We consider a portfolio selection problem in the presence of transaction costs. Transaction costs on each asset are assumed to be a convex function of the amount sold or bought. This function can be nondifferentiable in a finite number of points. The objective function of this problem is a sum of a convex twice differentiable function and a separable convex nondifferentiable function. We first consider the problem in the presence of linear constraints and later generalize the results to the case when the constraints are given by the convex piece-wise linear functions. <br /><br /> Due to the special structure, this problem can be replaced by an equivalent differentiable problem in a higher dimension. It's main drawback is efficiency since the higher dimensional problem is computationally expensive to solve. <br /><br /> We propose several alternative ways to solve this problem which do not require introducing new variables or constraints. We derive the optimality conditions for this problem using subdifferentials. First, we generalize an active set method to this class of problems. We solve the problem by considering a sequence of equality constrained subproblems, each subproblem having a twice differentiable objective function. Information gathered at each step is used to construct the subproblem for the next step. We also show how the nonsmoothness can be handled efficiently by using spline approximations. The problem is then solved using a primal-dual interior-point method. <br /><br /> If a higher accuracy is needed, we do a crossover to an active set method. Our numerical tests show that we can solve large scale problems efficiently and accurately.

We consider a portfolio selection problem in the presence of transaction costs. Transaction costs on each asset are assumed to be a convex function of the amount sold or bought. This function can be nondifferentiable in a finite number of points. The objective function of this problem is a sum of a convex twice differentiable function and a separable convex nondifferentiable function. We first consider the problem in the presence of linear constraints and later generalize the results to the case when the constraints are given by the convex piece-wise linear functions. <br /><br /> Due to the special structure, this problem can be replaced by an equivalent differentiable problem in a higher dimension. It's main drawback is efficiency since the higher dimensional problem is computationally expensive to solve. <br /><br /> We propose several alternative ways to solve this problem which do not require introducing new variables or constraints. We derive the optimality conditions for this problem using subdifferentials. First, we generalize an active set method to this class of problems. We solve the problem by considering a sequence of equality constrained subproblems, each subproblem having a twice differentiable objective function. Information gathered at each step is used to construct the subproblem for the next step. We also show how the nonsmoothness can be handled efficiently by using spline approximations. The problem is then solved using a primal-dual interior-point method. <br /><br /> If a higher accuracy is needed, we do a crossover to an active set method. Our numerical tests show that we can solve large scale problems efficiently and accurately.