Equilibrium problems on Riemannian manifolds with applications

We study the equilibrium problem on general Riemannian manifolds. The results on existence of solutions and on the convex structure of the solution set are established. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. As applications, the corresponding results for the mixed variational inequality and the Nash equilibrium are obtained. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388, 61-77, 2012 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on Hadamard manifolds.National Natural Science Foundation of ChinaNatural Science Foundation of Guizhou Province (China)Dirección General de Enseñanza SuperiorJunta de AndalucíaNational Science Council of Taiwa

Proximal point algorithms on Hadamard manifolds: linear convergence and finite termination

In the present paper, we consider inexact proximal point algorithms for finding singular points of multivalued vector fields on Hadamard manifolds. The rate of convergence is shown to be linear under the mild assumption of metric subregularity. Furthermore, if the sequence of parameters associated with the iterative scheme converges to 0, then the convergence rate is superlinear. At the same time, the finite termination of the inexact proximal point algorithm is also provided under a weak sharp minima-like condition. Applications to optimization problems are provided. Some of our results are new even in Euclidean spaces, while others improve and/or extend some known results in Euclidean spaces. As a matter of fact, in the case of exact proximal point algorithm, our results improve the corresponding results in [G. C. Bento and J. X. Cruz Neto, Optim., 63 (2014), pp. 1281–1288]. Finally, several examples are provided to illustrate that our results are applicable while the corresponding results in the Hilbert space setting are not.National Natural Science Foundation of ChinaZhejiang Provincial Natural Science Foundation of ChinaDirección General de Enseñanza SuperiorJunta de AndalucíaNational Science Council of Taiwa

Weak Sharp Minima on Riemannian Manifolds

This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and obtain their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and their Hadamard counterparts. A number of the results obtained in this paper are also new for the case of conventional problems in linear spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper

Impact of Aerosols on Convective Clouds and Precipitation

Aerosols are a critical factor in the atmospheric hydrological cycle and radiation budget. As a major reason for clouds to form and a significant attenuator of solar radiation, aerosols affect climate in several ways. Current research suggests that aerosol effects on clouds could further extend to precipitation, both through the formation of cloud particles and by exerting persistent radiative forcing on the climate system that disturbs dynamics. However, the various mechanisms behind these effects, in particular the ones connected to precipitation, are not yet well understood. The atmospheric and climate communities have long been working to gain a better grasp of these critical effects and hence to reduce the significant uncertainties in climate prediction resulting from such a lack of adequate knowledge. The central theme of this paper is to review past efforts and summarize our current understanding of the effect of aerosols on precipitation processes from theoretical analysis of microphysics, observational evidence, and a range of numerical model simulations. In addition, the discrepancy between results simulated by models, as well as that between simulations and observations will be presented. Specifically, this paper will address the following topics: (1) fundamental theories of aerosol effects on microphysics and precipitation processes, (2) observational evidence of the effect of aerosols on precipitation processes, (3) signatures of the aerosol impact on precipitation from large-scale analyses, (4) results from cloud-resolving model simulations, and (5) results from large-scale numerical model simulations. Finally, several future research directions on aerosol - precipitation interactions are suggested