Abstract
We first discuss some properties of the solution set of a pseudomonotone second-order cone linear complementarity problem (SOCLCP), and then analyse the limiting behavior of a sequence of strictly feasible solutions within a new wide neighborhood of the central trajectory for the pseudomonotone SOCLCP under assumptions of strict complementarity. Based on this, we derive four different characterizations of an error bound for the pseudomonotone SOCLCP.
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References
Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program., Ser. B 95(1), 3–51 (2003)
Baes, M.: Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras. Linear Algebra Appl. 422(2–3), 664–700 (2007)
Baes, M., Lin, H.L.: A Lipschitzian error bound for monotone symmetric cone linear complementarity problem. Optimization 64(11), 2395–2416 (2015)
Bai, Y.Q., Wang, G.Q., Roos, C.: Primal-dual interior-point algorithms for second-order cone optimization based on kernel functions. Nonlinear Anal. 70(10), 3584–3602 (2009)
Chua, C.B.: Analyticity of weighted central path and error bounds for semidefinite programming. Math. Program. 115(2), 239–271 (2008)
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. I. Springer, New York (2003)
Faraut, J., Koranyi, A.: Analysis on Symmetric Cones. Oxford Press, New York (1994)
Faybusovich, L.: Euclidean Jordan algebras and interior-point algorithms. Positivity 1(4), 331–357 (1997)
Faybusovich, L.: Linear systems in Jordan algebras and primal-dual interior-point algorithms. J. Comput. Appl. Math. 86(1), 149–175 (1997)
Feng, Z.Z., Fang, L., He, G.P.: An \(O(\sqrt{n}L)\) iteration primal-dual path-following method, based on wide neighbourhood and large update, for second-order cone programming. Optimization 63(5), 679–691 (2014)
Giannessi, F., Maugeri, A.: Variational Analysis and Applications. Nonconvex Optimization and Its Applications, vol. 79. Springer, New York (2005)
Goldfarb, D., Scheinberg, K.: Product-form Cholesky factorization in interior point methods for second-order cone programming. Math. Program., Ser. A 103(1), 153–179 (2005)
Karamardian, S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18(4), 445–454 (1976)
Lesaja, G., Roos, C.: Kernel-based interior-point methods for monotone linear complementarity problems over symmetric cones. J. Optim. Theory Appl. 150(3), 444–474 (2011)
Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284(1–3), 193–228 (1998)
Lu, Z., Monteiro, R.D.C.: Limiting behavior of the Alizadeh–Haeberly–Overton weighted paths in semidefinite programming. Optim. Methods Softw. 22(5), 849–870 (2007)
Monteiro, R.D.C., Tsuchiya, T.: Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions. Math. Program., Ser. A 88(1), 61–83 (2000)
Pang, J.S.: Error bounds in mathematical programming. Math. Program. 79(1–3), 299–332 (1997)
Peng, J., Roos, C., Terlaky, T.: Primal-dual interior-point methods for second-order conic optimization based on self-regular proximities. SIAM J. Optim. 13(1), 179–203 (2002)
Sayadi Shahraki, M., Mansouri, H., Zangiabadi, M.: Two wide neighborhood interior-point methods for symmetric cone optimization. Comput. Optim. Appl. 68(1), 29–55 (2017)
Sturm, J.F.: Error bounds for linear matrix inequalities. SIAM J. Optim. 10(4), 1228–1248 (2000)
Sturm, J.F., Zhang, S.: On sensitivity of central solutions in semidefinite programming. Math. Program. 90(2), 205–227 (2001)
Sun, H.C., Wang, Y.J., Qi, L.Q.: Global error bound for the generalized linear complementarity problem over a polyhedral cone. J. Optim. Theory Appl. 142(2), 417–429 (2009)
Tao, J.Y.: Pseudomonotonicity and related properties in Euclidean Jordan algebras. Electron. J. Linear Algebra 22, 225–251 (2011)
Tao, J.Y.: Linear complementarity problem with pseudomonotonicity on Euclidean Jordan algebras. J. Optim. Theory Appl. 159(1), 41–56 (2013)
Wang, G.Q., Bai, Y.Q.: A primal-dual interior-point algorithm for second-order cone optimization with full Nesterov–Todd step. Appl. Math. Comput. 215(3), 1047–1061 (2009)
Wang, G.Q., Zhu, D.T.: A class of polynomial interior-point algorithms for the Cartesian \(P_{*}(\kappa )\) second-order cone linear complementarity problem. Nonlinear Anal. 73(12), 3705–3722 (2010)
Yang, W.H., Zhang, L.H., Shen, C.G.: Solution analysis for the pseudomonotone second-order cone linear complementarity problem. Optimization 65(9), 1703–1715 (2016)
Yang, W.H., Zhang, L.H., Shen, C.G.: On the range of the pseudomonotone second-order cone linear complementarity problem. J. Optim. Theory Appl. 173(2), 504–522 (2017)
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China (11526053), the Natural Science Foundation of Fujian Province of China (2016J05003), and the Foundation of the Education Department of Fujian Province of China (JA15106).
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Supported by Foundation of the Education Department of Fujian Province (No. JA15106), Natural Science Foundation of Fujian Province (No. 2016J05003) and National Natural Science Foundations of China (No. 11301080).
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Lin, H. Conditions for Error Bounds of Linear Complementarity Problems over Second-Order Cones with Pseudomonotonicity. Acta Appl Math 156, 159–176 (2018). https://doi.org/10.1007/s10440-018-0158-1
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DOI: https://doi.org/10.1007/s10440-018-0158-1