This paper focuses on robust multi-criteria traffic network equilibrium problems with path capacity constraints. The concepts of robust vector equilibrium and robust vector equilibrium with respect to worst case are introduced, respectively. To generate a subset of robust vector equilibrium, an equivalent min-max optimization problem is constructed. In addition, a direct search algorithm, in which the step size without derivatives and redundant parameters, for solving this min-max problem is proposed. To find a subset of robust vector equilibrium with respect to worst case, an equivalent smoothing optimization problem based on a variant version of ReLU activation function is established. Numerical experiments are also presented to expound the advantages of the algorithms.

In this paper, a distributed algorithm is proposed to solve a consensus convex optimization problem. It is a Jacobi-proximal alternating direction method of multipliers with a damping parameter $\gamma$ in the iteration of multiplier. Compared with existing algorithms, it has the following nice properties: (1) The restriction on proximal matrix is relaxed substantively, thus alleviating the weight of the proximal term. Therefore, the algorithm has a faster convergence speed. (2) The convergence analysis of the algorithm is established for any damping parameter $\gamma\in(0,2]$, which is larger ones in the literature. In addition, some numerical experiments and an application to a logistic regression problem are provided to validate the effectiveness and the characteristics of the proposed algorithm.

Osteosarcoma (OSA) is the most common primary malignant bone tumor. More than 40% of patients with OSA have poor prognoses. We aimed to discover a biomarker for patient stratification and therapeutic targets for these high-risk patients. Using Single Sample Gene Set Enrichment Analysis (ssGSEA) and univariate Cox analysis, six hallmarks were identified as significant prognostic factors for overall survival (OS). Three were selected to construct a multivariate Cox model. Then, WGCNA, univariate Cox regression, Kaplan-Meier (KM) survival analyses, and multivariate Cox analyses were combined to filter promising candidates and establish a seven-gene signature to predict OS, whose prognostic value was validated internally and externally. Subsequently, Differential Expression Analysis was conducted between high- and low-risk patients, and the Robust Rank Aggregation algorithm was used to determine the robust DEGs. Metascape was used to perform pathway and process enrichment analyses as well as construct protein-protein interaction (PPI) networks. Finally, RPS28 was identified as an independent risk factor by using univariate and multivariate Cox regression, which was preliminarily validated as a promising therapeutic target by using RNA interference. In conclusion, we might contribute to optimizing risk stratification and an excellent therapeutic target for high-risk patients with OSA.