In this paper, we present two robust reward‐risk ratio optimization models. Two new models contain the worst case of not only conditional value‐at‐risk (CVaR), but also standard deviation (SD). Using properties of reward measure, CVaR measure, and standard deviation measure, new models can be proved to equivalent to min‐max problems. When the uncertainty set is an ellipsoid, new models can be further rewritten as second‐order cone problems step by step. Finally, we implement new models to portfolio problems. It shows that new models are robust and comparable with mean‐CVaR ratio model. Since considering standard deviation, allocation decision obtained by new models can give reasonable rewards according to personal preferences.

In this paper, we focus on a class of robust optimization problems whose objectives and constraints share the same uncertain parameters. The existing approaches separately address the worst cases of each objective and each constraint, and then reformulate the model by their respective dual forms in their worst cases. These approaches may result in that the value of uncertain parameters in the optimal solution may not be the same one as in the worst case of each constraint, since it is highly improbable to reach their worst cases simultaneously. In terms of being too conservative for this kind of robust model, we propose a new robust optimization model with shared uncertain parameters involving only the worst case of objectives. The proposed model is evaluated for the multi-stage logistics production and inventory process problem. The numerical experiment shows that the proposed robust optimization model can give a valid and reasonable decision in practice.

In this paper, elliptic optimal control problems with $L^1$-control cost and box constraints on the control are considered. To numerically solve the optimal control problems, we use the First optimize, then discretize approach. We focus on the inexact alternating direction method of multipliers (iADMM) and employ the standard piecewise linear finite element approach to discretize the subproblems in each iteration. However, in general, solving the subproblems is expensive, especially when the discretization is at a fine level. Motivated by the efficiency of the multigrid method for solving large-scale problems, we combine the multigrid strategy with the iADMM algorithm. Instead of fixing the mesh size before the computation process, we propose the strategy of gradually refining the grid. Moreover, to overcome the difficulty whereby the $L^1$-norm does not have a decoupled form, we apply nodal quadrature formulas to approximately discretize the $L^1$-norm and $L^2$-norm. Based on these strategies, an efficient multilevel heterogeneous ADMM (mhADMM) algorithm is proposed. The total error of the mhADMM consists of two parts: the discretization error resulting from the finite-element discretization and the iteration error resulting from solving the discretized subproblems. Both errors can be regarded as the error of inexactly solving infinite-dimensional subproblems. Thus, the mhADMM can be regarded as the iADMM in function space. Furthermore, theoretical results on the global convergence, as well as the iteration complexity results $o(1/k)$ for the mhADMM, are given. Numerical results show the efficiency of the mhADMM algorithm.

A new distributionally robust ratio optimization model is proposed under the known first and second moments of the uncertain distributions. In this article, both standard deviation (SD) and conditional value-at-risk (CVaR) are used to measure the risk, avoiding both fat-tail and volatility. The new model can be reduced to a simple distributionally robust model under assumptions on the measurements of reward, CVaR and SD. Furthermore, it can be rewritten as a tractable semi-definite programming problem by the duality theorem under partially known information of the uncertain parameters. Finally, the model is tested on portfolio problems and verified from numerical results that it can give a reasonable decision under only the first and second moments.

The study of ship floating balance plays a very important role in the safety of ships. According to certain conditions of ship statics, the nonlinear equations of ship balance can be established with relevant parameters such as ship weight, draft, heeling angle, longitudinal inclination angle and static moment. In this paper, motivated by the least square method, the nonlinear equations of floating equilibrium are transformed to a nonlinear programming problem. The draught, heel and trim of the ship are taken as variables, and inequality constrained by the actual situation. The penalty function for nonlinear programming is considered and minimized by particle swarm optimization. The proposed optimization method simplifies the calculation process by avoiding the calculation of parameters. Numerical results verify the efficiency of the proposed method.