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

Well-posedness of a class of perturbed optimization problems in Banach spaces

AbstractLet X be a Banach space and Z a nonempty subset of X. Let J:Z→R be a lower semicontinuous function bounded from below and p⩾1. This paper is concerned with the perturbed optimization problem of finding z0∈Z such that ‖x−z0‖p+J(z0)=infz∈Z{‖x−z‖p+J(z)}, which is denoted by minJ(x,Z). The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x∈X for which every minimizing sequence of the problem minJ(x,Z) has a converging subsequence is a dense Gδ-subset of X∖Z0, where Z0 is the set of all points z∈Z such that z is a solution of the problem minJ(z,Z). If additionally p>1 and X is J-strictly convex with respect to Z, then the set of all x∈X for which the problem minJ(x,Z) is well-posed is a dense Gδ-subset of X∖Z0

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

New strong convergence method for the sum of two maximal monotone operators

This paper aims to obtain a strong convergence result for a Douglas–Rachford splitting method with inertial extrapolation step for finding a zero of the sum of two set-valued maximal monotone operators without any further assumption of uniform monotonicity on any of the involved maximal monotone operators. Furthermore, our proposed method is easy to implement and the inertial factor in our proposed method is a natural choice. Our method of proof is of independent interest. Finally, some numerical implementations are given to confirm the theoretical analysis

Comparison of the Offspring Sex Ratio Between Cleavage Stage Embryo Transfer and Blastocyst Transfer

SummaryObjectiveTo compare the sex ratio of offspring born after cleavage stage embryo transfer and blastocyst transfer.Materials and MethodsIn this retrospective study of embryo transfer (ET), we included 473 offspring from 446 deliveries during the period January 2002 to December 2007. Statistical analysis was performed on the sex ratio of offspring resulting from day 3 cleavage stage embryo transfer and from sequential blastocyst culture transfer.ResultsIn total, 446 patient deliveries were included in this analysis. There were 251 singleton pregnancies, 109 twin pregnancies, and four triplet pregnancies. The total number of offspring was 473, of which 118 resulted from day 3 ETs, and 355 resulted from blastocyst ETs. At our center, the influence on the sex ratio of cleavage stage ET and blastocyst-stage ET showed a bias towards males in both cases. The overall female to male ratio for offspring resulting from day 3 ETs was not significantly higher than the same ratio for offspring resulting from blastocyst ETs (p = 0.24; odds ratio, 0.762). The female to male ratio for either singleton births or multiple deliveries was also not significantly different between day 3 ETs and blastocyst ETs.ConclusionThe sex ratio was influenced by cleavage stage ET and blastocyst-stage ET. In both cases, there was a bias towards males. In addition, when blastocyst ET was compared with day 3 ET, there was no further increase in the percentage of male offspring

The Inhibitory Effect of Ellagic Acid on Cell Growth of Ovarian Carcinoma Cells

Ellagic acid (EA) is able to inhibit the growth of several cancer cells; however, its effect on human ovarian carcinoma cells has not yet been investigated. Ovarian carcinoma ES-2 and PA-1 cells were treated with EA (10~100 μM) and assessed for viability, cell cycle, apoptosis, anoikis, autophagy, and chemosensitivity to doxorubicin and their molecular mechanisms. EA inhibited cell proliferation in a dose- and time-dependent manner by arresting both cell lines at the G1 phase of the cell cycle, which were from elevating p53 and Cip1/p21 and decreasing cyclin D1 and E levels. EA also induced caspase-3-mediated apoptosis by increasing the Bax : Bcl-2 ratio and restored anoikis in both cell lines. The enhancement of apoptosis and/or inhibition of autophagy in these cells by EA assisted the chemotherapy efficacy. The results indicated that EA is a potential novel chemoprevention and treatment assistant agent for human ovarian carcinoma