Top

Numerical Algorithms

Published in:

11-12-2019 | Original Paper

An incremental bundle method for portfolio selection problem under second-order stochastic dominance

Authors: Jian Lv, Ze-Hao Xiao, Li-Ping Pang

Published in: Numerical Algorithms | Issue 2/2020

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this paper, we propose an incremental bundle method with inexact oracle for solving the portfolio optimization with stochastic second-order dominance (SSD) constraints. We first relax the SSD problem as a stochastic semi-infinite programming (SIP) problem. For the particular case of SIP problem, we exploit the improvement function and the idea of incremental technique for dealing with the infinitely many constraints. In the stochastic model, as an adding-rules, the “inexact oracle” is introduced in this algorithm. Therefore, the algorithm does not need all the information about the constraints, but only needs the inexact information of one component function to update the bundle and produces the search direction. Our numerical results on solving the academic problems have shown advantages of the incremental bundle method over three existing algorithms. Finally, numerical results on a part of portfolio optimization problem are presented by using the FTSE100 Index.

previous article Numerical methods for static shallow shells lying over an obstacle

next article The asymptotic approximations to linear weakly singular Volterra integral equations via Laplace transform

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Ackooij, W, Van Sagastizábal, C.: Constrained bundle methods for upper inexact oracles with application to joint chance constrained energy problems. SIAM J. Optim. 24, 733–765 (2014)MathSciNetMATH

An, L. T. H., Tao, P. D.: The DC (Difference of Convex Functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133, 23–46 (2005)MathSciNetMATH

An, L. T. H., Belghiti, M. T., Tao, P. D.: A new efficient algorithm based on DC programming and DCA for clustering. J. Glob. Optim. 37, 593–608 (2007)MathSciNetMATH

Dentcheva, D., Ruszczyński, A.: Optimization with stochastic dominance constraints. SIAM J. Optim. 14, 548–566 (2003)MathSciNetMATH

Dentcheva, D., Ruszczyński, A.: Optimality and duality theory for stochastic optimization with nonlinear dominance constraints. Math. Program. 99, 329–350 (2004)MathSciNetMATH

Dentcheva, D., Ruszczyński, A.: Semi-infinite probabilistic optimization: first-order stochastic dominance constrain. Optimization 53, 583–601 (2004)MathSciNetMATH

Dentcheva, D., Ruszczyński, A.: Portfolio optimization with stochastic dominance constraints. J. Banking Finance 30, 433–451 (2006)MATH

Dentcheva, D., Ruszczyński, A.: Optimization with multivariate stochastic dominance constraints. Math. program. 117, 111–127 (2009)MathSciNetMATH

Emiel, G., Sagastizábal, C.: Incremental-like bundle methods with application to energy planning. Comput. Optim. Appl. 46, 305–332 (2010)MathSciNetMATH

10.

Fábián, C., Mitra, G., Roman, D.: Processing second-order stochastic dominance models using cutting-plane representations. Math. Program. 130, 33–57 (2011)MathSciNetMATH

11.

Gaudioso, M., Giallombardo, G., Miglionico, G.: An incremental method for solving convex finite Min-Max problems. Math. of Oper. Res. 31, 173–187 (2006)MathSciNetMATH

12.

Hiriart-Urruty, J. B., Lemarechal, C.: Convex Analysis and Minimization Algorithms, I: Fundamentals. Springer, Berlin (1993)MATH

13.

Hadar, J., Russell, W. R.: Rules for ordering uncertain prospects. Am. Econ. Rev. 59, 25–34 (1969)

14.

Hare, W., Sagastizábal, C., Solodov, M.: A proximal bundle method for nonsmooth nonconvex functions with inexact information. Comput. Optim. Appl. 63, 1–28 (2016)MathSciNetMATH

15.

Hanoch, G., Levy, H.: The efficiency analysis of choices involving risk. Handbook of the Fundamentals of Financial Decision Making: Part I., 287–29 (2013)

16.

Helmberg, C., Rendl, F: A spectral bundle method for semi-definite programming. SIAM J. Optim. 10, 673–696 (2000)MathSciNetMATH

17.

Homem-de-Mello, T., Mehrotra, S.: A cutting-surface method for uncertain linear programs with polyhedral stochastic dominance constraints. SIAM J. Optim. 20, 1250–1273 (2009)MathSciNetMATH

18.

Hu, J., Homen-De-Mello, T., Mehrotra, S.: Sample average approximation of stochastic dominance constrained programs. Math. Program. 133, 171–201 (2012)MathSciNetMATH

19.

Karas, E., Ribeiro, A., Sagastizȧbal, C., Solodov, M.: A bundle-filter method for nonsmooth convex constrained optimization. Math. Program. 116, 297–320 (2009)MathSciNetMATH

20.

Kibardin, V. M.: Decomposition into functions in the minimization problem. Avtomatika i Telemekhanika 9, 66–79 (1979)MathSciNetMATH

21.

Kiwiel, K. C.: Methods of Descent for Nondifferentiable Optimization. Springer, Berlin (2006)MATH

22.

Kiwiel, K. C.: A proximal bundle method with approximate subgradient linearizations. SIAM J. Optim. 16, 1007–1023 (2006)MathSciNetMATH

23.

Kiwiel, K. C.: A method of centers with approximate subgradient linearizations for nonsmooth convex optimization. SIAM J. Optim. 18, 1467–1489 (2008)MathSciNetMATH

24.

Kleywegt, A. J., Shapiro, A., Homem-De-Mello, T.: The sample average approximation method for stochastic discrete optimization. SIAM J. Optimiz. 12, 479–502 (2002)MathSciNetMATH

25.

