Optimal portfolio construction/rebalancing under nonconvex transaction cost

This paper is concerned with a minimal cost portfolio construction/rebalancing problem under nonconvex transaction costs and minimal transaction unit constraints. Recently, we proposed a branch and bound algorithm for solving a concave cost portfolio optimization problem under the mean-absolute deviation framework. We used a piecewise linear underestimating function for the nonconvex cost function and solved the resulting linear subproblems in a branch and bound method using a well-designed problem reduction technique. We showed that this algorithm generates an ε-optimal solution in a very efficient manner. We formulate the rebalancing problem as the piecewise-concave cost minimization problem and apply a similar branch and bound algorithm using a piecewise linear convex underestimation strategy. Also, we extend this algorithm to a more difficult class of problems where the transaction cost is a DC function and the amount of transaction is constrained to be an integer multiple of minimal transaction unit. The problem thus becomes a nonconvex minimization problem with integer constraints on the variables