{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:13:57Z","timestamp":1760213637748,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2016,12,6]],"date-time":"2016-12-06T00:00:00Z","timestamp":1480982400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Our purpose of this paper is to solve a class of stochastic linear complementarity problems (SLCP) with finitely many elements. Based on a new stochastic linear complementarity problem function, a new semi-smooth least squares reformulation of the stochastic linear complementarity problem is introduced. For solving the semi-smooth least squares reformulation, we propose a feasible nonsmooth Levenberg\u2013Marquardt-type method. The global convergence properties of the nonsmooth Levenberg\u2013Marquardt-type method are also presented. Finally, the related numerical results illustrate that the proposed method is efficient for the related refinery production problem and the large-scale stochastic linear complementarity problems.<\/jats:p>","DOI":"10.3390\/a9040083","type":"journal-article","created":{"date-parts":[[2016,12,6]],"date-time":"2016-12-06T10:07:17Z","timestamp":1481018837000},"page":"83","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Nonsmooth Levenberg-Marquardt Type Method for Solving a Class of Stochastic Linear Complementarity Problems with Finitely Many Elements"],"prefix":"10.3390","volume":"9","author":[{"given":"Zhimin","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Qingdao University, 308 Qingdao Ningxia Road, Qingdao 266071, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3854-3479","authenticated-orcid":false,"given":"Shouqiang","family":"Du","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Qingdao University, 308 Qingdao Ningxia Road, Qingdao 266071, China"}]},{"given":"Ruiying","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Qingdao University, 308 Qingdao Ningxia Road, Qingdao 266071, China"}]}],"member":"1968","published-online":{"date-parts":[[2016,12,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s101070050024","article-title":"Sample-path solution of stochastic variational inequalities","volume":"84","author":"Robinson","year":"1999","journal-title":"Math. Progr."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1022","DOI":"10.1287\/moor.1050.0160","article-title":"Expected residual minimization method for stochastic linear complementarity problems","volume":"30","author":"Chen","year":"2005","journal-title":"Math. Oper. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1016\/j.orl.2008.01.010","article-title":"The SC1 property of an expected residual function arising from stochastic complementarity problems","volume":"36","author":"Ling","year":"2008","journal-title":"Oper. Res. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1016\/j.ejor.2007.12.046","article-title":"Solving stochastic complementarity problems in energy market modeling using scenario reduction","volume":"197","author":"Gabriel","year":"2009","journal-title":"Eur. J. Oper. Res."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Facchinei, F., and Pang, J.S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems. I and II, Spring.","DOI":"10.1007\/b97544"},{"key":"ref_6","unstructured":"Cottle, R.W., Pang, J.S., and Stone, R.E. (1992). The Liner Complementarity Problem, Academic Press."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1007\/s10957-009-9606-4","article-title":"Stochastic Nonlinear Complementarity Problems: Stochastic Programming Reformulation and Penalty Based Approximation Method","volume":"144","author":"Wang","year":"2010","journal-title":"J. Optim. Theory Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1007\/s10957-009-9534-3","article-title":"Convergence Results of the ERM Method for Nonlinear Stochastic Variational Inequality Problems","volume":"142","author":"Luo","year":"2009","journal-title":"J. Optim. Theory Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s10957-008-9439-6","article-title":"Expected Residual Minimization Method for Stochastic Variational Inequality Problems","volume":"140","author":"Luo","year":"2009","journal-title":"J. Optim. Theory Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s10107-007-0163-z","article-title":"Robust solution of monotone stochastic linear complementarity problems","volume":"117","author":"Chen","year":"2009","journal-title":"Math. Progr."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/s10957-008-9358-6","article-title":"Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty","volume":"137","author":"Zhang","year":"2008","journal-title":"J. Optim. Theory Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s10957-008-9406-2","article-title":"Feasible Semismooth Newton Method for a Class of Stochastic Linear Complementarity Problems","volume":"139","author":"Zhou","year":"2008","journal-title":"J. Optim. Theory Appl."},{"key":"ref_13","unstructured":"Kall, P., and Wallace, S.W. (1994). Stochastic Programming, John Wiley & Sons."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"197","DOI":"10.4208\/jcm.1107-m3559","article-title":"A feasible semi-smooth Gauss-Newton method for solving a class of SLCPS","volume":"30","author":"Ma","year":"2012","journal-title":"J. Comput. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1080\/10556780410001683096","article-title":"On a semi-smooth least squares formulation of complementarity problems with gap reduction","volume":"19","author":"Kanzow","year":"2004","journal-title":"Optim. Methods Softw."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/s10589-007-9086-0","article-title":"A family of NCP-functions and a descent method for the nonlinear complementarity problem","volume":"40","author":"Chen","year":"2008","journal-title":"Comput. Optim. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2072","DOI":"10.1016\/j.nonrwa.2008.03.013","article-title":"The convergence of a smoothing damped Gauss-Newton method for nonlinear complementarity problem","volume":"10","author":"Ma","year":"2009","journal-title":"Nonlinear Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1007\/s10898-006-9027-y","article-title":"The semi-smooth-related properties of a merit function and a descent method for the nonlinear complementarity problem","volume":"36","author":"Chen","year":"2006","journal-title":"J. Glob. Optim."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1142\/S0217595907001292","article-title":"On some NCP-functions based on the generalized Fischer-Burmeister function","volume":"24","author":"Chen","year":"2007","journal-title":"Asia-Pac. J. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1080\/10556780701296455","article-title":"Projected filter trust region method for a semi-smooth least squares formulation of mixed complementarity problems","volume":"22","author":"Kanzow","year":"2007","journal-title":"Optim. Methods Softw."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1007\/BF02614395","article-title":"A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems","volume":"76","author":"Facchinei","year":"1997","journal-title":"Math. Progr."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1007\/BF01581275","article-title":"A nonsmooth version of Newton method","volume":"58","author":"Qi","year":"1993","journal-title":"Math. Progr."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s10589-009-9315-9","article-title":"A new class of penalized NCP-function and its properties","volume":"50","author":"Chen","year":"2011","journal-title":"Comput. Optim. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1007\/s101070050101","article-title":"Feasible descent algorithms for mixed complementarity problems","volume":"86","author":"Ferris","year":"1999","journal-title":"Math. Progr."},{"key":"ref_25","first-page":"326","article-title":"Global convergence properties of some iterative methods for linear complementarity problems","volume":"6","author":"Kanzow","year":"1996","journal-title":"Optimization"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1137\/S0895479894273134","article-title":"Some noninterior continuation methods for linear complementarity problems","volume":"17","author":"Kanzow","year":"1996","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1007\/s101070050009","article-title":"The global linear convergence of an infeasible noninterior path-following algorithm for complementarity problems with uniform P-functions","volume":"87","author":"Xu","year":"2000","journal-title":"Math. Progr."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1287\/moor.23.3.719","article-title":"The global linear convergence of a non-interior path-following algorithm for linear complementarity problems","volume":"23","author":"Burke","year":"1998","journal-title":"Math. Oper. Res."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s10898-009-9489-9","article-title":"A new smoothing and regularization Newton method for P0- NCP","volume":"48","author":"Ma","year":"2010","journal-title":"J. Glob. Optim."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/9\/4\/83\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:28:02Z","timestamp":1760210882000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/9\/4\/83"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,6]]},"references-count":29,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,12]]}},"alternative-id":["a9040083"],"URL":"https:\/\/doi.org\/10.3390\/a9040083","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2016,12,6]]}}}