Based on the risk control of conditional value-at-risk, distributionally robust return-risk optimization models with box constraints of random vector are proposed. They describe uncertainty in both the distribution form and moments (mean and covariance matrix of random vector). It is difficult to solve them directly. Using the conic duality theory and the minimax theorem, the models are reformulated as semidefinite programming problems, which can be solved by interior point algorithms in polynomial time. An important theoretical basis is therefore provided for applications of the models. Moreover, an application of the models to a practical example of portfolio selection is considered, and the example is evaluated using a historical data set of four stocks. Numerical results show that proposed methods are robust and the investment strategy is safe.
In this paper, an active set smoothing function based on the plus function is constructed for the maximum function. The active set strategy used in the smoothing function reduces the number of gradients and Hessians evaluations of the component functions in the optimization. Combing the active set smoothing function, a simple adjustment rule for the smoothing parameters, and an unconstrained minimization method, an active set smoothing method is proposed for solving unconstrained minimax problems. The active set smoothing function is continuously differentiable, and its gradient is locally Lipschitz continuous and strongly semismooth. Under the boundedness assumption on the level set of the objective function, the convergence of the proposed method is established. Numerical experiments show that the proposed method is feasible and efficient, particularly for the minimax problems with very many component functions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.