Minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint

References

• Fukushima M, Yamamoto Y. A second-order algorithm for continuous-time nonlinear optimal control problems. IEEE Trans Autom Control. 1986;AC-31(7):673–676.
   Web of Science ®Google Scholar
• Liao A. Solving unconstrained discrete-time optimal-control-problems using trust method. Technical Report CTC95 TR230. Ithaca, NY, USA: Advanced Computing Research Institute, Cornell Theory Center; 1995.
   Google Scholar
• Bockmann C. Curve-fitting and identification of physical spectra. J Comput Appl Math. 1996;70(2):207–224.
   Web of Science ®Google Scholar
• Kojima F. Back-propagation learning using the trust region algorithm and application to nondestructive testing in applied electromagnetics. Int J Appl Electromagnet Mater. 1993;4(7):27–33.
   Google Scholar
• Bofill JM. A conjugate-gradient algorithm with a trust region for molecular geometry optimization. J Mol Model. 1995;1(1):11–17.
   Web of Science ®Google Scholar
• Holstad A. Numerical solution of nonlinear equations in chemical speciation calculations. Comput Geosci. 1999;3(3–4):229–257.
   Google Scholar
• Jensen JJA, Agren H. A direct, restricted-step, second-order MC SCF program for large scale ab initio calculations. Chem Phys. 1986;104(2):229–250.
   Web of Science ®Google Scholar
• Furey BP. A sequential quadratic programming-based algorithm for optimization of gas networks. Automatica. 1993;29(6):1439–1450.
   Web of Science ®Google Scholar
• Lewis RM. A trust region framework for managing approximation models in engineering optimization. AIAA Paper 96-4101, presented at the Sixth AIAA/NASA/ISSMO Symposium on Multidisplinary Analysis and Design, Bellevue, WA, USA, 1996.
   Google Scholar
• Shiina T. Numerical solution technique for joint chance-constrained programming problem -- an application to electric power capacity expansion. J Oper Res Soc Jpn. 1999;42(2):128–140.
   Google Scholar
• Allen DM. Tailoring nonlinear least squares algorithms for the analysis of compartment models. In: Sail J, Lehman A, editors. Computationally Intensive Statistical Methods. Proceedings of the 26th Symposium on the Interface, Fairfax Station, VA, USA; Vol. 26, 1995. p. 533–535.
   Google Scholar
• Budil DE, Lee S, Saxena S, Freed JH. Nonlinear-least-squares analysis of slow-motion EPR-spectra in one and two dimensions using a modified Levenberg-Marquardt algorithm. J Magn Reson Ser A. 1996;120(2):155–189.
   Web of Science ®Google Scholar
• Studer G. Risk measurement with maximum loss. Math Methods Oper Res. 1999;50(1):121–134.
   Web of Science ®Google Scholar
• Di S, Sun WY. A trust region method for conic model to solve unconstrained optimization. Optim Methods Softw. 1996;6:237–263.
   Web of Science ®Google Scholar
• Moré J. Generalizatons of the trust region problem. Optim Methods Softw. 1993;2:189–209.
   Google Scholar
• Ben-Tal A, Teboulle M. Hidden convexity in some nonconvex quadratically constrained quadratic programming. Math Program. 1996;72:51–63.
   Web of Science ®Google Scholar
• Jeyakumar V, Rubinov AM, Wu ZY. Non-convex quadratic minimization problemswith quadratic constraints: Global optimality conditions. Math Program Ser A. 2007;110:521–541.
   Web of Science ®Google Scholar
• Jeyakumar V, Srisatkunarajah S. Lagrange multiplier necessary conditions for global optimality for non-convex minimization over a quadratic constraint via S-lemma. Optim Lett. 2009;3(1):23–33.
   Web of Science ®Google Scholar
• Stern RJ, Wolkowicz H. Indefinite trust region subproblems and nonsymmetric perturbations. SIAM J Optim. 1995;5(2):286–313.
   Web of Science ®Google Scholar
• Jin Q, Fang SC, Xing W. On the global optimality of generalized trust region subproblems. Optimization. 2010;59(8):1139–1151.
   Web of Science ®Google Scholar
• Feng J, Lin G, Sheu RL, Xia Y. Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint. J Global Optim. 2012;54:275–293.
   Web of Science ®Google Scholar
• Ben-Tal A, den Hertog D. Hidden conic quadratic representation of some nonconvex quadratic optimization problems. Math Program Ser A. 2014;143:1–29.
   Web of Science ®Google Scholar
• Pólik I, Terlaky T. A survey of the S-lemma. SIAM Rev. 2007;49:371–418.
   Web of Science ®Google Scholar
• Nesterov YuE. Semidefinite relaxation and nonconvex quadratic optimization. Optim Methods Softw. 1998;9:141–160.
   Web of Science ®Google Scholar
• Salahi M. Convex optimization approach to a single quadratically constrained quadratic minimization problem. Central Eur J Oper Res. 2010;18(2):181–187.
   Web of Science ®Google Scholar
• Ye Y. Approximating quadratic programming with bound and quadratic constraints. Math Program. 1999;84:219–226.
   Web of Science ®Google Scholar
• Pong TK, Wolkowicz H. The generalized trust region subproblem. Comput Optim Appl. 2014;58(2):273–322.
   Web of Science ®Google Scholar
• Uhlig F. Definite and semidefinite matrices in a real symmetric matrix pencil. Pac J Math. 1972;49(2):561–568.
   Web of Science ®Google Scholar
• Bazaraa MS, Sherali HD, Shetty CM. Nonlinear programming. New York (NY): Wiley; 1993.
   Google Scholar
• Ye Y, Zhang S. New results on quadratic minimization. SIAM J Optim. 2003;14:245–267.
   Web of Science ®Google Scholar
• Ben-Tal A, Nemirovski A. Lectures on modern convex optimization. Philadelphia: SIAM; 2001.
   Google Scholar
• Fradkov AL, Yakubovich VA. The S-procedure and the duality relation in convex quadratic programming problems. Vestnik Leningrad Univ. 1973;1:81–87.
   Google Scholar
• Beck A, Eldar YC. Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J Optim. 2006;17:844–860.
   Web of Science ®Google Scholar
• Zheng XJ, Sun XL, Li D, Xu YF. On zero duality gap in nonconvex quadratic programming problems. J Global Optim. 2012;52:229–242.
   Web of Science ®Google Scholar
• Sturm JF, Zhang S. On cones of nonnegative quadratic functions. Math Oper Res. 2003;28:246–267.
   Web of Science ®Google Scholar