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.

This paper addresses the numerical solution of large-scale stable Lyapunov matrix equations, which occupy a central position in control theory, model reduction, and signal processing. While the classical Bartels-Stewart method holds a central position in control theory and related fields and is efficient for small-scale problems, it suffers from poor scalability as problem sizes increase, motivating the need for robust iterative techniques. To this end, we propose the extrapolated alternating direction implicit (EADI) method, an accelerated variant of the standard ADI approach that incorporates a low-cost refinement step governed by an extrapolation parameter \(\omega\) . We establish a rigorous convergence theory, deriving sufficient conditions for the shift parameters \(\alpha\) , \(\beta\) and the extrapolation parameter \(\omega\) to ensure that the spectral radius of the iteration matrix remains strictly less than unity. Numerical experiments demonstrate that the EADI algorithm significantly outperforms existing methods, including the standard ADI, GADI and the recently developed R-GADI iteration. The results indicate that EADI consistently achieves the target tolerance with substantially fewer iteration steps and reduced CPU time. Furthermore, the proposed method exhibits remarkable stability across varying problem dimensions, making it a highly effective and robust tool for high-dimensional sparse systems.