Abstract
In this paper we investigate the applicability of a recently introduced primal-dual splitting method in the context of solving portfolio optimization problems which assume the minimization of risk measures associated to different convex utility functions. We show that, due to the splitting characteristic of the used primal-dual method, the main effort in implementing it constitutes in the calculation of the proximal points of the utility functions, which assume explicit expressions in a number of cases. When quantifying risk via the meanwhile classical conditional value-at-risk, an alternative approach relying on the use of its dual representation is presented as well. The theoretical results are finally illustrated via some numerical experiments on real and synthetic data sets.
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References
Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, New York (2011)
Ben-Tal, A., Teboulle, M.: Expected utility, penalty functions and duality in stochastic nonlinear programming. Manage. Sci. 32(11), 1445–1466 (1986)
Ben-Tal, A., Teboulle, M.: An old-new concept of risk measures: the optimized certainty equivalent. Math. Finance 17(3), 449–476 (2007)
Boţ, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin (2010)
Boţ, R.I., Csetnek, E.R., Heinrich, A.: A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators. SIAM J. Optim. 23(4), 2011–2036 (2013)
Boţ, R.I., Frătean, A.R.: Looking for appropriate qualification conditions for subdifferential formulae and dual representations for convex risk measures. Math. Meth. Oper. Res. 74(2), 191–215 (2011)
Boţ, R.I., Grad, S.M., Wanka, G.: Duality in Vector Optimization. Springer, Berlin (2009)
Boţ, R.I., Hendrich, C.: Convergence analysis for a primal-dual monotone \(+\) skew splitting algorithm with applications to total variation minimization. J. Math. Imaging Vis. 49(3), 551–568 (2014)
Boţ, R.I., Hendrich, C.: A Douglas–Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators. SIAM J. Optim. 23(4), 2541–2565 (2013)
Briceño-Arias, L.M., Combettes, P.L.: A monotone \(+\) skew splitting model for composite monotone inclusions in duality. SIAM J. Optim. 21(4), 1230–1250 (2011)
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)
Combettes, P.L., Pesquet, J.-C.: Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued Var. Anal. 20(2), 307–330 (2012)
Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158(2), 460–479 (2013)
Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stochast. 6(4), 429–447 (2002)
Föllmer, H., Schied, A.: Robust representation of convex measures of risk. In: Sandmann, K., Schönbucher, P. (eds.) Advances in Finance and Stochastics, pp. 39–56. Springer, Berlin (2002)
Föllmer, H., Schied, A.: Stochastic Finance. A Introduction in Discrete Time. Walter de Gruyter, Berlin (2002)
Lüthi, H., Doege, J.: Convex risk measures for portfolio optimization and concepts of flexibility. Math. Program. 104(2–3), 541–559 (2005)
Rockafellar, R.T.: On the maximal monotonicity of subdiferential mappings. Pacif. J. Math. 33(1), 209–216 (1970)
Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–42 (2000)
Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank Finance 26(7), 1443–1471 (2002)
Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Deviation measures in risk analysis and optimization. Report 2002–7, ISE Depatment, University of Florida (2002)
Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Generalized deviations in risk analysis. Finance Stoch. 10(1), 51–74 (2006)
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)
Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comp. Math. 38(3), 667–681 (2013)
Acknowledgments
The authors are thankful to an anonymous reviewer for remarks and suggestions which have improved the quality of the paper. R. I. Boţ’s research partially supported by DFG (German Research Foundation), project BO 2516/4-1. C. Hendrich’s research supported by a Graduate Fellowship of the Free State Saxony, Germany.
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Boţ, R.I., Hendrich, C. Convex risk minimization via proximal splitting methods. Optim Lett 9, 867–885 (2015). https://doi.org/10.1007/s11590-014-0809-8
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DOI: https://doi.org/10.1007/s11590-014-0809-8