Solving the mean-variance customer portfolio in Markov chains using iterated quadratic/Lagrange programming: A credit-card customer limits approach

In this paper we present a new mean-variance customer portfolio optimization algorithm for a class of ergodic finite controllable Markov chains. In order to have a realistic result we propose an iterated two-step method for solving the given portfolio constraint problem: (a) the first step is designed to optimize the nonlinear problem using a quadratic programming method for finding the long run fraction of the time that the system is in a given state (segment) and an action (promotion) is chosen and, (b) the second step is designed to find the optimal number of customers using a Lagrange programming approach. Both steps are based on the c-variable method to make the problem computationally tractable and obtain the optimal solution for the customer portfolio. The Tikhonov's regularization method is used to ensure the convergence of the objective-function to a single optimal portfolio solution. We prove that the proposed method converges by the Weierstrass theorem: the objective function of the mean-variance customer portfolio problem decreases, it is monotonically non-decreasing and bounded from above. In addition, for solving the customer portfolio problem we consider both, a constant risk-aversion restriction and budget limitations. The constraints imposed by the system produce mixed strategies. Effectiveness of the proposed method is successfully demonstrated theoretically and by a simulated experiment related with credit-card and customer-credit limits approach for a bank.