Evolutionary Optimization of Constrained $k$-Means Clustered Assets for Diversification in Small Portfolios

The problem of portfolio optimization has been rendered complex for direct solving by traditional and numerical approaches when constraints that model investor preferences and/or market friction are included in the mathematical model, and for such cases, heuristic approaches have been sought for their solution. In this paper, we discuss the solution of a subclass of portfolio optimization problems, which include basic, bounding, cardinality, and class constraints in its fold, with the investor targeting diversification in small portfolios. The strategy employs $k$-means cluster analysis to eliminate the cardinality constraint and thereby simplify the mathematical model and the evolutionary optimization process. An evolution strategy which is a variant of the conventional $( \mu + \lambda )$ evolution strategy but employs real coded genes with genetic inheritance operators such as arithmetic variable point cross over and real number uniform mutation to initiate a fast converging reproduction process has been evolved to solve the simplified model. The strategy also employs refined weight standardization algorithms to tackle the bounding and class constraints. Experimental results have been demonstrated on the Bombay Stock Exchange, India (BSE200 index, Period: July 2001–July 2006) and on the Tokyo Stock Exchange, Japan (Nikkei225 index, Period: March 2002–March 2007) datasets and compared with those obtained by the Markowitz mean-variance, random matrix theory filtered, and quadratic programming-based solution models for the appropriate cases.