<p>Sparse optimization problems have drawn wide attention in recent years, which have been extensively applied to compressed sensing, signal and image processings. As a kind of important optimization problem, absolute value equations have been extensively studied. Motivated by the above two aspects, we investigate the sparse solutions of the tensor absolute value equations, which is an NP hard problem due to the nonconvexity and noncontinuity of the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40305_2025_605_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$l_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> norm. By the equivalency of the tensor absolute value equations and the generalized tensor complementarity problems, we transform this problem into solving the sparse solutions of generalized tensor complementarity problems. We make use of the smoothing function of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40305_2025_605_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$l_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> norm to relax <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40305_2025_605_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$l_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> norm. By transforming the complementarity constraints into a fixed point equation, we propose an <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40305_2025_605_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$l_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> regularization minimization model for relaxation and a gradient projection method to solve the transformed problem. Numerical results illustrate that the proposed method can find the sparse solutions of the tensor absolute value equations.</p>

Sparse Solutions to Tensor Absolute Value Equations

期刊论文
发表日期：

2025年6月18日

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Shou-Qiang Du,
Jing-Jing Sun,
Yi-Min Wei

摘要

Sparse optimization problems have drawn wide attention in recent years, which have been extensively applied to compressed sensing, signal and image processings. As a kind of important optimization problem, absolute value equations have been extensively studied. Motivated by the above two aspects, we investigate the sparse solutions of the tensor absolute value equations, which is an NP hard problem due to the nonconvexity and noncontinuity of the $l_0$ $l_{0}$ norm. By the equivalency of the tensor absolute value equations and the generalized tensor complementarity problems, we transform this problem into solving the sparse solutions of generalized tensor complementarity problems. We make use of the smoothing function of $l_1$ $l_{1}$ norm to relax $l_0$ $l_{0}$ norm. By transforming the complementarity constraints into a fixed point equation, we propose an $l_1$ $l_{1}$ regularization minimization model for relaxation and a gradient projection method to solve the transformed problem. Numerical results illustrate that the proposed method can find the sparse solutions of the tensor absolute value equations.