Abstract
This paper considers the characterization and computation of sparse solutions and least-p-norm \((0<p<1)\) solutions of the linear complementarity problem \(\hbox {LCP}(q,M)\). We show that the number of non-zero entries of any least-p-norm solution of the \(\hbox {LCP}(q,M)\) is less than or equal to the rank of M for any arbitrary matrix M and any number \(p\in (0,1)\), and there is \(\bar{p}\in (0,1)\) such that all least-p-norm solutions for \(p\in (0, \bar{p})\) are sparse solutions. Moreover, we provide conditions on M such that a sparse solution can be found by solving convex minimization. Applications to the problem of portfolio selection within the Markowitz mean-variance framework are discussed.
This is a preview of subscription content, log in via an institution to check access.
Notes
When \(p\in [0,1), \Vert x\Vert _p\) is only a pseudo norm since it fails to satisfy the triangle inequality (and thus convexity). For simplicity, without the confusion, we call \(\Vert x\Vert _p\) a norm.
References
Adler, I., Verma, S.: The Linear Complementarity Problem, Lemke Algorithm, Perturbation, and the Complexity Class PPAD, Industrial Engineering and Operations Research. University of California Berkeley, Berkeley (2011)
Bruckstein, A.M., Donoho, D.L., Elad, M.: From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51, 34–81 (2009)
Brodie, J., Daubechies, I., De Mol, C., Giannone, D., Loris, I.: Sparse and stable Markowitz portfolios. Proc. Natl. Acad. Sci. 106, 12267–12272 (2009)
Candes, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52, 489–509 (2006)
Candes, E., Tao, T.: Decoding by linear programming. IEEE Trans. Inform. Theory 51, 4203–4215 (2005)
Cesarone, F., Scozzari, A., Tardella, F.: Efficient algorithms for mean-variance portfolio optimization with hard real-world constraints. Giornale dell’Istituto Italiano degli Attuari 72, 37–56 (2009)
Chen, X., Ge, D., Wang, Z., Ye, Y.: Complexity of unconstrained \(L_2\)-\(L_p\) minimization. Math. Program. 143, 371–383 (2014)
Chen, X., Xiang, S.: Implicit solution function of \(P_0\) and \(Z\) matrix linear complementarity constraints. Math. Program. 128, 1–18 (2011)
Chen, X., Xiang, S.: Newton iterations in implicit time-stepping scheme for differential linear complementarity systems. Math. Program. 138, 579–606 (2013)
Chen, X., Xu, F., Ye, Y.: Lower bound theory of nonzero entries in solutions of \(l_2\)-\(l_p\) minimization. SIAM J. Sci. Comput. 32, 2832–2852 (2010)
Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston (1992)
Cottle, R.W., Pang, J.-S., Venkateswaran, V.: Sufficient matrices and the linear complementarlty problem. Linear Algebra Appl. 114(115), 231–249 (1989)
Donoho, D.L., Elad, M.: Optimally sparse representation in general (non-orthogonal) dictionaries via \(L_1\) minimization. Proc. Natl. Acad. Sci. 100, 2197–2202 (2003)
Ferris, M.C., Pang, J.-S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)
Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. Springer, Basel (2013)
Ge, D., Jiang, X., Ye, Y.: A note on the complexity of \(L_p\) minimization. Math. Program. 129, 285–299 (2011)
Han, L., Tiwari, A., Camlibel, M.K., Pang, J.-S.: Convergence of time-stepping schemes for passive and extended linear complementarity systems. SIAM J. Numer. Anal. 47, 3768–3796 (2009)
Ingleton, A.W.: A problem in linear inequalities. Proc. Lond. Math. Soc. 16, 519–536 (1966)
Markowitz, H.M.: Portfolio Selection: Efficient Diversification of Investments. Wiley, New York (1959)
Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24, 227–234 (1995)
Xu, S.: On local w-uniqueness of solutions to linear complementarity problem. Linear Algebra Appl. 290, 23–29 (1999)
Ye, Y.: On homogeneous and self-dual algorithms for LCP. Math. Program. 76, 211–221 (1996)
Acknowledgments
The authors are grateful for the comments of two very knowledgeable and thorough referees.
Author information
Authors and Affiliations
Corresponding author
Additional information
Xiaojun Chen’s work is supported partly by Hong Kong Research Grant Council grant PolyU5003/11p.
Shuhuang Xiang’s work is supported partly by NSF of China (No.11371376), the Innovation-Driven Project and Mathematics and Interdisciplinary Sciences Project of Central South University.
Rights and permissions
About this article
Cite this article
Chen, X., Xiang, S. Sparse solutions of linear complementarity problems. Math. Program. 159, 539–556 (2016). https://doi.org/10.1007/s10107-015-0950-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-015-0950-x