Quadratic programming with transaction costs

We consider the problem of maximizing the mean-variance utility function of n$n$ assets. Associated with a change in an asset's holdings from its current or target value is a transaction cost. These must be accounted for in practical problems. A straightforward way of doing so results in a 3n$3n$-dimensional optimization problem with 3n$3n$ additional constraints. This higher dimensional problem is computationally expensive to solve. We present an algorithm for solving the 3n$3n$-dimensional problem by modifying an active set quadratic programming (QP) algorithm to solve the 3n$3n$-dimensional problem as an n$n$-dimensional problem accounting for the transaction costs implicitly rather than explicitly. The method is based on deriving the optimality conditions for the higher dimensional problem solely in terms of lower dimensional quantities and requires substantially less computational effort than any active set QP algorithm applied directly on a 3n$3n$-dimensional problem.     

Continuous-time mean variance portfolio with transaction costs: a proximal approach involving time penalization

This paper proposes a new continuous-time optimization solution that enables the computation of the portfolio problem (based on the utility option pricing and the shortfall risk minimization). We first propose a dynamical stock price process, and then, we transform the solution to a continuous-time discrete-state Markov decision processes. The market behavior is characterized by considering arbitrage-free and assessing transaction costs. To solve the problem, we present a proximal optimization approach, which considers time penalization in the transaction costs and the utility. In order to include the restrictions of the market, as well as those that imposed by the continuous-time space, we employ the Lagrange multipliers approach. As a result, we obtain two different equations: one for computing the portfolio strategies and the other for computing the Lagrange multipliers. Each equation in the portfolio is an optimization problem, for which the necessary condition of a maximum/minimum is solved employing the gradient method approach. At each step of the iterative proximal method, the functional increases and finally converges to a final portfolio. We show the convergence of the method. A numerical example showing the effectiveness of the proposed approach is also developed and presented.     

Fuzzy portfolio selection using genetic algorithm

This paper presents the development of fuzzy portfolio selection model in investment. Fuzzy logic is utilized in the estimation of expected return and risk. Using fuzzy logic, managers can extract useful information and estimate expected return by using not only statistical data, but also economical and financial behaviors of the companies and their business strategies. In the formulated fuzzy portfolio model, fuzzy set theory provides the possibility of trade-off between risk and return. This is obtained by assigning a satisfaction degree between criteria and constraints. Using the formulated fuzzy portfolio model, a Genetic Algorithm (GA) is applied to find optimal values of risky securities. Numerical examples are given to demonstrate the effectiveness of proposed method.     

基于灵敏度分析的含比例型手续费的投资组合优化

研究含比例型手续费的离散时间投资组合优化问题. 基于马尔可夫决策过程模型和性能灵敏度分析方法, 推导两个不同投资策略之间的资产长期平均增值率的差分公式, 利用差分公式的结构特点, 证明了最优性方程, 并设计出可在线应用的策略迭代算法. 仿真实例验证了所提出算法的有效性.

     

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.     

Rebalancing an investment portfolio in the presence of convex transaction costs,including market impact costs

The inclusion of transaction costs is an essential element of any realistic portfolio optimization. We extend the standard portfolio optimization problem to consider convex transaction costs incurred when rebalancing an investment portfolio. Market impact costs measure the effect on the price of a security that result from an effort to buy or sell the security, and they can constitute a large part of the total transaction costs. The loss to a portfolio from market impact costs is often modelled with a convex function that can be expressed using second-order cone constraints. The Markowitz framework of mean-variance efficiency is used. In order to properly represent the variance of the resulting portfolio, we suggest rescaling by the funds available after paying the transaction costs. This results in a fractional programming problem, which we show can be reformulated as an equivalent convex program of size comparable to the model without transaction costs. We show that an optimal solution to the convex program can always be found that does not discard assets.     

Robust optimization framework for cardinality constrained portfolio problem

One of the primary concerns on any asset allocation problem is to maintain a limited number of assets from the market. The problem becomes more complicated when the return of all risky assets are subject to uncertainty. In this paper, we propose a new portfolio modeling approach with uncertain data and it is also analyzed using different robust optimization techniques. The proposed formulations are solved using genetic algorithm. The implementation of the proposed method is examined on variety of well known benchmark data sets.     

A linear programming algorithm for optimal portfolio selection with transaction costs

We study the optimal portfolio selection problem with transaction costs. In general, the efficient frontier can be determined by solving a parametric non-quadratic programming problem. In a general setting, the transaction cost is a V-shaped function of difference between the existing and the new portfolio. We show how to transform this problem into a quadratic programming model. Hence a linear programming algorithm is applicable by establishing a linear approximation on the utility function of return and variance.     

A memetic algorithm for the Minimum Sum Coloring Problem

Given an undirected graph G, the Minimum Sum Coloring Problem (MSCP) is to find a legal assignment of colors (represented by natural numbers) to each vertex of G such that the total sum of the colors assigned to the vertices is minimized. This paper presents a memetic algorithm for MSCP based on a tabu search procedure with two neighborhoods and a multi-parent crossover operator. Experiments on a set of 77 well-known DIMACS and COLOR 2002–2004 benchmark instances show that the proposed algorithm achieves highly competitive results in comparison with five state-of-the-art algorithms. In particular, the proposed algorithm can improve the best known results for 15 instances.     

