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Program."],"published-print":{"date-parts":[[2026,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We consider the complex cut polytope: the convex hull of Hermitian rank\u00a01 matrices\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$xx^\\textrm{H}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mtext>H<\/mml:mtext>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , where the elements of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$x \\in \\mathbb {C}^n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mi>C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    are\n                    <jats:italic>m<\/jats:italic>\n                    th unit roots. These polytopes have applications in MAX-3-CUT, digital communication technology, angular synchronization and more generally, complex quadratic programming. For\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$m=2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , the complex cut polytope corresponds to the well-known cut polytope. We generalize valid cuts for this polytope to cuts for any complex cut polytope with finite\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$m&gt;2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and provide a framework to compare them. Further, we consider a second semidefinite lifting of the complex cut polytope for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$m=\\infty $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>\u221e<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . This lifting is proven to be equivalent to other complex Lasserre-type liftings of the same order proposed in the literature, while being of smaller size. Our theoretical findings are supported by numerical experiments on various optimization problems.\n                  <\/jats:p>","DOI":"10.1007\/s10107-024-02147-3","type":"journal-article","created":{"date-parts":[[2024,10,9]],"date-time":"2024-10-09T11:14:11Z","timestamp":1728472451000},"page":"119-168","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Cuts and semidefinite liftings for the complex cut polytope"],"prefix":"10.1007","volume":"218","author":[{"given":"Lennart","family":"Sinjorgo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3298-7255","authenticated-orcid":false,"given":"Renata","family":"Sotirov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Miguel F.","family":"Anjos","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,10,9]]},"reference":[{"issue":"1\u20132","key":"2147_CR1","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1007\/s10107-009-0304-7","volume":"128","author":"W Ai","year":"2011","unstructured":"Ai, W., Huang, Y., Zhang, S.: New results on Hermitian matrix rank-one decomposition. 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