{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T18:34:28Z","timestamp":1779906868162,"version":"3.53.1"},"reference-count":36,"publisher":"American Mathematical Society (AMS)","issue":"229","license":[{"start":{"date-parts":[[2000,2,19]],"date-time":"2000-02-19T00:00:00Z","timestamp":950918400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Recently, based upon the Chen-Harker-Kanzow-Smale smoothing function and the trajectory and the neighbourhood techniques, Hotta and Yoshise proposed a noninterior point algorithm for solving the nonlinear complementarity problem. Their algorithm is globally convergent under a relatively mild condition. In this paper, we modify their algorithm and combine it with the superlinear convergence theory for nonlinear equations. We provide a globally linearly convergent result for a slightly updated version of the Hotta-Yoshise algorithm and show that a further modified Hotta-Yoshise algorithm is globally and superlinearly convergent, with a convergence\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q\">\n                        <mml:semantics>\n                          <mml:mi>Q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">Q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1 plus t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">1+t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , under suitable conditions, where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t element-of left-parenthesis 0 comma 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t\\in (0,1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is an additional parameter.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01082-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"283-304","source":"Crossref","is-referenced-by-count":57,"title":["Improving the convergence of non-interior point algorithms for nonlinear complementarity problems"],"prefix":"10.1090","volume":"69","author":[{"given":"Liqun","family":"Qi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Defeng","family":"Sun","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[1999,2,19]]},"reference":[{"key":"1","unstructured":"J. Burke and S. Xu, \u201cThe global linear convergence of a non-interior path-following algorithm for linear complementarity problem\", to appear in Mathematics of Operations Research."},{"key":"2","unstructured":"J. Burke and S. Xu, \u201cA non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem\", Preprint, Department of Mathematics, University of Washington, Seattle, WA 98195, September, 1997."},{"key":"3","doi-asserted-by":"crossref","unstructured":"J. Burke and S. Xu, \u201cA non-interior predictor-corrector path following method for LCP\u201d, in: M. Fukushima and L. Qi, eds., Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publisher, Nowell, Maryland, pp. 45\u201364, 1998.","DOI":"10.1007\/978-1-4757-6388-1_3"},{"key":"4","unstructured":"B. Chen and X. Chen, \u201cA global linear and local quadratic continuation smoothing method for variational inequalities with box constraints\u201d, Preprint, Department of Management and Systems, Washington State University, Pullman, March 1997."},{"key":"5","unstructured":"B. Chen and X. Chen, \u201cA global and local superlinear continuation-smoothing method for \ud835\udc43\u2080+\ud835\udc45\u2080 and monotone NCP\u201d, to appear in SIAM Journal on Optimization."},{"issue":"4","key":"6","doi-asserted-by":"publisher","first-page":"1168","DOI":"10.1137\/0614081","article-title":"A non-interior-point continuation method for linear complementarity problems","volume":"14","author":"Chen, Bintong","year":"1993","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/BF01585559","article-title":"A continuation method for monotone variational inequalities","volume":"69","author":"Chen, Bintong","year":"1995","journal-title":"Math. Programming","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5610","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1137\/S1052623495280615","article-title":"Smooth approximations to nonlinear complementarity problems","volume":"7","author":"Chen, Bintong","year":"1997","journal-title":"SIAM J. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/1052-6234","issn-type":"print"},{"key":"9","unstructured":"B. Chen and N. Xiu, \u201cA global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing function\u201d, to appear in SIAM Journal on Optimization."},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/BF01592244","article-title":"Smoothing methods for convex inequalities and linear complementarity problems","volume":"71","author":"Chen, Chun Hui","year":"1995","journal-title":"Math. Programming","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5610","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/BF00249052","article-title":"A class of smoothing functions for nonlinear and mixed complementarity problems","volume":"5","author":"Chen, Chunhui","year":"1996","journal-title":"Comput. Optim. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0926-6003","issn-type":"print"},{"issue":"222","key":"12","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1090\/S0025-5718-98-00932-6","article-title":"Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities","volume":"67","author":"Chen, X.","year":"1998","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"13","unstructured":"X. Chen and Y. Ye, \u201cOn homotopy-smoothing methods for variational inequalities\u201d, to appear in SIAM Journal on Control and Optimization."},{"key":"14","series-title":"Canadian Mathematical Society Series of Monographs and Advanced Texts","isbn-type":"print","volume-title":"Optimization and nonsmooth analysis","author":"Clarke, Frank H.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/047187504X"},{"key":"15","unstructured":"F. Facchinei, H. Jiang, and L. Qi, \u201cA smoothing method for mathematical programming with equilibrium constraints\u201d, to appear in Mathematical Programming."},{"key":"16","doi-asserted-by":"crossref","unstructured":"M.C. Ferris and J.-S. Pang, \u201cEngineering and economic applications of complementarity problems\u201d, SIAM Review, 39 (1997), 669\u2013713.","DOI":"10.1137\/S0036144595285963"},{"key":"17","doi-asserted-by":"crossref","unstructured":"M. Fukushima, Z.-Q. Luo, and J.