Portfolio selection using neural networks

In this paper we apply a heuristic method based on artificial neural networks (NN) in order to trace out the efficient frontier associated to the portfolio selection problem. We consider a generalization of the standard Markowitz mean-variance model which includes cardinality and bounding constraints. These constraints ensure the investment in a given number of different assets and limit the amount of capital to be invested in each asset. We present some experimental results obtained with the NN heuristic and we compare them to those obtained with three previous heuristic methods. The portfolio selection problem is an instance from the family of quadratic programming problems when the standard Markowitz mean-variance model is considered. But if this model is generalized to include cardinality and bounding constraints, then the portfolio selection problem becomes a mixed quadratic and integer programming problem. When considering the latter model, there is not any exact algorithm able to solve the portfolio selection problem in an efficient way. The use of heuristic algorithms in this case is imperative. In the past some heuristic methods based mainly on evolutionary algorithms, tabu search and simulated annealing have been developed. The purpose of this paper is to consider a particular neural network (NN) model, the Hopfield network, which has been used to solve some other optimisation problems and apply it here to the portfolio selection problem, comparing the new results to those obtained with previous heuristic algorithms.     

Evolutionary multi-objective portfolio optimization in practical context

This paper addresses evolutionary multi-objective portfolio optimization in the practical context by incorporating realistic constraints into the problem model and preference criterion into the optimization search process. The former is essential to enhance the realism of the classical mean-variance model proposed by Harry Markowitz, since portfolio managers often face a number of realistic constraints arising from business and industry regulations, while the latter reflects the fact that portfolio managers are ultimately interested in specific regions or points along the efficient frontier during the actual execution of their investment orders. For the former, this paper proposes an order-based representation that can be easily extended to handle various realistic constraints like floor and ceiling constraints and cardinality constraint. An experimental study, based on benchmark problems obtained from the OR-library, demonstrates its capability to attain a better approximation of the efficient frontier in terms of proximity and diversity with respect to other conventional representations. The experimental results also illustrated its viability and practicality in handling the various realistic constraints. A simple strategy to incorporate preferences into the multi-objective optimization process is highlighted and the experimental study demonstrates its capability in driving the evolutionary search towards specific regions of the efficient frontier.     

A hybrid algorithm for constrained portfolio selection problems

Since Markowitz’s seminal work on the mean-variance model in modern portfolio theory, many studies have been conducted on computational techniques and recently meta-heuristics for portfolio selection problems. In this work, we propose and investigate a new hybrid algorithm integrating the population based incremental learning and differential evolution algorithms for the portfolio selection problem. We consider the extended mean-variance model with practical trading constraints including the cardinality, floor and ceiling constraints. The proposed hybrid algorithm adopts a partially guided mutation and an elitist strategy to promote the quality of solution. The performance of the proposed hybrid algorithm has been evaluated on the extended benchmark datasets in the OR Library. The computational results demonstrate that the proposed hybrid algorithm is not only effective but also efficient in solving the mean-variance model with real world constraints.     

Distributed Multi-Parametric Quadratic Programming

One of the fundamental problems in the area of large-scale optimization is to study locality features of spatially distributed optimization problems in which the variables are coupled in the cost function as well as constraints. Such problems can motivate the development of fast and well-conditioned distributed algorithms. In this paper, we study spatial locality features of large-scale multi-parametric quadratic programming (MPQP) problems with linear inequality constraints. Our main application focus is receding horizon control of spatially distributed linear systems with input and state constraints. We propose a new approach for analysis of large-scale MPQP problems by blending tools from duality theory with operator theory. The class of spatially decaying matrices is introduced to capture couplings between optimization variables in the cost function and the constraints. We show that the optimal solution of a convex MPQP is piecewise affine- represented as convolution sums. More importantly, we prove that the kernel of each convolution sum decays in the spatial domain at a rate proportional to the inverse of the corresponding coupling function of the optimization problem.     

