Mean-variance portfolio optimization problem under concave transaction costs and minimal transaction unit constraints

This paper addresses itself to a mean-variance portfolio optimization problem under concave transaction costs and minimal transaction unit constraints. Associated with portfolio selection is a fee for purchasing assets. Unit transaction fee is larger as the amount of transaction is smaller, so the transaction cost can be expressed as a concave function up to the certain point. But the transaction cost function becomes convex beyond the certain bound when the amount of transaction increases. The unit price of assets increases due to illiquidity/market impact effects. Minimal transaction unit is required in some market, for example, in China. Therefore, for the problem, we present a nonlinear integer programming model and give a genetic algorithm based on integer genetic coding to solve the model. It is shown by a series of numerical experiments that the model is effective and the proposed algorithm can solve the problem for practical scale in acceptable time.