-S. Pang, \u201cA globally convergent sequential quadratic programming algorithm for mathematical programming problems with linear complementarity constraints\u201d, Computational Optimization and Applications, 10 (1998), 5\u201334.","DOI":"10.1023\/A:1018359900133"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1007\/BF01582255","article-title":"Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications","volume":"48","author":"Harker, Patrick T.","year":"1990","journal-title":"Math. Programming","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5610","issn-type":"print"},{"key":"19","unstructured":"K. Hotta and A. Yoshise \u201cGlobal convergence of a class of non-interior-point algorithms using Chen-Harker-Kanzow functions for nonlinear complementarity problems\u201d, Discussion Paper Series No. 708, Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Ibaraki 305, Japan, December 1996."},{"issue":"4","key":"20","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1137\/S0895479894273134","article-title":"Some noninterior continuation methods for linear complementarity problems","volume":"17","author":"Kanzow, Christian","year":"1996","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"key":"21","doi-asserted-by":"crossref","unstructured":"C. Kanzow and H. Jiang, \u201cA continuation method for (strongly) monotone variational inequalities\", Mathematical Programming 81 (1998), 103\u2013125.","DOI":"10.1007\/BF01584847"},{"issue":"4","key":"22","doi-asserted-by":"publisher","first-page":"754","DOI":"10.1287\/moor.16.4.754","article-title":"Homotopy continuation methods for nonlinear complementarity problems","volume":"16","author":"Kojima, Masakazu","year":"1991","journal-title":"Math. Oper. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/0364-765X","issn-type":"print"},{"key":"23","series-title":"Lecture Notes in Computer Science","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-54509-3","volume-title":"A unified approach to interior point algorithms for linear complementarity problems","volume":"538","author":"Kojima, M.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/3540545093"},{"key":"24","volume-title":"Iterative solution of nonlinear equations in several variables","author":"Ortega, J. M.","year":"1970"},{"key":"25","isbn-type":"print","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1007\/978-1-4615-2025-2_6","article-title":"Complementarity problems","author":"Pang, Jong-Shi","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0792331206"},{"issue":"1","key":"26","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1287\/moor.18.1.227","article-title":"Convergence analysis of some algorithms for solving nonsmooth equations","volume":"18","author":"Qi, Li Qun","year":"1993","journal-title":"Math. Oper. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/0364-765X","issn-type":"print"},{"issue":"2","key":"27","doi-asserted-by":"publisher","first-page":"402","DOI":"10.1137\/S036301299223619X","article-title":"A globally convergent successive approximation method for severely nonsmooth equations","volume":"33","author":"Qi, Li Qun","year":"1995","journal-title":"SIAM J. Control Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0363-0129","issn-type":"print"},{"issue":"3","key":"28","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1007\/BF01581275","article-title":"A nonsmooth version of Newton\u2019s method","volume":"58","author":"Qi, Li Qun","year":"1993","journal-title":"Math. Programming","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5610","issn-type":"print"},{"key":"29","isbn-type":"print","first-page":"346","article-title":"Generalized equations","author":"Robinson, Stephen M.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/3540120823"},{"key":"30","isbn-type":"print","first-page":"172","article-title":"Algorithms for solving equations","author":"Smale, Steve","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/0821801104"},{"issue":"2","key":"31","doi-asserted-by":"publisher","first-page":"386","DOI":"10.1137\/S105262349427409X","article-title":"An infeasible path-following method for monotone complementarity problems","volume":"7","author":"Tseng, Paul","year":"1997","journal-title":"SIAM J. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/1052-6234","issn-type":"print"},{"key":"32","doi-asserted-by":"crossref","unstructured":"P. Tseng, \u201cAnalysis of a non-interior continuation method based on Chen-Mangasarian functions for complementarity problems\u201d, in: M. Fukushima and L. Qi, eds., Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publisher, Nowell, Maryland, pp. 381\u2013404, 1998.","DOI":"10.1007\/978-1-4757-6388-1_20"},{"key":"33","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971453","volume-title":"Primal-dual interior-point methods","author":"Wright, Stephen J.","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/089871382X"},{"issue":"4","key":"34","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1287\/moor.21.4.815","article-title":"A superlinear infeasible-interior-point algorithm for monotone complementarity problems","volume":"21","author":"Wright, Stephen","year":"1996","journal-title":"Math. Oper. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/0364-765X","issn-type":"print"},{"key":"35","unstructured":"S. Xu, \u201cThe global linear convergence of an infeasible non-interior path-following algorithm for complementarity problems with uniform \ud835\udc43-functions\", Preprint, Department of Mathematics, University of Washington, Seattle, WA 98195, December 1996."},{"key":"36","unstructured":"S. Xu, \u201cThe global linear convergence and complexity of a non-interior path-following algorithm for monotone LCP based on Chen-Harker-Kanzow-Smale smooth functions\", Preprint, Department of Mathematics, University of Washington, Seattle, WA 98195, February 1997."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2000-69-229\/S0025-5718-99-01082-0\/S0025-5718-99-01082-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-229\/S0025-5718-99-01082-0\/S0025-5718-99-01082-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:12:11Z","timestamp":1776723131000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-229\/S0025-5718-99-01082-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,2,19]]},"references-count":36,"journal-issue":{"issue":"229","published-print":{"date-parts":[[2000,1]]}},"alternative-id":["S0025-5718-99-01082-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01082-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1999,2,19]]}}}