The paper outlines novel variational technique for finding microstructures of optimal multimaterial composites, bounds of composites properties, and multimaterial optimal designs. The translation method that is used for the exact two-material bounds is complemented by additional pointwise inequalities on stresses in materials within an optimal composites. The method leads to exact multimaterial bounds and provides a hint for optimal structures that may be multi-rank laminates or, for isotropic composites, ``wheel assemblages". The Lagrangian of the formulated nonconvex multiwell variational problem is equal to the energy of the best adapted to the loading microstructures plus the cost of the used materials; the technique improves both the lower and upper bounds for the quasiconvex envelope of that Lagrangian. The problem of 2d elastic composites is described in some details; on particular, the isotropic component of the quasiconvex envelope of three-well Lagrangian for elastic energy is computed. The obtained results are applied for computing of optimal multimaterial elastic designs; an example of such a design is demonstrated. Finally, the optimal ``wheel assemblages" are generalized and novel types of exotic microstructures with unusual properties are described.
High-Probability Convergence for Composite and Distributed Stochastic Minimization and Variational Inequalities with Heavy-Tailed Noise
Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Université de Montréal（蒙特利尔大学）；Mila（米拉研究所）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机过程研究所）；Ivannikov Institute for System Programming RAS（伊万尼科夫系统编程研究所（俄罗斯科学院））；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）
New Classes of Non-monotone Variational Inequality Problems Solvable via Proximal Gradient on Smooth Gap Functions
Antai College of Economics and Management, Shanghai Jiao Tong University（上海交通大学安泰经济与管理学院）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）；Department of Industrial and Systems Engineering, University of Minnesota（明尼苏达大学工业与系统工程系）
Optimal Rates for Federated Variational Inequalities under different relaxations of bounded variance assumption
Innopolis University（伊诺波利斯大学）；Lomonosov Moscow State University（莫斯科国立罗蒙诺索夫大学）；HSE University（高等经济大学）；Trusted AI Research Center, RAS（俄罗斯科学院可信人工智能研究中心）；Steklov Mathematical Institute, RAS（俄罗斯科学院斯捷克洛夫数学研究所）
