{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:09:27Z","timestamp":1760238567077,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,17]],"date-time":"2020-08-17T00:00:00Z","timestamp":1597622400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004252","name":"Qatar University","doi-asserted-by":"publisher","award":["NCBP-QUCP-CAS-2020-1"],"award-info":[{"award-number":["NCBP-QUCP-CAS-2020-1"]}],"id":[{"id":"10.13039\/501100004252","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different conditions, that are minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints. Then, we discuss cases where the CQRs return the optimal solution of the problem, revealing new conditions under which the underlying problem admits strong Lagrangian duality and enjoys exact semidefinite optimization relaxation. Finally, under the given sufficient conditions, we present necessary and sufficient conditions for global optimality of the problem and obtain a form of S-lemma for a system of two quadratic and a fixed number of linear inequalities.<\/jats:p>","DOI":"10.3390\/sym12081369","type":"journal-article","created":{"date-parts":[[2020,8,17]],"date-time":"2020-08-17T21:58:53Z","timestamp":1597701533000},"page":"1369","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Generalized Trust-Region Sub-Problem with Additional Linear Inequality Constraints\u2014Two Convex Quadratic Relaxations and Strong Duality"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0308-2422","authenticated-orcid":false,"given":"Temadher A.","family":"Almaadeed","sequence":"first","affiliation":[{"name":"Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Akram","family":"Taati","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht 4199613776, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maziar","family":"Salahi","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht 4199613776, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1950-8907","authenticated-orcid":false,"given":"Abdelouahed","family":"Hamdi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Conn, A.R., Gould, N.I.M., and Toint, P.L. (2000). Trust Region Methods.","DOI":"10.1137\/1.9780898719857"},{"key":"ref_2","first-page":"1","article-title":"Sequential quadratic programming","volume":"4","author":"Boggs","year":"1995","journal-title":"Math. Program. Numer."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3384","DOI":"10.1137\/100791841","article-title":"Strong duality in robust convex programming: Complete characterizations","volume":"20","author":"Jeyakumar","year":"2010","journal-title":"SIAM J. Optim."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.orl.2011.02.007","article-title":"A robust von-Neumann minimax theorem for zero-sum games under bounded payoff uncertainty","volume":"39","author":"Jeyakumar","year":"2011","journal-title":"Oper. Res. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1016\/j.orl.2003.12.007","article-title":"Robust linear optimization under general norms","volume":"32","author":"Bertsimas","year":"2004","journal-title":"Oper. Res. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1137\/16M1058200","article-title":"Solving the trust region subproblem by a generalized eigenvalue problem","volume":"27","author":"Adachi","year":"2017","journal-title":"SIAM J. Optim."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1080\/10556780410001647186","article-title":"The trust region subproblem and semidefinite programming","volume":"19","author":"Fortin","year":"2004","journal-title":"Optim. Methods Softw."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1137\/S1052623497322735","article-title":"Solving the trust-region subproblem using the Lanczos method","volume":"9","author":"Gould","year":"1999","journal-title":"SIAM J. Optim."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1137\/0904038","article-title":"Computing a trust region step","volume":"4","author":"Sorensen","year":"1983","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1137\/S105262349928887X","article-title":"A new matrix-free algorithm for the large-scale trust-region subproblem","volume":"11","author":"Rojas","year":"2001","journal-title":"SIAM J. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1137\/S105262340139001X","article-title":"New results on quadratic minimization","volume":"14","author":"Ye","year":"2003","journal-title":"SIAM J. Optim."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"844","DOI":"10.1137\/050644471","article-title":"Strong duality in nonconvex quadratic optimization with two quadratic constraints","volume":"17","author":"Beck","year":"2006","journal-title":"SIAM J. Optim."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1287\/moor.28.2.246.14485","article-title":"On cones of nonnegative quadratic functions","volume":"28","author":"Sturm","year":"2003","journal-title":"Math. Oper. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s10589-016-9867-4","article-title":"Local nonglobal minima for solving large-scale extended trust-region subproblems","volume":"66","author":"Salahi","year":"2017","journal-title":"Comput. Optim. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s10107-013-0716-2","article-title":"Trust-region problems with linear inequality constraints: Exact SDP relaxation, global optimality and robust optimization","volume":"147","author":"Jeyakumar","year":"2014","journal-title":"Math. Program."},{"key":"ref_16","unstructured":"Hsia, Y., and Sheu, R. (2013). Trust region subproblem with a fixed number of additional linear inequality constraints has polynomial complexity. arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1485","DOI":"10.1137\/16M1065197","article-title":"A Second-order cone based approach for solving the trust-region subproblem and its variants","volume":"27","year":"2017","journal-title":"SIAM J. Optim."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1007\/s10107-017-1145-4","article-title":"SOCP reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices","volume":"169","author":"Jiang","year":"2018","journal-title":"Math. Program."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/BF02592331","article-title":"Hidden convexity in some nonconvex quadratically constrained quadratic programming","volume":"72","author":"Teboulle","year":"1996","journal-title":"Math. Program."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s10898-010-9625-6","article-title":"Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint","volume":"54","author":"Feng","year":"2012","journal-title":"J. Glob. Optim."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1080\/10556789308805542","article-title":"Generalizations of the trust region problem","volume":"2","year":"1993","journal-title":"Optim. Methods Softw."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/s10589-013-9635-7","article-title":"The generalized trust region subproblem","volume":"58","author":"Pong","year":"2014","journal-title":"Comput. Optim. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s10107-017-1206-8","article-title":"Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint","volume":"173","author":"Adachi","year":"2019","journal-title":"Math. Program."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/s10589-019-00105-w","article-title":"A conjugate gradient-based algorithm for large-scale quadratic programming problem with one quadratic constraint","volume":"74","author":"Taati","year":"2019","journal-title":"Comput. Optim. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s40314-016-0349-1","article-title":"An efficient algorithm for solving the generalized trust region subproblem","volume":"37","author":"Salahi","year":"2018","journal-title":"Comput. Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10107-013-0710-8","article-title":"Hidden conic quadratic representation of some nonconvex quadratic optimization problems","volume":"143","author":"Hertog","year":"2014","journal-title":"Math. Program."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.orl.2014.12.002","article-title":"Some results for quadratic problems with one or two quadratic constraints","volume":"43","author":"Locatelli","year":"2015","journal-title":"Oper. Res. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.1007\/s11590-016-1001-0","article-title":"Exactness conditions for an SDP relaxation of the extended trust region problem","volume":"10","author":"Locatelli","year":"2016","journal-title":"Optim. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1603","DOI":"10.1137\/18M1174313","article-title":"Novel reformulations and efficient algorithms for the generalized trust region subproblem","volume":"29","author":"Jiang","year":"2019","journal-title":"SIAM J. Optim."},{"key":"ref_30","unstructured":"Grant, M., and Boyd, S. (2020, June 15). CVX: Matlab Software for Disciplined Convex Programming, Version 2.1. Available online: http:\/\/cvxr.com\/cvx."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1137\/S003614450444556X","article-title":"Canonical forms for hermitian matrix pairs under strict equivalence and congruence","volume":"47","author":"Lancaster","year":"2005","journal-title":"SIAM Rev."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Ben-Tal, A., and Nemirovski, A. (2001). Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications.","DOI":"10.1137\/1.9780898718829"},{"key":"ref_33","first-page":"461","article-title":"A revisit to quadratic programming with one inequality quadratic constraint via matrix pencil","volume":"10","author":"Hsia","year":"2014","journal-title":"Pac. J. Optim."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1369\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:02:07Z","timestamp":1760176927000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1369"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,17]]},"references-count":33,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["sym12081369"],"URL":"https:\/\/doi.org\/10.3390\/sym12081369","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,8,17]]}}}