Abstract
The paper discusses a general framework for outer approximation type algorithms for the canonical DC optimization problem. The algorithms rely on a polar reformulation of the problem and exploit an approximated oracle in order to check global optimality. Consequently, approximate optimality conditions are introduced and bounds on the quality of the approximate global optimal solution are obtained. A thorough analysis of properties which guarantee convergence is carried out; two families of conditions are introduced which lead to design six implementable algorithms, whose convergence can be proved within a unified framework.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Al-Khayyal F., Sherali H.: On finitely terminating branch-and-bound algorithms for some global optimization problems. SIAM J. Optim. 10, 1049–1057 (2000)
Androulakis I., Maranas C., Floudas C.: α–BB: a global optimization method for general constrained nonconvex problems. J. Global Optim. 7, 337–363 (1995)
Ben Saad S.: A new cutting plane algorithm for a class of reverse convex 0–1 integer programs. In: Floudas, C.A., Pardalos, P.M. (eds) Recent advances in global optimization, pp. 152–164. Princeton University Press, Princeton (1992)
Ben Saad S., Jacobsen S.E.: A level set algorithm for a class of reverse convex programs. Ann. Oper. Res. 25, 19–42 (1990)
Ben Saad S., Jacobsen S.E.: Comments on a reverse convex programming algorithm. J. Global Optim. 5, 95–96 (1994)
Chapelle O., Sindhwani V., Keerthi S.S.: Optimization techniques for semi-supervised support vector machines. J. Mach. Learn. Res. 9, 203–233 (2008)
Fulop J.: A finite cutting plane method for solving linear programs with an additional reverse constraint. European J. Oper. Res. 44, 395–409 (1990)
Grippo L., Sciandrone M.: On the convergence of the block nonlinear Gauss–Seidel method under convex constraints. Oper. Res. Lett. 26, 127–136 (2000)
Horst R., Tuy H.: Global optimization. Springer, Berlin (1990)
Horst, R., Pardalos, P.M. (eds.): Handbook of global optimization. Kluwer, Dordrecht (1995)
Nghia M.D., Hieu N.D.: A method for solving reverse convex programming problems. Acta Math. Vietnam. 11, 241–252 (1986)
Pham D.T., El Bernoussi S.: Numerical methods for solving a class of global nonconvex optimization problems. Int. Ser. Numer. Math. 87, 97–132 (1989)
Pintér J.D. (ed.): Global optimization: scientific and engineering case studies. Springer, Berlin (2006)
Rikun A.D.: A convex envelope formula for multilinear functions. J. Global Optim. 10, 425–437 (1997)
Ryoo H., Sahinidis N.: Global optimization of multiplicative programs. J. Global Optim. 26, 387–418 (2003)
Rockafellar R.T.: Convex analysis. Princeton University Press, Princeton (1970)
Strekalovsky A.S., Tsevendorj I.: Testing the \({\mathbb{R}}\) -strategy for a reverse convex problem. J. Global Optim. 13, 61–74 (1998)
Thach P.T.: Convex programs with several additional reverse convex constraints. Acta Math. Vietnam. 10, 35–57 (1985)
Thoai N.V.: A modified version of Tuy’s method for solving DC programming problems. Optimization 19, 665–674 (1988)
Tuan H.D.: Remarks on an algorithm for reverse convex programs. J. Global Optim. 16, 295–297 (2000)
Tuy H.: Global minimization of a difference of two convex functions. Math. Program. Stud. 30, 150–182 (1987)
Tuy H.: A general deterministic approach to global optimization via DC programming. In: Hiriart- Urruty, J.B. (eds) FERMAT Days 85: mathematics for optimization, pp. 273–303. North-Holland, Amsterdam (1986)
Tuy H.: Convex programs with an additional reverse convex constraint. J. Optim. Theory Appl. 52, 463–486 (1987)
Tuy H., Horst R.: Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and DC optimization problems. Math. Program. 41, 161–183 (1988)
Tuy H.: Normal conical algorithm for concave minimization over polytopes. Math. Program. 51, 229–245 (1991)
Tuy H.: Effect of the subdivision strategy on convergence and efficiency of some global optimization algorithms. J. Global Optim. 1, 23–36 (1991)
Tuy H.: On nonconvex optimization problems with separated nonconvex variables. J. Global Optim. 2, 133–144 (1992)
Tuy H.: Canonical DC programming problem: outer approximation methods revisited. Oper. Res. Lett. 18, 99–106 (1995)
Tuy H.: DC optimization: theory, methods and algorithms. In: Horst, R., Pardalos, P.M. (eds) Handbook of global optimization, pp. 149–216. Kluwer, Dordrecht (1995)
Tuy H.: Convex analysis and global optimization. Kluwer, Dordrecht (1998)
Tuy H., Al-Khayyal F.A.: Global optimization of a nonconvex single facility location problem by sequential unconstrained convex minimization. J. Global Optim. 2, 61–71 (1992)
Tuy H., Migdalas A., Varbrand P.: A quasiconcave minimization method for solving linear two-level programs. J. Global Optim. 4, 243–263 (1994)
Tuy H., Tam B.T.: Polyhedral annexation vs outer approximation for the decomposition of monotonic quasiconcave minimization problems. Acta Math. Vietnam. 20, 99–114 (1995)
Wen Y.-W., Ng M.K., Huang Y.-M.: Efficient total variation minimization methods for color image restoration. IEEE Trans. Image Process. 17, 2081–2088 (2008)
Zhang, Q.H.: Outer approximation algorithms for DC programs and beyond. PhD Thesis, Università di Pisa, Pisa. http://etd.adm.unipi.it/theses/available/etd-07022008-181656/unrestricted/Thesis.pdf (2008). Accessed 02 July 2008
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bigi, G., Frangioni, A. & Zhang, Q. Outer approximation algorithms for canonical DC problems. J Glob Optim 46, 163–189 (2010). https://doi.org/10.1007/s10898-009-9415-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-009-9415-1