Proximal point algorithms （PPA） are attractive methods for solving monotone variational inequalities （MVI）. Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approx...Proximal point algorithms （PPA） are attractive methods for solving monotone variational inequalities （MVI）. Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approximate versions ofPPA （APPA） are developed for practical applications. In this paper, we compare two APPA methods, both of which can be viewed as prediction-correction methods. The only difference is that they use different search directions in the correction-step. By extending the general forward-backward splitting methods, we obtain Algorithm Ⅰ; in the same way, Algorithm Ⅱ is proposed by spreading the general extra-gradient methods. Our analysis explains theoretically why Algorithm Ⅱ usually outperforms Algorithm Ⅰ. For computation practice, we consider a class of MVI with a special structure, and choose the extending Algorithm Ⅱ to implement, which is inspired by the idea of Gauss-Seidel iteration method making full use of information about the latest iteration. And in particular, self-adaptive techniques are adopted to adjust relevant parameters for faster convergence. Finally, some numerical experiments are reported on the separated MVI. Numerical results showed that the extending Algorithm II is feasible and easy to implement with relatively low computation load.展开更多
The known results on comparison of extended mean values are used to compare power means, Stolarsky means and Heron mean. Some proofs of well known results are simplified with several new results obtained.
Briand et al. gave a counterexample showing that given g, Jensen's inequality for g-expectation usually does not hold in general This paper proves that Jensen's inequality for g-expectation holds in general if...Briand et al. gave a counterexample showing that given g, Jensen's inequality for g-expectation usually does not hold in general This paper proves that Jensen's inequality for g-expectation holds in general if and only if the generator g (t, z) is super-homogeneous in z. In particular, g is not necessarily convex in z.展开更多
This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Caratheodory spaces. The authors show some comparison results for solutions of the rele...This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Caratheodory spaces. The authors show some comparison results for solutions of the relevant differential inequalities and use them to get some symmetry and monotonicity properties of solutions, in bounded or unbounded domains.展开更多
Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative co...Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative constant, Li-Yau's estimation of the first eigenvalue is also given.展开更多
Recently, double projection methods for solving variational inequalities havereceived much attention due to their fewer projection times at each iteration. In this paper, weunify these double projection methods within...Recently, double projection methods for solving variational inequalities havereceived much attention due to their fewer projection times at each iteration. In this paper, weunify these double projection methods within two unified frameworks, which contain the existingdouble projection methods as special cases. On the basis of this unification, theoretical andnumerical comparison between these double projection methods is presented.展开更多
基金Project (No. 1027054) supported by the National Natural Science Foundation of China
文摘Proximal point algorithms （PPA） are attractive methods for solving monotone variational inequalities （MVI）. Since solving the sub-problem exactly in each iteration is costly or sometimes impossible, various approximate versions ofPPA （APPA） are developed for practical applications. In this paper, we compare two APPA methods, both of which can be viewed as prediction-correction methods. The only difference is that they use different search directions in the correction-step. By extending the general forward-backward splitting methods, we obtain Algorithm Ⅰ; in the same way, Algorithm Ⅱ is proposed by spreading the general extra-gradient methods. Our analysis explains theoretically why Algorithm Ⅱ usually outperforms Algorithm Ⅰ. For computation practice, we consider a class of MVI with a special structure, and choose the extending Algorithm Ⅱ to implement, which is inspired by the idea of Gauss-Seidel iteration method making full use of information about the latest iteration. And in particular, self-adaptive techniques are adopted to adjust relevant parameters for faster convergence. Finally, some numerical experiments are reported on the separated MVI. Numerical results showed that the extending Algorithm II is feasible and easy to implement with relatively low computation load.
文摘The known results on comparison of extended mean values are used to compare power means, Stolarsky means and Heron mean. Some proofs of well known results are simplified with several new results obtained.
基金Project supported by the National Natural Science Foundation of China (No.10131030)
文摘Briand et al. gave a counterexample showing that given g, Jensen's inequality for g-expectation usually does not hold in general This paper proves that Jensen's inequality for g-expectation holds in general if and only if the generator g (t, z) is super-homogeneous in z. In particular, g is not necessarily convex in z.
基金National Natural Science Foundation of China(No.10071023)MOST and Foundation for University Key TeacherShanghai Priority Academic Discipline
文摘This paper studies the properties of solutions of quasilinear equations involving the p-laplacian type operator in general Carnot-Caratheodory spaces. The authors show some comparison results for solutions of the relevant differential inequalities and use them to get some symmetry and monotonicity properties of solutions, in bounded or unbounded domains.
基金supported by the National Natural Science Foundation of China(Nos.11471246,11171253)the Natural Science Foundation of the Anhui Higher Education Institutions(No.KJ2014A257)
文摘Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative constant, Li-Yau's estimation of the first eigenvalue is also given.
基金This work is supported by the Natural Science Foundation of China (Grant No. 10171055, 10231060).
文摘Recently, double projection methods for solving variational inequalities havereceived much attention due to their fewer projection times at each iteration. In this paper, weunify these double projection methods within two unified frameworks, which contain the existingdouble projection methods as special cases. On the basis of this unification, theoretical andnumerical comparison between these double projection methods is presented.
