A Unified Single-loop Alternating Gradient Projection Algorithm for Nonconvex-Concave and Convex-Nonconcave Minimax Problems

Much recent research effort has been directed to the development of efficient algorithms for solving minimax problems with theoretical convergence guarantees due to the relevance of these problems to a few emergent applications. In this paper, we propose a unified single-loop alternating gradient projection (AGP) algorithm for solving nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. AGP employs simple gradient projection steps for updating the primal and dual variables alternatively at each iteration. We show that it can find an $\varepsilon$-stationary point of the objective function in $\mathcal{O}\left( \varepsilon ^{-2} \right)$ (resp. $\mathcal{O}\left( \varepsilon ^{-4} \right)$) iterations under nonconvex-strongly concave (resp. nonconvex-concave) setting. Moreover, its gradient complexity to obtain an $\varepsilon$-stationary point of the objective function is bounded by $\mathcal{O}\left( \varepsilon ^{-2} \right)$ (resp., $\mathcal{O}\left( \varepsilon ^{-4} \right)$) under the strongly convex-nonconcave (resp., convex-nonconcave) setting. To the best of our knowledge, this is the first time that a simple and unified single-loop algorithm is developed for solving both nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. Moreover, the complexity results for solving the latter (strongly) convex-nonconcave minimax problems have never been obtained before in the literature

Molecular dynamics simulation of graphene sinking during chemical vapor deposition growth on semi-molten Cu substrate

Copper foil is the most promising catalyst for the synthesis of large-area, high-quality monolayer graphene. Experimentally, it has been found that the Cu substrate is semi-molten at graphene growth temperatures. In this study, based on a self-developed C-Cu empirical potential and density functional theory (DFT) methods, we performed systematic molecular dynamics simulations to explore the stability of graphene nanostructures, i.e., carbon nanoclusters and graphene nanoribbons, on semi-molten Cu substrates. Many atomic details observed in the classical MD simulations agree well with those seen in DFT-MD simulations, confirming the high accuracy of the C-Cu potential. Depending on the size of the graphene island, two different sunken-modes are observed: (i) graphene island sinks into the first layer of the metal substrate and (ii) many metal atoms surround the graphene island. Further study reveals that the sinking graphene leads to the unidirectional alignment and seamless stitching of the graphene islands, which explains the growth of large single-crystal graphene on Cu foil. This study deepens our physical insights into the CVD growth of graphene on semi-molten Cu substrate with multiple experimental mysteries well explained and provides theoretic references for the controlled synthesis of large-area single-crystalline monolayer graphene