风险资产收益序列相关的多阶段均值-方差投资组合选择

基于多阶段均值-方差框架, 研究任意多种风险资产存在一般收益序列相关时的投资组合选择问题. 首先, 采用Lagrange 对偶原理与动态规划相结合的方法对模型进行求解, 得到多阶段均值-方差模型的有效投资策略和有效边界的解析表达式; 然后, 证明在含有无风险资产的情形下有效边界仍为均值-标准差平面上的一条射线; 最后, 应用所得结论给出一个具体的实例分析.

     

A computational intelligence method for solving a class of portfolio optimization problems

In this paper, we revisit the mean-variance model of Markowitz and the construction of the risk-return efficient frontier. A few other models, such as the mean absolute deviation, the minimax and maximin, and models with diagonal quadratic form as objectives, which use alternative metrics for risk are also introduced. Then we present a neurodynamic model for solving these kinds of problems. By employing Lyapunov function approach, it is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the original problem. The validity and transient behavior of the neural network are demonstrated by using several examples of portfolio selection.     

An artificial bee colony algorithm with feasibility enforcement and infeasibility toleration procedures for cardinality constrained portfolio optimization

One of the most studied variant of portfolio optimization problems is with cardinality constraints that transform classical mean–variance model from a convex quadratic programming problem into a mixed integer quadratic programming problem which brings the problem to the class of NP-Complete problems. Therefore, the computational complexity is significantly increased since cardinality constraints have a direct influence on the portfolio size. In order to overcome arising computational difficulties, for solving this problem, researchers have focused on investigating efficient solution algorithms such as metaheuristic algorithms since exact techniques may be inadequate to find an optimal solution in a reasonable time and are computationally ineffective when applied to large-scale problems. In this paper, our purpose is to present an efficient solution approach based on an artificial bee colony algorithm with feasibility enforcement and infeasibility toleration procedures for solving cardinality constrained portfolio optimization problem. Computational results confirm the effectiveness of the solution methodology.     

A portfolio selection model with borrowing constraint based on possibility theory

Compared with the conventional probabilistic mean-variance methodology, fuzzy number can better describe an uncertain environment with vagueness and ambiguity. In this paper, the portfolio selection model with borrowing constraint is proposed by means of possibilistic mean, possibilistic variance, and possibilistic covariance under the assumption that the returns of assets are fuzzy numbers. And a quadratic programming model with inequality constraints is presented when the returns of assets are trapezoid fuzzy numbers. Furthermore, Lemke algorithm is utilized to solve the model. Finally, a numerical example of the portfolio selection problem is given to illustrate our proposed effective means and variances. The results of the numerical example also show that the investor can make different decisions according to different requirements for the values of expected returns. And the efficient portfolio frontier of the model with borrowing constraints can be easily obtained.     

考虑通货膨胀因素下的连续时间均值-方差投资组合选择

考虑通货膨胀因素,利用均值-方差模型研究连续时间投资组合选择问题.利用 Lagrange 乘子技术将原均值-方差模型转化为一个标准的随机最优控制问题,应用动态规划的方法得到问题的解析解,进而求解出原均值-方差模型的有效投资策略和有效边界的解析表达式.通过实证分析进一步表明了结论的正确性.     

Efficient implementation of an active set algorithm for large-scale portfolio selection

This paper deals with the efficient implementation of parametric quadratic programming that is specialized for large-scale mean-variance portfolio selection with a dense covariance matrix. The aim is to calculate the whole Pareto front of solutions that represent the trade-off between maximizing expected return and minimizing variance of return.We describe and compare in a uniform framework several techniques to speed up the necessary matrix operations, namely the initial matrix decomposition, the solution process in each iteration, and the matrix updates. Techniques considered include appropriate ordering of the matrix rows and columns, reducing the size of the system of linear equations, and dividing the system into two parts. Regarding implementation, we suggest to simultaneously use two different matrix representations that are specifically adapted to certain parts of the algorithm and propose a technique that prevents algorithm stalling due to numerical errors. Finally, we analyse and compare the runtime of these algorithm variants on a set of benchmark problems. As we demonstrate, the most sophisticated variant is several orders of magnitude faster than the standard implementation on all tested problem instances.     

