An incremental bundle method for portfolio selection problem under second-order stochastic dominance
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An incremental bundle method for portfolio selection problem under second-order stochastic dominance Jian Lv1 · Ze-Hao Xiao2 · Li-Ping Pang2 Received: 16 October 2018 / Accepted: 15 October 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
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. Keywords Stochastic dominance · Portfolio optimization · Incremental bundle method · Inexact oracle
1 Introduction Stochastic dominance is a fundamental concept in decision theory and economics (see, e.g., [29, 30, 44]). Hadar and Russell [13] and Hanoch and Levy [15] discussed Li-Ping Pang
[email protected] Jian Lv [email protected] Ze-Hao Xiao [email protected] 1
School of Finance, Zhejiang University of Finance and Economics, Hangzhou, 310018, China
2
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
Numerical Algorithms
second-order stochastic dominance (SSD) and decision theory, respectively. The wide application of SSD is due to its consistency with risk-averse preferences. In portfolio selection, SSD is of interest because that, for any decision-maker with non-satiable preferences and risk-averse, a portfolio which dominates a benchmark portfolio is preferred to the benchmark. Dentcheva and Ruszczy´nski [4–7] first introduced optimization problems with stochastic dominance constraints, which is an attractive approach for managing risks in an optimization setting. Their optimization model can be viewed as an expected profit maximization problem subject to the constraint that the profit dominates the benchmark profit in second order. Roman et al. [40] proposed a multiobjective portfolio selection model with SSD constraints. Dentcheva and Ruszczy´nski [8] introduced the concept of positive linear multivariate stochastic dominance and gained necessary conditions of optimality for non-convex problems. Kopa and Post [25] developed a linear programming test to analyze whether a given investment portfolio is efficient in terms of SSD about all possible portfolios formed from a set of base assets. Let v(x, ξ ) be a concave function with ra
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