Necessary and Sufficient Conditions for Rank-One-Generated Cones

References

• [1] Agler J, Helton W, McCullough S, Rodman L (1988) Positive semidefinite matrices with a given sparsity pattern. Linear Algebra Its Appl. 107:101–149.Crossref, Google Scholar
• [2] Anstreicher KM, Burer S (2010) Computable representations for convex hulls of low-dimensional quadratic forms. Math. Programming 124(1–2):33–43.Crossref, Google Scholar
• [3] Bao X, Sahinidis NV, Tawarmalani M (2011) Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons. Math. Programming 129(1):129.Crossref, Google Scholar
• [4] Barker GP, Carlson D (1975) Cones of diagonally dominant matrices. Pacific J. Math. 57(1):15–32.Crossref, Google Scholar
• [5] Barvinok A (2002) A Course in Convexity. Graduate Studies in Mathematics, vol. 54 (American Mathematical Society, Providence, RI).Crossref, Google Scholar
• [6] Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust Optimization. Princeton Series in Applied Mathematics, vol. 28 (Princeton University Press, Princeton, NJ).Crossref, Google Scholar
• [7] Blekherman G, Sinn R, Velasco M (2017) Do sums of squares dream of free resolutions? SIAM J. Appl. Algebra Geometry. 1:175–199.Crossref, Google Scholar
• [8] Borwein J, Wolkowicz H (1981) Regularizing the abstract convex program. J. Math. Anal. Appl. 83(2):495–530.Crossref, Google Scholar
• [9] Burer S (2015) A gentle, geometric introduction to copositive optimization. Math. Programming 151:89–116.Crossref, Google Scholar
• [10] Burer S, Ye Y (2019) Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs. Math. Programming 181(4):1–17.Google Scholar
• [11] Ceria S, Soares J (1999) Convex programming for disjunctive convex optimization. Math. Programming 86:595–614.Crossref, Google Scholar
• [12] de Carli Silva MK, Tunçel L (2020) A notion of total dual integrality for convex, semidefinite, and extended formulations. SIAM J. Discrete Math. 34(1):470–496.Crossref, Google Scholar
• [13] Dempster AP (1972) Covariance selection. Biometrics 28(1):157–175.Crossref, Google Scholar
• [14] Dines LL (1941) On the mapping of quadratic forms. Bull. Amer. Math. Soc. 47(6):494–498.Crossref, Google Scholar
• [15] Fradkov AL, Yakubovich VA (1979) The S-procedure and duality relations in nonconvex problems of quadratic programming. Vestnik Leningrad Univ. Math. 6:101–109.Google Scholar
• [16] Frangioni A, Gentile C (2006) Perspective cuts for a class of convex 0–1 mixed integer programs. Math. Programming 106:225–236.Crossref, Google Scholar
• [17] Grone R, Johnson CR, Sá EM, Wolkowicz H (1984) Positive definite completions of partial Hermitian matrices. Linear Algebra Its Appl. 58:109–124.Crossref, Google Scholar
• [18] Günlük O, Linderoth J (2010) Perspective reformulations of mixed integer nonlinear programs with indicator variables. Math. Programming 124:183–205.Crossref, Google Scholar
• [19] Hildebrand R (2016) Spectrahedral cones generated by rank 1 matrices. J. Global Optim. 64(2):349–397.Crossref, Google Scholar
• [20] Kılınç-Karzan F, Wang AL (2021) Exactness in semidefinite program relaxations of quadratically constrained quadratic programs: Theory and applications. Carlsson JG, ed. Emerging Optimization Methods and Modeling Techniques with Applications. INFORMS TutORials in Operations Research (INFORMS, Catonsville, MD), 312–345.Link, Google Scholar
• [21] Laurent M, Poljak S (1995) On a positive semidefinite relaxation of the cut polytope. Linear Algebra Its Appl. 223–224:439–461.Crossref, Google Scholar
• [22] Liu M, Pataki G (2018) Exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming. Math. Programming 167:435–480.Crossref, Google Scholar
• [23] Pataki G (1998) On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Math. Oper. Res. 23(2):339–358.Link, Google Scholar
• [24] Pataki G (2013) Strong duality in conic linear programming: Facial reduction and extended duals. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol. 50 (Springer, New York), 613–634.Crossref, Google Scholar
• [25] Paulsen VI, Power SC, Smith RR (1989) Schur products and matrix completions. J. Functional Anal. 85(1):151–178.Crossref, Google Scholar
• [26] Phan-huy-Hao E (1982) Quadratically constrained quadratic programming: Some applications and a method for solution. Zeitschrift Oper. Res. 26:105–119.Google Scholar
• [27] Rockafellar RT (1970) Convex Analysis. Princeton Mathematical Series, vol. 28 (Princeton University Press, Princeton, NJ).Crossref, Google Scholar
• [28] Shor NZ (1990) Dual quadratic estimates in polynomial and Boolean programming. Ann. Oper. Res. 25(1–4):163–168.Crossref, Google Scholar
• [29] Sturm JF (2000) Error bounds for linear matrix inequalities. SIAM J. Optim. 10(4):1228–1248.Crossref, Google Scholar
• [30] Sturm JF, Zhang S (2003) On cones of nonnegative quadratic functions. Math. Oper. Res. 28(2):246–267.Link, Google Scholar
• [31] Wang AL, Kılınç-Karzan F (2020) A geometric view of SDP exactness in QCQPs and its applications. Preprint, 2021, https://arxiv.org/abs/2011.07155.Google Scholar
• [32] Wang AL, Kılınç-Karzan F (2022) The generalized trust region subproblem: Solution complexity and convex hull results. Math. Programming. 191:445–486.Crossref, Google Scholar
• [33] Wang AL, Kılınç-Karzan F (2022) On the tightness of SDP relaxations of QCQPs. Math. Programming. Forthcoming.Crossref, Google Scholar
• [34] Yakubovich VA (1971) S-procedure in nonlinear control theory. Vestnik Leningrad Univ. Math. 1:62–77.Google Scholar
• [35] Ye Y, Zhang S (2003) New results on quadratic minimization. SIAM J. Optim. 14(1):245–267.Crossref, Google Scholar