On the use of optimization models for portfolio selection: A review and some computational results

Portfolio theory deals with the question of how to allocate resources among several competing alternatives (stocks, bonds), many of which have an unknown outcome. In this paper we provide an overview of different portfolio models with emphasis on the corresponding optimization problems. For the classical Markowitz mean-variance model we present computational results, applying a dual algorithm for constrained optimization.     

A new Probe Guided Mutation operator and its application for solving the cardinality constrained portfolio optimization problem

This paper revisits the classical Polynomial Mutation (PLM) operator and proposes a new probe guided version of the PLM operator designed to be used in conjunction with Multiobjective Evolutionary Algorithms (MOEAs). The proposed Probe Guided Mutation (PGM) operator is validated by using data sets from six different stock markets. The performance of the proposed PGM operator is assessed in comparison with the one of the classical PLM with the assistance of the Non-dominated Sorting Genetic Algorithm II (NSGAII) and the Strength Pareto Evolutionary Algorithm 2 (SPEA2). The evaluation of the performance is based on three performance metrics, namely Hypervolume, Spread and Epsilon indicator. The experimental results reveal that the proposed PGM operator outperforms with confidence the performance of the classical PLM operator for all performance metrics when applied to the solution of the cardinality constrained portfolio optimization problem (CCPOP). We also calculate the True Efficient Frontier (TEF) of the CCPOP by formulating the CCPOP as a Mixed Integer Quadratic Program (MIQP) and we compare the relevant results with the approximate efficient frontiers that are generated by the proposed PGM operator. The results confirm that the PGM operator generates near optimal solutions that lie very close or in certain cases overlap with the TEF.     

Applying combinatorial optimization metaheuristics to the golf scramble problem

One typical golf tournament format is termed a 'Scramble,' comprised of four-person teams. The participants are rank-ordered into four equally sized 'flights' based on integer-valued handicaps determined by skill level. One participant from each flight is selected to make up a team. Of interest is the assignment of teams in an 'equitable' fashion, where equitable is defined as minimizing the difference between the largest and smallest sum of the handicaps. For a typical tournament of 36 teams there are over 10 ¹²⁴ unique assignments. Since in general there are duplicate handicap values, the number of 'equivalent' assignments is reduced (but still very large). Various heuristics are explored for efficiently identifying an optimal or near optimal solution. These include descent heuristics, simulated annealing, tabu search, and genetic algorithms. Genetic algorithms outperform other heuristics by taking advantage of the problem structure.     

Evolutionary multi-objective optimization algorithms for fuzzy portfolio selection

In this paper, we consider a recently proposed model for portfolio selection, called Mean-Downside Risk-Skewness (MDRS) model. This modelling approach takes into account both the multidimensional nature of the portfolio selection problem and the requirements imposed by the investor. Concretely, it optimizes the expected return, the downside-risk and the skewness of a given portfolio, taking into account budget, bound and cardinality constraints. The quantification of the uncertain future return on a given portfolio is approximated by means of LR-fuzzy numbers, while the moments of its return are evaluated using possibility theory. The main purpose of this paper is to solve the MDRS portfolio selection model as a whole constrained three-objective optimization problem, what has not been done before, in order to analyse the efficient portfolios which optimize the three criteria simultaneously. For this aim, we propose new mutation, crossover and reparation operators for evolutionary multi-objective optimization, which have been specially designed for generating feasible solutions of the cardinality constrained MDRS problem. We incorporate the operators suggested into the evolutionary algorithms NSGAII, MOEA/D and GWASF-GA and we analyse their performances for a data set from the Spanish stock market. The potential of our operators is shown in comparison to other commonly used genetic operators and some conclusions are highlighted from the analysis of the trade-offs among the three criteria.     

An approximation algorithm for interval data minmax regret combinatorial optimization problems

The general problem of minimizing the maximal regret in combinatorial optimization problems with interval data is considered. In many cases, the minmax regret versions of the classical, polynomially solvable, combinatorial optimization problems become NP-hard and no approximation algorithms for them have been known. Our main result is a polynomial time approximation algorithm with a performance ratio of 2 for this class of problems.     

A new Chance-Variance optimization criterion for portfolio selection in uncertain decision systems

