Portfolio selection with an alternative measure of risk: Computational performances of particle swarm optimization and genetic algorithms

Particle Swarm Optimization for Preference Disaggregation in Multicriteria Credit Scoring Problems

“…For more details about this method and about the relationships between the solutions of the constrained problem (8) and those of the unconstrained problem (13), see [5,8,14,17]. We only remark here that the penalty function P(w, q, t, s; ε) is clearly nondifferentiable because of the 1 -norm in (14).…”

“…To avoid this problem, different strategies have been proposed in the literature, and most of them involve the repositioning of the particles [18] or the introduction of some external criteria to rearrange the components of the particles [6,16]. In this paper we follow the same approach adopted in [5], which consists in keeping PSO as in its original formulation and reformulating the optimization problem into an unconstrained one: min w,q,t,s P(w, q, t, s; ε) (13) where the objective function P(w, q, t, s; ε) is defined as follows:…”

An Evolutionary Approach to Preference Disaggregation in a MURAME-Based Credit Scoring Problem

“…To avoid this problem, different strategies have been proposed in the literature, and most of them involve the repositioning of the particles ( [22]) or the introduction of some external criteria to rearrange the components of the particles ( [8] and [20]). In this paper we follow the same approach adopted in [6], which consists in keeping PSO as in its original formulation and reformulating the optimization problem into an unconstrained one: min w,q,t,s P (w, q, t, s; ε) (17) where the objective function P (w, q, t, s; ε) is defined as follows:…”

“…The approach adopted is called ℓ 1 penalty function method; for more details about this method and about the relationships between the solutions of the constrained problem (12) and those of the unconstrained problem (17), see [21,11] and [17,6]. We may observe that the penalty function P (w, q, t, s; ε) is clearly nondifferentiable because of the ℓ 1 -norm in (18); this feature contributes to motivate the choice of using PSO, since it does not require the derivatives of P (w, q, t, s; ε).…”

Distance Measures for Portfolio Selection

“…The asset whose share is to be increased must be chosen so that x i C step Ä 1; if there is no asset that satisfy this constraint, step value is modified accordingly. -Initial solution The starting solution is created randomly in order to satisfy all constraints in the formulation; -Cost Function For Markowitz portfolios we use a penalty approach (Corazza et al 2012) in which the cost function is given by the sum of the portfolio variance (risk) and the degree of violation of the return constraint; for Index Tracking portfolios we do not add any penalty to the objective function, which is the tracking error defined as (19); -Local Search Strategies Threshold Accepting algorithm was implemented with the following settings: Iterations D 10,000, Restart D 20, Epochs D 5. These parameters have been estimated by F-Race (Birattari et al 2010).…”