A bicriterion approach to time/cost trade-offs in scheduling with convex resource-dependent job processing times and release dates

This paper presents a bicriterion analysis of time/cost trade-offs for the single-machine scheduling problem where both job processing times and release dates are controllable by the allocation of a continuously nonrenewable resource. Using the bicriterion approach, we distinguish between our sequencing criterion, namely the makespan, and the cost criterion, the total resource consumed, in order to construct an efficient time/cost frontier. Although the computational complexity of the problem of constructing this frontier remains an open question, we show that the optimal job sequence is independent of the total resource being used; thereby we were able to reduce the problem to a sequencing one. We suggest an exact dynamic programming algorithm for solving small to medium sizes of the problem, while for large-scale problems we present some heuristic algorithms that turned out to be very efficient. Five different special cases that are solvable by using polynomial time algorithms are also presented.     

Expansion of the cross-efficiency evaluation by using the equations forming the efficient frontier in DEA

In this paper, we propose an algorithm to calculate cross-efficiency scores which used the equations forming the efficient frontier in data envelopment analysis (DEA). In many standard DEA models, each decision-making unit (DMU) is evaluated by using the advantageous weight for itself. Then, many DMUs are evaluated as efficient, and those efficient DMUs are not ranked by the models. The cross-efficiency evaluation is a method to rank DMUs by using the advantageous weights for all DMUs. Previously, the cross-efficiency scores based on different ideas are calculated by solving multiple linear or nonlinear programming problems. However, it is often hard to solve such a nonlinear programming problem. Therefore, by analysing the efficient frontier, we construct an algorithm to calculate alternative cross-efficiency scores.     

A class of continuous-time portfolio selection with liability under jump-diffusion processes

A continuous-time mean-variance portfolio selection model is formulated with multiple risky assets and one liability under discontinuous prices which follow jump-diffusion processes in an incomplete market. The correlations between the risky assets and the liability are considered. The corresponding Hamilton–Jacobi–Bellman equation of the problem is presented. The optimal dynamic strategy and the efficient frontier in closed forms are derived explicitly by using stochastic linear-quadratic control technique. Finally, the effects on efficient frontier under the value-at-risk constraint are illustrated.     

跳跃扩散股价的最优投资组合选择

假定股票价格服从跳跃扩散过程.在传统均值-方差组合投资模型基础上,最大化最终收益的期望及最小化最终财富的方差.引进一个随机线性二次最优控制问题作为原问题的近似问题.证明了一个状态为跳跃扩散过程的一般最优控制问题的验证性定理.应用验证性定理求解HJB(Hamilton-Jacobi-Bellman)方程得到了原问题的最优策略.最后还给出了原问题有效前沿的表达式.     

A working-set approach for sizing optimization of frame-structures subjected to time-dependent constraints

In this paper, we propose a working-set approach for sizing optimization of structures subjected to time-dependent loads. The optimization problems we consider have a very large number of constraints while relatively few design variables and degrees of freedom. Instead of solving the original problem directly, we solve a sequence of smaller sub-problems. The sub-problems consider only constraints in the working set, which is a small sub-set of all constraints. After each sub-problem, we compute all constraint function values for the current design and add critical constraints to the working set. The algorithm terminates once an optimal point to a sub-problem is found that satisfies all constraints of the original problem. We tested the approach on several reproducible problem instances and demonstrate that the approach finds optimal points to the original problem by only considering a very small fraction of all constraints. The proposed approach drastically reduces the memory storage requirements and computational expenses of the linear algebra in the optimization solver and the computational cost of the design sensitivity analysis. Consequently, the approach can efficiently solve large-scale optimization problems with several hundred millions of constraints.     