The Markowitz’s mean-variance (M-V) model has received widespread acceptance as a practical tool for portfolio optimization, and his seminal work has been widely extended in the literature. The aim of this article is to extend the M-V method in hybrid decision systems. We suggest a new Chance-Variance (C-V) criterion to model the returns characterized by fuzzy random variables. For this purpose, we develop two types of C-V models for portfolio selection problems in hybrid uncertain decision systems. Type I C-V model is to minimize the variance of total expected return rate subject to chance constraint; while type II C-V model is to maximize the chance of achieving a prescribed return level subject to variance constraint. Hence the two types of C-V models reflect investors’ different attitudes toward risk. The issues about the computation of variance and chance distribution are considered. For general fuzzy random returns, we suggest an approximation method of computing variance and chance distribution so that C-V models can be turned into their approximating models. When the returns are characterized by trapezoidal fuzzy random variables, we employ the variance and chance distribution formulas to turn C-V models into their equivalent stochastic programming problems. Since the equivalent stochastic programming problems include a number of probability distribution functions in their objective and constraint functions, conventional solution methods cannot be used to solve them directly. In this paper, we design a heuristic algorithm to solve them. The developed algorithm combines Monte Carlo (MC) method and particle swarm optimization (PSO) algorithm, in which MC method is used to compute probability distribution functions, and PSO algorithm is used to solve stochastic programming problems. Finally, we present one portfolio selection problem to demonstrate the developed modeling ideas and the effectiveness of the designed algorithm. We also compare the proposed C-V method with M-V one for our portfolio selection problem via numerical experiments.     

基于改进粒子群算法的投资组合选择模型

研究了在实际投资决策中存在交易成本(税收和交易费用)和投资数量约束下的投资组合选择问题,并进一步设计了一种求解该问题的改进粒子群算法.最后,给出了一个数值例子,说明该模型和方法的有效性.     

Using genetic algorithm to support clustering-based portfolio optimization by investor information

A clustering-based portfolio optimization scheme that employs a genetic algorithm (GA) based on investor information for active portfolio management is presented. Whereas numerous studies have investigated trading behaviors, investor performance, and portfolio investment strategies, few works have developed investment strategies based on investor information. This study is conducted in two phases. First, a basket of portfolio (i.e., a collection of stocks held in individual portfolios) is developed through a cluster analysis of investor information. A GA is then employed to optimize the weights of the selected stocks. And the optimized portfolio is rebalanced to get excess return. It is concluded that the proposed multistage portfolio optimization scheme for active portfolio management generates superior results than previously proposed methods for the Korean stock market.     

An effective asexual genetic algorithm for solving the job shop scheduling problem

By using the notion of elite pool, this paper presents an effective asexual genetic algorithm for solving the job shop scheduling problem. Based on mutation operations, the algorithm selectively picks the solution with the highest quality from the pool and after its modification, it can replace the solution with the lowest quality with such a modified solution. The elite pool is initially filled with a number of non-delay schedules, and then, in each iteration, the best solution of the elite pool is removed and mutated in a biased fashion through running a limited tabu search procedure. A decision strategy which balances exploitation versus exploration determines (i) whether any intermediate solution along the run of tabu search should join the elite pool, and (ii) whether upon joining a new solution to the pool, the worst solution should leave the pool. The genetic algorithm procedure is repeated until either a time limit is reached or the elite pool becomes empty. The results of extensive computational experiments on the benchmark instances indicate that the success of the procedure significantly depends on the employed mechanism of updating the elite pool. In these experiments, the optimal value of the well-known 10 × 10 instance, ft10, is obtained in 0.06 s. Moreover, for larger problems, solutions with the precision of less than one percent from the best known solutions are achieved within several seconds.     

Finding low autocorrelation binary sequences with memetic algorithms

This paper deals with the construction of binary sequences with low autocorrelation, a very hard problem with many practical applications. The paper analyzes several metaheuristic approaches to tackle this kind of sequences. More specifically, the paper provides an analysis of different local search strategies, used as stand-alone techniques and embedded within memetic algorithms. One of our proposals, namely a memetic algorithm endowed with a Tabu Search local searcher, performs at the state-of-the-art, as it consistently finds optimal sequences in considerably less time than previous approaches reported in the literature. Moreover, this algorithm is also able to provide new best-known solutions for large instances of the problem. In addition, a variant of this algorithm that explores only a promising subset of the whole search space (known as skew-symmetric sequences) is also analyzed. Experimental results show that this new algorithm provides new best-known solutions for very large instances of the problem.     

Comprehensive learning particle swarm optimization based memetic algorithm for model selection in short-term load forecasting using support vector regression

BackgroundShort-term load forecasting is an important issue that has been widely explored and examined with respect to the operation of power systems and commercial transactions in electricity markets. Of the existing forecasting models, support vector regression (SVR) has attracted much attention. While model selection, including feature selection and parameter optimization, plays an important role in short-term load forecasting using SVR, most previous studies have considered feature selection and parameter optimization as two separate tasks, which is detrimental to prediction performance.ObjectiveBy evolving feature selection and parameter optimization simultaneously, the main aims of this study are to make practitioners aware of the benefits of applying unified model selection in STLF using SVR and to provide one solution for model selection in the framework of memetic algorithm (MA).MethodsThis study proposes a comprehensive learning particle swarm optimization (CLPSO)-based memetic algorithm (CLPSO-MA) that evolves feature selection and parameter optimization simultaneously. In the proposed CLPSO-MA algorithm, CLPSO is applied to explore the solution space, while a problem-specific local search is proposed for conducting individual learning, thereby enhancing the exploitation of CLPSO.ResultsCompared with other well-established counterparts, benefits of the proposed unified model selection problem and the proposed CLPSO-MA for model selection are verified using two real-world electricity load datasets, which indicates the SVR equipped with CLPSO-MA can be a promising alternative for short-term load forecasting.