Robust Toland-Fenchel-Lagrange duality for DC problem under uncertainty

Articles in Press

Document Type : Original paper

Authors

1 Ecole Normale Supérieure, Institut des Sciences et Technologies, Ouagadougou, Burkina Faso

2 Département de mathématiques, Université Houphouet-Boigny, Abidjant, Côte D'Ivoire

Abstract

This paper concerns the robust duality theory for a  DC problem under uncertainty. We use a "robust" qualification condition to establish  robust Toland-Fenchel-Lagrange duality property. We deduce robust strong Lagrange duality to the case when we have an uncertain conical convex problem.

Keywords

Main Subjects

References

[1] A.D. Alexandrov, On surfaces which may be represented by a difference of convex functions, Izvestiya Akademii Nauk Kazakhskoj SSR, Seria Fiziko Matematicheskikh 3 (1949), 3–20.
[2] A.D. Alexandrov, On surfaces which may be represented by differences of convex functions, Dokl. Akad. Nauk SSR 72 (1950), 613–616.
[3] M. Amara, A. Obeid, and G. Vallet, Existence results for a degenerated nonlinear elliptic partial differential equation, J. Math. Anal. Appl. 310 (2005), no. 2, 641–656. https://doi.org/10.1016/j.jmaa.2005.02.033
[4] L.T.H. An, An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints, Math. Program. 87 (2000), 401–426. https://doi.org/10.1007/s101070050003
[5] L.T.H. An and P.D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world non-convex optimization problems, Ann. Oper. Res. 133 (2005), 23–46. https://doi.org/10.1007/s10479-004-5022-1
[6] L.T.H. An, P.D. Tao, and D.N. Hao, Solving an inverse problem for an elliptic equation by d.c. programming, J. Global Optim. 25 (2005), 407–423. https://doi.org/10.1023/A:1022530520406
[7] M. Barro, A. Ouédraogo, and S. Traoré, On uncertain conical convex optimization problem, Pac. J. Optim. 13 (2017), no. 1, 29–42.
[8] A. Beck and A. Ben-Tal, Duality in robust optimization: Primal worst equals dual best, Oper. Res. Lett. 37 (2009), no. 1, 1–6. https://doi.org/10.1016/j.orl.2008.09.010
[9] A. Ben-Tal, L.E. Ghaoui, and A. Nemirovski, Robust Optimization, Princeton Series in Applied Mathematics, Princeton University Press, 2009.
[10] A. Ben-Tal and A. Nemirovski, Robust optimization - methodology and applications, Math. Program. 92 (2002), 453–480. https://doi.org/10.1007/s101070100286
[11] A. Ben-Tal and A. Nemirovski, Selected topics in robust convex optimization, Math. Program. 112 (2008), 125–158.
https://doi.org/10.1007/s10107-006-0092-2
[12] D. Bertsimas and D.B. Brown, Constructing uncertainty sets for robust linear optimization, Oper. Res. 57 (2009), no. 6, 1483–1495. https://doi.org/10.1287/opre.1080.0646
[13] D. Bertsimas, D. Pachamanova, and M. Sim, Robust linear optimization under general norms, Oper. Res. Lett. 32 (2004), no. 6, 510–516. https://doi.org/10.1016/j.orl.2003.12.007
[14] R.I. Bot¸ and G. Wanka, An alternative formulation for a new closed cone constraint qualification, Nonlinear Anal. 64 (2006), no. 6, 1367–1381. https://doi.org/10.1016/j.na.2005.06.041
[15] R.S. Burachik and V. Jeyakumar, Dual condition for the convex subdifferential sum formula with applications, J. Convex Anal. 15 (2005), 540–554.
[16] R.S. Burachik and V. Jeyakumar, A new geometric condition for Fenchel’s duality in infinite dimensional spaces, Math. Program. 104 (2005), 229–233. https://doi.org/10.1007/s10107-005-0614-3
[17] C. Combari, M. Laghdir, and L. Thibault, Sous-différentiels de fonctions convexes composées, Ann. Sci. Math. Québec 18 (1994), no. 2, 119–148.
[18] N. Dinh, M.A. Goberna, and M.A. López, From linear to convex systems: Consistency, Farkas’ Lemma and applications, J. Convex Anal. 13 (2006), 113–133.
[19] N. Dinh, T.T.A. Nghia, and G. Vallet, A closedness condition and its applications to DC programs with convex constraints, Optimization: A Journal of Mathematical Programming and Operations Research 59 (2010), no. 4, 541–560.
[20] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, NorthHolland Publishing Company, 1976.
[21] R. Horst and N.V. Thoai, DC programming: Overview, J. Optim. Theory Appl. 103 (1999), 1–43. https://doi.org/10.1023/A:1021765131316
[22] V. Jeyakumar, N. Dinh, and G.M. Lee, A new closed cone constraint qualification for convex optimization, Applied Mathematics Research Report AMR04/8, Tech. report, School of Mathematics, University of New South Wales, 2004.
[23] V. Jeyakumar, G.M. Lee, and N. Dinh, New sequential Lagrange multiplier conditions characterizing optimality without constraint qualifications for convex programs, SIAM J. Optim. 14 (2003), no. 2, 534–547. https://doi.org/10.1137/S1052623402417699
[24] V. Jeyakumar and G. Li, Characterizing robust set containments and solutions of uncertain linear programs without qualifications, Oper. Res. Lett. 38 (2010), no. 3, 188–194. https://doi.org/10.1016/j.orl.2009.12.004
[25] V. Jeyakumar and G. Li, Strong duality in robust convex programming: Complete characterizations, SIAM J. Optim. 20 (2010), no. 6, 3384–3407. https://doi.org/10.1137/100791841
[26] M. Laghdir, Optimality conditions and Toland’s duality for a non-convex minimization problem, Math. Vesnik 55 (2003), no. 1-2, 21–30.
[27] E.M. Landis, On functions representable as the difference of two convex functions, Dokl. Akad. Nauk SSSR 80 (1951), 9–11.
[28] G.Y. Li, V. Jeyakumara, and G.M. Lee, Robust conjugate duality for convex optimization under uncertainty with application to data classification, Nonlinear Analysis 74 (2011), no. 6, 2327–2341. https://doi.org/10.1016/j.na.2010.11.036
[29] J.E. Martiez-Legaz and M. Volle, Duality in D.C. programming: The case of several D.C. constraints, J. Math. Anal. Appl. 237 (1999), no. 2, 657–671. https://doi.org/10.1006/jmaa.1999.6496
[30] A. Shapiro, Stochastic programming approach to optimization under uncertainty, Math. Program. 112 (2008), 183–220. https://doi.org/10.1007/s10107-006-0090-4
[31] A. Shapiro, D. Dentcheva, and A.P. Ruszczynski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, 2009.
[32] J.F. Toland, Duality in non-convex optimization, J. Math. Anal. Appl. 66 (1978), no. 2, 399–415. https://doi.org/10.1016/0022-247X(78)90243-3
[33] H. Tuy, Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht, 1998.
[34] G. Wanka and R.I. Bot¸, On the relations between different dual problems in convex mathematical programming, Operations Research Proceedings 2001 (Berlin) (P. Chamoni, R. Leisten, A. Martin, J. Minnemann, and H. Stadtler, eds.),
Springer, 2002, pp. 255–262. https://doi.org/10.1007/978-3-642-50282-8_32
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 27 June 2026

Files

Share

How to cite

Statistics