Abstract
This paper extends and completes the discussion by Xing et al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted) about the quadratic programming over one quadratic constraint (QP1QC). In particular, we relax the assumption to cover more general cases when the two matrices from the objective and the constraint functions can be simultaneously diagonalizable via congruence. Under such an assumption, the nonconvex (QP1QC) problem can be solved through a dual approach with no duality gap. This is unusual for general nonconvex programming but we can explain by showing that (QP1QC) is indeed equivalent to a linearly constrained convex problem, which happens to be dual of the dual of itself. Another type of hidden convexity can be also found in the boundarification technique developed in Xing et al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted).
Similar content being viewed by others
References
Ben-Tal A., Teboulle M.: Hidden convexity in some nonconvex quadratically constrained quadratic programming. Math. Program. 72, 51–63 (1996)
Fang S.C., Gao D.Y., Sheu R.L., Wu S.Y.: Canonical dual approach to solve 0-1 quadratic programming problems. J. Ind. Manag. Optim. 4(1), 125–142 (2008)
Fang S.C., Gao D.Y., Sheu R.L., Xing W.: Global optimization for a class of fractional programming problems. J. Global Optim. 45(3), 337–353 (2009)
Fang, S.C., Lin, G.X., Sheu, R.L., Xing, W.: Canonical dual solutions for the double well potential problem (preprint)
Fehmers G.C., Kamp L.P.J., Sluijter F.W.: An algorithm for quadratic optimization with one quadratic constraint and bounds on the variables. Inverse Probl. 14, 893–901 (1998)
Fortin C., Wolkowicz H.: The trust region subproblem and semidefinite programming. Optim. Methods Softw. 19(1), 41–67 (2004)
Gao D.Y.: Canonical dual transformation method and generalized triality theory in nonsmooth global optimization. J. Global Optim. 17, 127–160 (2000)
Gao D.Y.: Canonical duality theory and solutions to constrainted nonconvex quadratic programming. J. Global Optim. 29, 377–399 (2004)
Gay D.M.: Computing optimal locally constrained steps. SIAM J. Sci. Stat. Comput. 2(2), 186–197 (1981)
Golub G.H., Von Matt U.: Quadratically constrained least squares and quadratic problems. Numer. Math. 59, 186–197 (1991)
Hiriart-Urruty J.-B.: Potpourri of conjectures and open questions in nonlinear analysis and optimization. SIAM Rev. 49(2), 255–273 (2007)
Horn R., Johnson C.R.: Matrix analysis. Cambridge University Press, Cambridge (1985)
Jeyakumar V., Srisatkunarajah S.: Lagrange multiplier necessary conditions for global optimality for non-convex minimization over a quadratic constraint via S-lemma. Optim. Lett. 3(1), 23–33 (2009)
Martínez J.M.: Local minimizers of quadratic functions on euclidean balls and spheres. SIAM J. Optim. 4, 159–176 (1994)
More J.J., Sorensen D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4, 553–572 (1983)
Palanthandalam-Madapusi H.J., Van Pelt T.H., Bernstein D.S.: Matrix pencils and existence conditions for quadratic programming with a sign-indefinite quadratic equality constraint. J. Global Optim. 45(4), 533–549 (2009)
Pardalos P.M., Resende M.G.C.: Interior point methods for global optimization problems. In: Terlaky, T. (ed.) Interior point methods of mathematical programming, pp. 467–500. Kluwer, Dordrecht (1996)
Pardalos, P.M., Resende, M.G.C. (eds): Handbook of applied optimization. Oxford University Press, Oxford (2002)
Stern R.J., Wolkowicz H.: Indefinite trust region subproblems and nonsymmetric perturbations. SIAM J. Optim. 5(2), 286–313 (1995)
Sturm J.F., Zhang S.: On cones of nonnegtive quadratic functions. Math. Oper. Res. 28(2), 246–267 (2003)
Wang Z., Fang S.C., Gao D.Y., Xing W.: Global extremal conditions for multi-integer quadratic programming. J. Ind. Manag. Optim. 4(2), 213–225 (2008)
Xing, W., Fang, S.C., Gao, D.Y., Sheu, R.L., Zhang, L.: Canonical dual solutions to the quadratic programming over a quadratic constraint, (submitted)
Ye Y., Zhang S.: New results on quadratic minimization. SIAM J. Optim. 14(1), 245–267 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Feng, JM., Lin, GX., Sheu, RL. et al. Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint. J Glob Optim 54, 275–293 (2012). https://doi.org/10.1007/s10898-010-9625-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-010-9625-6