A DISTRIBUTIONAL ROBUST MINIMAX REGRET OPTIMIZATION APPROACH FOR MULTI-OBJECTIVE PORTFOLIO PROBLEMS

A distributional robust minimax regret optimization approach is proposed to deal with uncertain optimization problems. We reformulate the distributional robust minimax regret optimization problem as a tri-level optimization problem, where the set of distributions is constructed by the moment information which contains the true distribution on the uncertain data and the optimal decision in the worst case. Optimality conditions of the moment problem is obtained by the strong duality of the inner maximization problem and its Lagrange dual problem. Based on a randomization method, the distributional robust minimax regret optimization model is transformed into a semi-infinite min-max-min problem. By an adaptive smoothing method, we construct a family of finite min-max problems, which, together with their optimality functions are consistent approximations to the original pair. The proposed distributionally robust minimax regret approach is applied to solve the uncertain multi-objective portfolio problem in which the two objective functions are the portfolio return to be maximized and the mean absolute deviation as a risk measure to be minimized. An algorithm based on smoothing technique and Armijo linear search is proposed to find the distributionally robust minimax regret optimal solution. Convergence of the proposed algorithm is also established.