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The sparsity problems have attracted a great deal of attention in recent years, which aim to find the sparsest solution of a representation or an equation. In the paper, we mainly study the sparsity of underdetermined linear system via lp minimization for 0<p<1. We show, for a given underdetermined linear system of equations pm×np = p, that although it is not certain that the problem (pp) (i.e., minlx||X||plp subject to pp = b, where  0<p<1 ) generates sparser solutions as the value of p decreases and especially the problem (plp) generates sparser solutions than the problem (p1) (i.e., minlx||X||1 subject to AX = b ), there exists a sparse constant γ(A, p) > 0 such that the following conclusions hold when p < γ(A, b): (1) the problem (pp) generates sparser solution as the value of p decreases; (2) the sparsest optimal solution to the problem (pp) is unique under the sense of absolute value permutation; (3) let X1 and X2 be the sparsest optimal solution to the problems (pp1) and (pp2) , respectively, and let  X1 not be the absolute value permutation of  X2. Then there exist t1,t2 ε [p1,p2]  such that X1 is the sparsest optimal solution to the problem (pt) (∀t ε [p1, t1])  and X2 is the sparsest optimal solution to the problem (pt) (∀t ε (t2, p2])

mathematical problems in engineering

The Sparsity of Underdetermined Linear System via<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>Minimization for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>p</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:math>

The Sparsity of Underdetermined Linear System vialpMinimization for0<p<1