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Program."],"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    There has been growing interest in high-order tensor methods for nonconvex optimization, with adaptive regularization, as they possess better\/optimal worst-case evaluation complexity globally and faster convergence asymptotically. These algorithms crucially rely on repeatedly minimizing nonconvex multivariate Taylor-based polynomial sub-problems, at least locally. Finding efficient techniques for the solution of these sub-problems, beyond the second-order case, has been an open question. This paper proposes a second-order method, Quadratic Quartic Regularisation (QQR), for efficiently minimizing nonconvex quartically-regularized cubic polynomials, such as the AR\n                    <jats:italic>p<\/jats:italic>\n                    sub-problem (Birgin et al. Math Program 163:359\u2013368, 2017) with\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$p=3$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . Inspired by Nesterov (Quartic regularity, 2022), QQR approximates the third-order tensor term by a linear combination of quadratic and quartic terms, yielding (possibly nonconvex) local models that are solvable to global optimality. In order to achieve accuracy\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\epsilon $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03f5<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    in the first-order criticality of the sub-problem in finitely many iterations, we show that the error in the QQR method decreases either linearly or by at least\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {O}(\\epsilon ^{4\/3})$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>\u03f5<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mn>4<\/mml:mn>\n                                <mml:mo>\/<\/mml:mo>\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for locally convex iterations, while in the nonconvex case, by at least\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {O}(\\epsilon )$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03f5<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ; thus improving, on these types of iterations, the general cubic-regularization bound. Preliminary numerical experiments indicate that two QQR variants perform competitively with state-of-the-art approaches such as ARC (also known as AR\n                    <jats:italic>p<\/jats:italic>\n                    with\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$p=2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>p<\/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                    ), achieving either lower objective value or iteration counts.\n                  <\/jats:p>","DOI":"10.1007\/s10107-025-02235-y","type":"journal-article","created":{"date-parts":[[2025,7,31]],"date-time":"2025-07-31T08:59:29Z","timestamp":1753952369000},"page":"669-715","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Second-order methods for quartically-regularised cubic polynomials, with applications to high-order tensor methods"],"prefix":"10.1007","volume":"215","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0963-5550","authenticated-orcid":false,"given":"Coralia","family":"Cartis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenqi","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,7,31]]},"reference":[{"key":"2235_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2024.109808","volume":"452","author":"AA Ahmadi","year":"2024","unstructured":"Ahmadi, A.A., Chaudhry, A., Zhang, J.: Higher-order newton methods with polynomial work per iteration. 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