Kopa, M., Post, T.: A general test for SSD portfolio efficiency. OR Spectr. 37, 703–734 (2015)MathSciNetMATH

26.

Lemaréchal, C., Nemirovskii, A., Nesterov, Y.: New variants of bundle methods. Math. Program. 69, 111–147 (1995)MathSciNetMATH

27.

Lv, J., Pang, L. P., Wang, J. H.: Special backtracking proximal bundle method for nonconvex maximum eigenvalue optimization. Appl. Math. Comput. 265, 635–651 (2015)MathSciNetMATH

28.

Meskarian, R., Xu, H. F., Fliege, J.: Numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization. Eur. J. Oper. Res. 216, 376–385 (2012)MathSciNetMATH

29.

Müller, A., Scarsini, M.: Stochastic Orders and Decision Under Risk. Institute of mathematical statistics, Hayward (1991)

30.

Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York (2002)MATH

31.

Nedić, A., Bertsekas, D. P.: Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optim. 12, 109–138 (2001)MathSciNetMATH

32.

Nguyen, T. T. V., Strodiot, J. J., Nguyen, V. H.: A bundle method for solving equilibrium problems. Math. Program. 116, 529–552 (2009)MathSciNetMATH

33.

Oliveira, W., Sagastizábal, C., Scheimberg, S.: Inexact bundle methods for two-stage stochastic programming. SIAM J. Optim. 21, 517–544 (2011)MathSciNetMATH

34.

Ogryczak, W., Ruszczyński, A.: From stochastic dominance to mean-risk models: semideviations as risk measures. Eur. J. Oper. Res. 116, 33–50 (1999)MATH

35.

Ogryczak, W., Ruszczyński, A.: On consistency of stochastic dominance and mean-semideviation models. Math. Program. 89, 217–232 (2001)MathSciNetMATH

36.

Rinnooy Kan, A. H., Timmer, G. T.: Stochastic global optimization methods. Part II: multilevel methods. Math. Program. 39, 57–78 (1987)MATH

37.

Rockafellar, R. T.: Convex Analysis. Princeton University Press, Princeton (1970)MATH

38.

Rockafellar, R. T., Wets, J.J.-B.: Variational Analysis. Springer, Berlin (1998)MATH

39.

Rockafellar, R. T., Uryasev, S.: Optimization of conditional value-at-risk. Journal of Risk 2, 21–41 (2000)

40.

Roman, D., Darby-Dowman, K., Mitra, G.: Portfolio construction based on stochastic dominance and target return distributions. Math. Program. 108, 541–569 (2006)MathSciNetMATH

41.

Rudolf, G., Ruszczyński, A.: Optimization problems with second order stochastic dominance constraints: duality, compact formulations, and cut generation methods. SIAM J. Optim. 19, 1326–1343 (2008)MathSciNetMATH

42.

Sagastizábal, C., Solodov, M.: An infeasible bundle method for nonsmooth convex constrained optimization without a penalty function or a filter. SIAM J. Optim. 16, 146–169 (2005)MathSciNetMATH

43.

Salmon, G., Strodiot, J.-J., Nguyen, V. H.: A bundle method for solving variational inequalities. SIAM J. Optim. 14, 869–893 (2004)MathSciNetMATH

44.

Shaked, M., Shanthikumar, J. G.: Stochastic Orders and Their Applications. Academic Press, Boston (1994)MATH

45.

Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on stochastic programming: modeling and theory. Society for Industrial and Applied Mathematics (2009)

46.

Sun, H. L., Xu, H. F., Meskarian, R., Wang, Y.: Exact penalization, level function method, and modified cutting-plane method for stochastic programs with second order stochastic dominance constraints. SIAM J. Optim. 23, 602–631 (2013)MathSciNetMATH

47.

Sun, H. L., Xu, H. F., Wang, Y.: Smoothing penalized sample average approximation method for stochastic programs with second-order stochastic dominance constraints. Asia-Pac. J. Oper. Res. 30, 1340002 (2013)MathSciNetMATH

48.

Solodov, M. V.: A bundle method for a class of bilevel nonsmooth convex minimization problems. SIAM J. Optim. 18, 242–259 (2007)MathSciNetMATH

49.

Xu, M. W., Wu, S. -Y., Ye, J. J.: Solving semi-infinite programs by smoothing projected gradient method. Computat. Optim. Appl. 59, 591–616 (2014)MathSciNetMATH

50.

žaković, S., Rustem, B.: Semi-infinite programming and applications to minimax problem. Ann. Oper. Res. 124, 81–110 (2003)MathSciNetMATH

Title: An incremental bundle method for portfolio selection problem under second-order stochastic dominance
Authors: Jian Lv
Ze-Hao Xiao
Li-Ping Pang
Publication date: 11-12-2019
Publisher: Springer US
Published in: Numerical Algorithms / Issue 2/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI: https://doi.org/10.1007/s11075-019-00831-6

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Other articles of this Issue 2/2020

Efficient methods for nonlinear time fractional diffusion-wave equations and their fast implementations

Secant variable projection method for solving nonnegative separable least squares problems

An efficient third-order scheme for BSDEs based on nonequidistant difference scheme

Convergence study on the proximal alternating direction method with larger step size

Performance of compact and non-compact structure preserving algorithms to traveling wave solutions modeled by the Kawahara equation

Superconvergence analysis of a two-grid method for an energy-stable Ciarlet-Raviart type scheme of Cahn-Hilliard equation

Premium Partner