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In this paper, we establish new convergence results for the quantized distributed gradient descent and suggest a novel strategy of choosing the stepsizes for the high-performance of the algorithm. Under the strongly convexity assumption on the aggregate cost function and the smoothness assumption on each local cost function, we prove the algorithm converges exponentially fast to a small neighborhood of the optimizer whose radius depends on the stepsizes. Based on our convergence result, we suggest an effective selection of stepsizes which repeats diminishing the stepsizes after a number of specific iterations. Both the convergence results and the effectiveness of the suggested stepsize selection are also verified by the numerical experiments., 21 pages, 4 figures

Gradient-push algorithm has been widely used for decentralized optimization problems when the connectivity network is a direct graph. This paper shows that the gradient-push algorithm with stepsize $\alpha>0$ converges exponentially fast to an $O(\alpha)$-neighborhood of the optimizer under the assumption that each cost is smooth and the total cost is strongly convex. Numerical experiments are provided to support the theoretical convergence results.

In this work, we are concerned with the decentralized optimization problem: \begin{equation*} \min_{x \in \Omega}~f(x) = \frac{1}{n} \sum_{i=1}^n f_i (x), \end{equation*} where $\Omega \subset \mathbb{R}^d$ is a convex domain and each $f_i : \Omega \rightarrow \mathbb{R}$ is a local cost only known to agent $i$. A fundamental algorithm is the decentralized projected gradient method (DPG) given by \begin{equation*} x_i(t+1)=\mathcal{P}_\Omega\Big[\sum^n_{j=1}w_{ij} x_j(t) -\alpha(t)\nabla f_i(x_i(t))\Big] \end{equation*} where $\mathcal{P}_{\Omega}$ is the projection operator to $\Omega$ and $ \{w_{ij}\}_{1\leq i,j \leq n}$ are communication weight among the agents. While this method has been widely used in the literatures, its sharp convergence property has not been established well so far, except for the special case $\Omega = \mathbb{R}^n$. This work establishes new convergence estimates of DPG when the aggregate cost $f$ is strongly convex and each function $f_i$ is smooth. If the stepsize is given by constant $\alpha (t) \equiv\alpha >0$ and suitably small, we prove that each $x_i (t)$ converges to an $O(\sqrt{\alpha})$-neighborhood of the optimal point. In addition, we take a one-dimensional example $f_i$ and prove that the point $x_i (t)$ converges to an $O(\alpha)$-neighborhood of the optimal point. Also, we obtain convergence estimates for decreasing stepsizes. Numerical experiments are also provided to support the convergence results., Comment: 27 pages, 3 figures

In this work, we establish convergence results for the distributed proximal point algorithm (DPPA) for distributed optimization problems. We consider the problem on the whole domain Rd and find a general condition on the stepsize and cost functions such that the DPPA is stable. We prove that the DPPA with stepsize $\eta > 0$ exponentially converges to an $O(\eta)$-neighborhood of the optimizer. Our result clearly explains the advantage of the DPPA with respect to the convergence and stability in comparison with the distributed gradient descent algorithm. We also provide numerical tests supporting the theoretical results.

In this paper, we consider the online proximal mirror descent for solving the time-varying composite optimization problems. For various applications, the algorithm naturally involves the errors in the gradient and proximal operator. We obtain sharp estimates on the dynamic regret of the algorithm when the regular part of the cost is convex and smooth. If the Bregman distance is given by the Euclidean distance, our result also improves the previous work in two ways: (i) We establish a sharper regret bound compared to the previous work in the sense that our estimate does not involve $O(T)$ term appearing in that work. (ii) We also obtain the result when the domain is the whole space $\mathbb{R}^n$, whereas the previous work was obtained only for bounded domains. We also provide numerical tests for problems involving the errors in the gradient and proximal operator., Comment: 16 pages, 5 figures

Distributed optimization has received a lot of interest in recent years due to its wide applications in various fields. In this work, we revisit the convergence property of the decentralized gradient descent [A. Nedi{\'c}-A.Ozdaglar (2009)] on the whole space given by $$ x_i(t+1) = \sum^m_{j=1}w_{ij}x_j(t) - \alpha(t) \nabla f_i(x_i(t)), $$ where the stepsize is given as $\alpha (t) = \frac{a}{(t+w)^p}$ with $0< p\leq 1$. Under the strongly convexity assumption on the total cost function $f$ with local cost functions $f_i$ not necessarily being convex, we show that the sequence converges to the optimizer with rate $O(t^{-p})$ when the values of $a>0$ and $w>0$ are suitably chosen., Comment: 25 pages, fixed typos

In this paper, we propose an integral based event-triggering controller for bipartite consensus of the multi-agent systems whose dynamics are described by general linear system. We prove that the system achieves the bipartite consensus in asymptotic regime and there is a positive minimum inter-event time (MIET) between two consecutive triggering times of each agent. The proof of the asymptotive stability involves a novel argument used to bound the norm of the difference between the true state and its estimated state (for each time) by an integration of its square. Numerical results are provided supporting the effectiveness of the proposed controller.

