satisfy the necessary optimality conditions for the sparse bilinear least squares problem.
Paper Structure (11 sections, 32 theorems, 137 equations)

This paper contains 11 sections, 32 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First-order optimality conditions
  4. N-type and T-type stationary points
  5. Basic concepts and lemma
  6. First-order optimality conditions by Bouligand tangent and normal cones
  7. First-order optimality conditions by Clarke tangent and normal cones
  8. Coordinate-Wise minimum
  9. $L$-like stationary point
  10. M-stationary point
  11. Conclusion

Key Result

Lemma 3.1

For any $x\in F_1$ defined in (fsbl) and $\left\{x^k\right\}\subset F_1$ satisfying ${\rm lim}_{k\to\infty}x^k=x$, there must exist $k_0\in \mathbb{N}$ such that the index for the first nonzero element of $x^k$ is equal to $\gamma_x$ for all $k>k_0$.

Theorems & Definitions (48)

  • Example 1.1
  • Example 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.1
  • ...and 38 more