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On the Convexity of the Solution Set of Linear Complementarity Problem over Tensor Spaces

Sonali Sharma, V. Vetrivel, Jein-Shan Chen

Abstract

This paper investigates the convexity of the solution set of the linear complementarity problems over tensor spaces (TLCPs). We introduce the notion of a $T$-column sufficient tensor and study its properties and relationships with several structured tensors. An equivalent condition for the convexity of the solution set of the $\mathrm{TLCP}$ is established. In addition, sufficient conditions for uniqueness and for feasibility implying solvability are derived.

On the Convexity of the Solution Set of Linear Complementarity Problem over Tensor Spaces

Abstract

This paper investigates the convexity of the solution set of the linear complementarity problems over tensor spaces (TLCPs). We introduce the notion of a -column sufficient tensor and study its properties and relationships with several structured tensors. An equivalent condition for the convexity of the solution set of the is established. In addition, sufficient conditions for uniqueness and for feasibility implying solvability are derived.

Paper Structure

This paper contains 8 sections, 14 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. T-column sufficient tensor
  4. Properties of the Solution Set of the TLCP(M,Q)
  5. Convexity
  6. Feasibility implies Solvability
  7. Uniqueness
  8. Conclusions

Key Result

Lemma 2.2

Shang19062025 Let $\mathcal{M},\mathcal{N}$ be in $\mathbb{R}^{k_{1}\times\cdots \times k_{m}}$, and $\mathcal{X},\mathcal{Y}$ be $p$-th order and $({k_{m-p+1}\times\cdots \times k_{m})}$-dimensional real tensors ($p \in [m]$), and $\alpha\in\mathbb{R}$. Then the following relations hold: $(\mathcal

Figures (1)

  • Figure 1: Illustration of the results obtained for a $T$-column sufficient tensor.

Theorems & Definitions (38)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Definition 2.8
  • Proposition 2.9
  • Theorem 2.10
  • ...and 28 more