基于风险价值约束的动态均值-方差投资组合的研究

研究了基于风险价值约束的动态均值-方差项目投资组合的数学模型,该模型是控制带约束的随机线性二次型（LQ）控制问题.在讨论该随机LQ控制问题的解之后,给出投资组合动态数学模型对应的随机哈密顿-雅克比-贝尔曼方程的解,得出了有效边界和最佳策略,讨论了风险价值约束的影响.最后,针对某油田勘探开发项目的实际情况,应用上述结论求出该实例的解,并讨论了风险价值约束发挥的作用.     

Robust multiperiod portfolio management in the presence of transaction costs

We study the viability of different robust optimization approaches to multiperiod portfolio selection. Robust optimization models treat future asset returns as uncertain coefficients in an optimization problem, and map the level of risk aversion of the investor to the level of tolerance of the total error in asset return forecasts. We suggest robust optimization formulations of the multiperiod portfolio optimization problem that are linear and computationally efficient. The linearity of the optimization problems is an advantage when complex additional requirements need to be imposed on the portfolio structure, e.g., limitations on positions in certain assets or tax constraints. We compare the performance of our robust formulations to the performance of the traditional single period mean-variance formulation frequently employed in the financial industry.     

Mean-Extended Gini Portfolios: A 3D Efficient Frontier

Using a numerical optimization technique we construct the mean-extended Gini (MEG) efficient frontier as a workable alternative to the mean-variance efficient frontier. MEG enables the introduction of specific risk aversion into portfolio selection. The resulting portfolios are stochastically dominant for all risk-averse investors. Solving for MEG portfolios allows investors to tailor portfolios for specific risk aversion. The extended Gini is calculated by the covariance of asset returns with a weighing function of the cumulative distribution function (CDF) of these returns. In a sample of asset returns, the CDF is estimated by ranking returns. In this case, analytical optimization techniques using continuous gradient approaches are unavailable, thus the need to develop numerical optimization techniques. In this paper we develop a numerical optimization algorithm that finds the portfolio optimal frontier for arbitrarily large sets of shares. The result is a 3-dimension MEG efficient frontier in the space formed by mean, the extended Gini, and the risk aversion coefficient.     

Large scale portfolio optimization with piecewise linear transaction costs

We consider the fundamental problem of computing an optimal portfolio based on a quadratic mean-variance model for the objective function and a given polyhedral representation of the constraints. The main departure from the classical quadratic programming formulation is the inclusion in the objective function of piecewise linear, separable functions representing the transaction costs. We handle the non-smoothness in the objective function by using spline approximations. The problem is first solved approximately using a primal-dual interior-point method applied to the smoothed problem. Then, we crossover to an active set method applied to the original non-smooth problem to attain a high accuracy solution. Our numerical tests show that we can solve large scale problems efficiently and accurately.     

SCPIP – an efficient software tool for the solution of structural optimization problems

This paper describes SCPIP, a FORTRAN77 subroutine that has been proven to be a reliable implementation of convex programming methods in an industrial environment. Convex approximation methods like the method of moving asymptotes are used nowadays in many software packages for structural optimization. They are known to be efficient tools for the solution of design problems, in particular if displacement dependent constraints like stresses occur. A major advantage over many but not all classical approaches of mathematical programming is that at an iteration point a local model is formulated. For the solution of such a model no further function and gradient evaluations are necessary besides those at the current iteration point. The first versions of convex approximation methods used all a dual approach to solve the subproblems which is still a very efficient algorithm to solve problems with at most a medium number of constraints. But it is not efficient for problems with many constraints. An alternative is the use of an interior point method for the subproblem solution. This leads to more freedom in the definition of the linear systems where most of the computing time to solve the subproblems is spent. In consequence, large-scale problems can be handled more efficiently.