Abstract
The value-at-risk is an important risk measure that has been used extensively in recent years in portfolio selection and in risk analysis. This problem, with its known bilevel linear program, is reformulated as a polyhedral DC program with the help of exact penalty techniques in DC programming and solved by DCA. To check globality of computed solutions, a global method combining the local algorithm DCA with a well adapted branch-and-bound algorithm is investigated. An illustrative example and numerical simulations are reported, which show the robustness, the globality and the efficiency of DCA.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andersson F, Mausser H, Rosen D, Uryasev S (2001) Credit risk optimization with conditional value-at-risk criterion. Math Programm Ser B 89: 273–291
Gilli M, Kellezi A (2002) A global optimization heuristic for portfolio choice with VaR and expected shortfall. In: Kontoghiorghes EJ, Rustem B, Siokos S (eds) Computational Methods in Decision-making, Economics and Finance. Applied Optimization Series. Kluwer Academic Publishers, Dordrecht, pp 167–183
Hiriart-Urruty JB, Lemarechal C (1991) Convex analysis and minimization algorithms. Parts I&II. Springer, Heidelberg
Le Thi HA, Pham Dinh T (1997) Solving a class of linearly constrained indefinite quadratic problems by DC algorithms. J Glob Optim 11: 253–285
Le Thi HA, Pham Dinh T, Muu LD (1999) Exact penalty in DC programming. Vietnam J Math 27(2): 169–178
Hoang T (1995) DC optimization: theory, methods and algorithms. In: Horst R, Pardalos PM(eds) Handbook of global optimization. Kluwer, Dordecht
Hoang T (1998) Convex analysis and global optimization. Kluwer, Dordrecht
Horst R, Van Thoai N (1999) DC programming: overview. J Optim Theory Appl 103: 1–43
Horst R, Pardalos PM, Van Thoai N (1996) Global optimization: deterministic approaches. Kluwer, Dordrecht
Le Thi HA, Pham Dinh T (2002) DC Programming: theory, algorithms and applications. The State of the Art. A. (ed) The first international workshop on global constrained optimization and constraint satisbfaction (Cocos’ 02), Valbonne-Sophia Antipolis, France, October 2–4, 28 pp
Le Thi HA, Pham Dinh T (2003) Large scale molecular optimization from distances matrices by a DC optimization approach. SIAM J Optim 14(1): 77–116
Le Thi HA, Pham Dinh T (2004) DC programming approaches and DCA for globally solving linear complematarity problems. Research report. National Institute for Applied Sciences, Rouen (submitted)
Le Thi HA, Pham Dinh T (2005) The DC (Difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problem. Ann Oper Res 133: 23–46
Le Thi HA, Pham Dinh T, Ngai HV (2005) Exact penalty techniques in DC programming, Research Report. National Institute for Applied Sciences, Rouen, 38 p
Larsen N, Mausser H, Uryasev S (2002) Algorithms for optimization of value-at-risk. In: Pardalos P, Tsitsiringos VK(eds) Financial engineering, E-commerce and supply chain. Kluwer, Norwell, pp 129–157
Luo Z-Q, Pang J-S, Ralph D, Wu S-Q (1996) Exact penalization and stationary conditions of mathematical programs with equilibrium constraints. Math Programm 75: 19–76
Pang J-S, Leyffer S (2004) On the global minimization of the value-at-risk. Optim Methods Softw 19(5): 611–631
Pflug GC (2000) Some remarks on the value-at-risk and the conditional value-at-risk. In: Uryasev S(eds) Probabilistic constrained optimization: methodology and optimization. Kluwer, Norwell, pp 278–287
Rockafellar RT (1970) Convex analysis. Princeton University Press, Englewood Cliffs
Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2: 21–41
Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distribution. J Banking Financ 26: 1443–1471
Pham Dinh T, Le Thi HA (1997) Convex analysis approach to DC programming: theory, algorithms and applications. Acta Math Vietnamica, dedicated to Professor Hoang Tuy on the occasion of his 70th birthday 22(1): 289–355
Pham Dinh T, Le Thi HA (1998) DC optimization algorithms for solving the trust region subproblem. SIAM J Optim 8: 476–505
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pham Dinh, T., Nam, N.C. & Le Thi, H.A. DC programming and DCA for globally solving the value-at-risk. Comput Manag Sci 6, 477–501 (2009). https://doi.org/10.1007/s10287-009-0099-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10287-009-0099-2