{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T00:29:46Z","timestamp":1778718586407,"version":"3.51.4"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"288","license":[{"start":{"date-parts":[[2014,11,7]],"date-time":"2014-11-07T00:00:00Z","timestamp":1415318400000},"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, the globally uniquely solvable (GUS) property of the linear transformation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M element-of double-struck upper R Superscript n times n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>\n                                  \u00d7\n                                  \n                                <\/mml:mo>\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">M\\in \\mathbb {R}^{n\\times n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the second-order cone linear complementarity problem (SOCLCP) receives much attention and has been studied substantially. Yang and Yuan contributed a new characterization of the GUS property of the linear transformation, which is formulated by basic linear-algebra-related properties. In this paper, we consider efficient numerical algorithms to solve the SOCLCP where the linear transformation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M\">\n                        <mml:semantics>\n                          <mml:mi>M<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">M<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    has the GUS property. By closely relying on the new characterization of the GUS property, a globally convergent bisection method is developed in which each iteration can be implemented using only\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 n squared\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2n^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    flops. Moreover, we also propose an efficient Newton method to accelerate the bisection algorithm. An attractive feature of this Newton method is that each iteration only requires\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"5 n squared\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>5<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">5n^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    flops and converges quadratically. These two approaches make good use of the special structure contained in the SOCLCP and can be effectively combined to yield a fast and efficient bisection-Newton method. Numerical testing is carried out and very encouraging computational experiments are reported.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2013-02795-0","type":"journal-article","created":{"date-parts":[[2013,11,7]],"date-time":"2013-11-07T08:56:49Z","timestamp":1383814609000},"page":"1701-1726","source":"Crossref","is-referenced-by-count":9,"title":["An efficient algorithm for second-order cone linear complementarity problems"],"prefix":"10.1090","volume":"83","author":[{"given":"Lei-Hong","family":"Zhang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei Hong","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2013,11,7]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1080\/1055678031000122586","article-title":"Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone","volume":"18","author":"Auslender, Alfred","year":"2003","journal-title":"Optim. 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