In this paper, we propose a framework under which the decentralized optimization algorithms suggested in \cite{JKJJ,MA, NO,NO2} can be treated in a unified manner. More precisely, we show that the distributed subgradient descent algorithms \cite{JKJJ, NO}, the subgradient-push algorithm \cite{NO2}, and the distributed algorithm with row-stochastic matrix \cite{MA} can be derived by making suitable choices of consensus matrices, step-size and subgradient from the decentralized subgradient descent proposed in \cite{NO}. As a result of such unified understanding, we provide a convergence proof that covers the algorithms in \cite{JKJJ,MA, NO,NO2} under a novel algebraic condition that is strictly weaker than the conventional graph-theoretic condition in \cite{NO}. This unification also enables us to derive a new distributed optimization scheme.

In this paper, we are concerned with a nonlinear optimal control problem of ordinary differential equations. We consider a discretization of the problem with the discontinuous Galerkin method with arbitrary order $r \in \mathbb{N}\cup \{0\}$. Under suitable regularity assumptions on the cost functional and solutions of the state equations, we provide sharp estimates for the error of the approximate solutions. Numerical experiments are presented supporting the theoretical results.

In this paper, we are concerned with the convergence rate of a FEM based numerical scheme approximating extremal functions of the Sobolev inequality. We prove that when the domain is polygonal and convex in $\R^2$, the convergence of a finite element solution to an exact extremal function in $L^2$ and $H^1$ norms has the rates $O(h^2)$ and $O(h)$ respectively, where $h$ denotes the mesh size of a triangulation of the domain., 14 pages

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schr��dinger equation $$\left( \sqrt{-c^2��+ m^2 c^4}-mc^2\right) u + ��u = |u|^{p-1}u\quad \textrm{in}~\mathbb{R}^n~(n \geq 2)$$ involving an $H^{1/2}$-critical/supercritical power-type nonlinearity, i.e., $p \geq \frac{n+1}{n-1}$. We prove that in the non-relativistic regime, there exists a nontrivial solution provided that the nonlinearity is $H^{1/2}$-critical/supercritical but it is $H^1$-subcritical. On the other hand, we also show that there is no nontrivial bounded solution either $(i)$ if the nonlinearity is $H^{1/2}$-critical/supercritical in the ultra-relativistic regime or $(ii)$ if the nonlinearity is $H^1$-critical/supercritical in all cases.

This is the first of two papers which study asymptotic behavior of minimal energy solutions to the fractional Lane-Emden system in a smooth bounded domain $\Omega$ \[(-\Delta)^s u = v^p, \quad (-\Delta)^s v = u^q \text{ in } \Omega \quad \text{and} \quad u = v = 0 \text{ on } \pa \Omega \quad \text{for } 0 < s < 1\] under the assumption that the subcritical pair $(p,q)$ approaches to the critical Sobolev hyperbola. If $p = 1$, the above problem is reduced to the subcritical higher-order fractional Lane-Emden equation with the Navier boundary condition \[(-\Delta)^s u = u^{\frac{n+2s}{n-2s}-\ep} \text{ in } \Omega \quad \text{and} \quad u = (-\Delta)^{s \over 2} u = 0 \quad \text{for } 1 < s < 2.\] The main objective of this paper is to deduce the existence of minimal energy solutions, and to examine their (normalized) pointwise limits provided that $\Omega$ is convex. As a by-product of our study, a new approach for the existence of an extremal function for the Hardy-Littlewood-Sobolev inequality is provided., Comment: 25 pages

In this paper, we consider nonlocal Schr��dinger equations with certain potentials $V$ given by an integro-differential operator $L_K$ as follows; \begin{equation*}L_K u+V u=f\,\,\text{ in $\BR^n$ }\end{equation*} where $V\in\rh^q$ for $q>\f{n}{2s}$ and $0

In this paper, we prove the Malgrange-Ehrenpreis theorem for nonlocal Schr��dinger operators $L_K+V$ with nonnegative potentials $V\in L^q_{\loc}(\BR^n)$ for $q>\f{n}{2s}$ with $0

In this paper, we establish a convergence result for the operator splitting scheme $Z_��$ introduced by Ignat, with initial data in $H^1$, for the nonlinear Schr��dinger equation : $$ \partial_t u = i ��u + i��|u|^{p} u,\qquad u (x,0) =��(x), $$ where $(x,t) \in \mathbb{R}^d \times [0,\infty)$, with $0< p < 4/(d-2)$ for $d\geq3$ and $0< p

