Applications of Integer Programming to Financial Optimization

Abstract

The problems to be discussed in this chapter make up a class of nonconvex financial optimization problems that can be solved within a practical amount of time using the state-of-the-art integer programming methodologies. We will first discuss mean-risk portfolio optimization problems (Elton and Gruber, 1998; Konno and Yamazaki, 1991; Markowitz, 1959) subject to nonconvex constraints such as minimal transaction unit constraints and cardinality constraints on the number of assets to be included in the portfolio (Konno and Yamamoto, 2005b). Also, we will discuss problems with piecewise linear nonconvex transaction costs (Konno and Wijayanayake, 2001, 2002; Konno and Yamamoto, 2005a, 2005b). It will be shown that fairly large-scale problems can now be solved to optimality by formulating the problem as a mixed 0−1 integer linear programming problem if we use convex piecewise linear risk measure such as absolute deviation instead of variance. The second class of problems are so-called maximal predictability portfolio optimization problems (Lo and MacKinlay, 1997), where we maximize the coefficient of determination of the portfolio using factor models. This model, though very promising, was set aside long ago, since we need to maximize the ratio of convex quadratic functions, which is not a concave function. This problem can be solved to optimality by a hyper-rectangular subdivision algorithm (Gotoh and Konno, 2001; Phong et al., 1995) or by 0−1 integer programming approach (Yamamoto and Konno, to appear; Yamamoto et al., to appear) if the number of assets is relatively small. To solve larger problems, we employ absolute deviation as a measure of variation and define the coefficient of determination as the ratio of functions defined by the sum of absolute values of linear functions. The resulting nonconvex minimization problem can be reformulated as a linear complementarity problem that can be solved by using 0−1 integer programming algorithms (Konno et al., 2007, to appear).