Abstract
This paper studies the canonical duality theory for solving a class of quadrinomial minimization problems subject to one general quadratic constraint. It is shown that the nonconvex primal problem in \({\mathbb {R}^n}\) can be converted into a concave maximization dual problem over a convex set in \({\mathbb {R}^2}\), such that the problem can be solved more efficiently. The existence and uniqueness theorems of global minimizers are provided using the triality theory. Examples are given to illustrate the results obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ekeland I.: Legendre duality in nonconvex optimization and calculus of variations. SIAM J. Control Optim. 15, 905–934 (1977)
Ekeland, I.: Nonconvex duality. In: Gao, D.Y. (ed.) Proceedings of IUTAM Symposium on Duality, Complementarity and Symmetry in Nonlinear Mechanics. Kluwer Academic, Dordrecht/Boston/London (2003)
Ekeland I., Temam R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)
Fang S.-C., Gao D.Y., Sheu R.L., Wu S.Y.: Canonical dual approach for solving 0–1 quadratic programming problems. J. Ind. Manage. Optim. 4(1), 125–142 (2008)
Fang S.-C., Gao D.Y., Sheu R.L., Xing W.X.: Global optimization for a class of fractional programming problems. J. Glob. Optim. 45, 337–353 (2009)
Floudas C.A., Visweswaran V.: Quadratic optimization. In: Horst, R., Pardalos, P.M. (eds) Handbook of Global Optimization, pp. 217–270. Kluwer Academic Publishers, Dordrecht/Boston/London (1995)
Gao D.Y.: Dual extremum principles in finite deformation theory with applications to post-buckling analysis of extended nonlinear beam theory. Appl. Mech. Rev. 50(11), S64–S71 (1997)
Gao D.Y.: Duality-mathematics. Wiley Encyclopedia Electronical Electronical Eng. 6, 68–77 (1999)
Gao D.Y.: Duality Principles in Nonconvex Systems: Theory, Methods and Applications. Kluwer Academic Publishers, Dordrecht/Boston/London (2000a)
Gao D.Y.: Canonical dual transformation method and generalized triality theory in nonsmooth global optimization. J. Glob. Optim. 17(1/4), 127–160 (2000b)
Gao D.Y.: Perfect duality theory and complete set of, solutions to a class of global optimization. Optimization 52(4–5), 467–493 (2003)
Gao D.Y.: Complete solutions to constrained quadratic optimization problems. J. Glob. Optim. 29, 377–399 (2004)
Gao D.Y.: Sufficient conditions and canonical duality in nonconvex minimization with inequality constraints. J. Ind. Manage. Optim. 1(1), 53–63 (2005)
Gao D.Y.: Complete solutions and extremality criteria to polynomial optimization problems. J. Glob. Optim. 35, 131–143 (2006)
Gao D.Y., Sherali H.D.: Canonical duality theory: connections between nonconvex mechanics and global optimization. In: Gao, D.Y., Sherali, H. (eds) Advances in Applied Mathematics and Global Optimization, pp. 257–326. Springer, USA (2009)
Gao D.Y., Strang G.: Geometric nonlinearity: potential energy, complementary energy, and the gap function. Q. Appl. Math. 47(3), 487–504 (1989)
Gao, D.Y., Wu, C.Z.: On the triality theory in global optimization (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yuan, YB., Fang, SC. & Gao, D.Y. Global optimal solutions to a class of quadrinomial minimization problems with one quadratic constraint. J Glob Optim 52, 195–209 (2012). https://doi.org/10.1007/s10898-011-9658-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-011-9658-5