In this paper, we are concerned with the nonrelativistic limit of the following pseudo-relativistic equation with Hartree nonlinearity or power type nonlinearity \[ \left(\sqrt{-\hbar^2c^2 \Delta +m^2c^4} - mc^2 \right) u + \mu u = \mathcal{N}(u), \] where $c$ denotes the speed of light. We prove that the ground states of this equation converges to the ground state of its nonrelativistic counterpart \[ -\frac{\hbar^2}{2m}\Delta u + \mu u = \mathcal{N}(u) \] with an explicit convergence rate $1/c^2$ in arbitrary order as $c \to \infty$. Moreover, we show that this rate is optimal., Comment: 23 pages, title and abstract were modified

We study qualitative properties of solutions to the fractional Lane-Emden-Fowler equations with slightly subcritical exponents where the associated fractional Laplacian is defined in terms of either the spectra of Dirichlet Laplacian or the integral representation. As a consequence, we classify the asymptotic behavior of all finite energy solutions. Our method also provides a simple and unified approach to deal with the classical (local) Lane-Emden-Fowler equation for any dimension greater than 2., 32 pages, revised introduction

In this paper we study the asymptotic behavior of minimal energy solutions to the Lane-Emden system $-\Delta u = v^p$ and $-\Delta v = u^q$ on bounded domains as the index $(p,q)$ approaches to the critical hyperbola from below. Precisely, we remove the convexity assumption on the domain in the result of Guerra \cite{G}. The main task is to get the uniform boundedness of the solutions near the boundary because it is difficulty to adapt the moving plane method for the system on nonconvex domains if $\max \{p,q\} > \frac{n+2}{n-2}$. For the purpose, we shall derive a contradiction by exploiting carefully the Pohozaev type identity if the maximum point approaches to the boundary., Comment: 44 pages

In this paper we study standing waves for pseudo-relativistic nonlinear Schr��dinger equations. In the first part we find ground state solutions. We also prove that they have one sign and are radially symmetric. The second part is devoted to take nonrelativistic limit of the ground state solutions in $H^1 (\mathbb{R}^n)$ space., 17 pages

The objective of this paper is to obtain qualitative characteristics of multi-bubble solutions to the Lane-Emden-Fowler equations with slightly subcritical exponents given any dimension $n \ge 3$. By examining the linearized problem at each $m$-bubble solution, we provide a number of estimates on the first $(n+2)m$-eigenvalues and their corresponding eigenfunctions. Specifically, we present a new proof of the classical theorem due to Bahri-Li-Rey (Calc. Var. Partial Differential Equations 3 (1995) 67-93) which states that if $n \ge 4$, then the Morse index of a multi-bubble solution is governed by a certain symmetric matrix whose component consists of a combination of Green's function, the Robin function, and their first and second derivatives. Our proof also allows us to handle the intricate case $n = 3$., 31 pages

In this paper, we study a class of semilinear nonlocal elliptic equations posed on settings without compact Sobolev embedding. More precisely, we prove the existence of infinitely many solutions to the fractional Brezis-Nirenberg problems on bounded domain., Comment: 29 pages, Typos are fixed, Intro and refereces are extended

In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen as a nonlocal analog of the results of Han (Ann. Inst. H. Poincare Anal. Non Lineaire, 1991) and Rey (J. Funct. Anal., 1990)., Parts of Section 1, 3, 4 and 6 are revised, and typos are fixed. To appear in Journal of Functional Analysis

Hormander-Mihklin type multiplier theorem on compacts manifolds withour boundary has been obtained by using the wave kernels. We consider maximal multiplies on this setting. To obtain the result, we carefully deal with the remainder terms and find an additional information on the wave kernels., 18 pages

In this paper, we consider oscillating convolution operotors on the Heisenberg group $H^n_a$ with respect to the norm $\rho(x,t) = \rho_1(b x, b t)$ with $\rho_1(x,t)= (|x|^4 + t^2)^{1/4}$. We obtain $L^2$ boundedness properties using the oscillatory integral estimates for degenerate phases in the Euclidean setting. Our result contains an improvement of the Lyall's result for the cases $\frac{a^2}{b^2} \geq C_{\beta}$., Comment: 14pages, presentations has been changed

In the first part, we obtain sharp results for L^2 boundedness of strongly singular operators on the Heisenberg group. We also define the oscillating convolution operators on the Heisenberg group and study their boundedness properties. In the second part, we obtain the boundedness on Hardy spaces., Comment: 25 pages, This paper includes the author's paper 'arXiv:1204.5654', to appear in forum mathematicum

In this paper, we prove Lp boundedness of maximal multipliers on stratified groups and maximal multipliers on product spaces of those groups., Comment: 22 pages, the details for product spaces has been added