On the linear convergence of distributed Nash equilibrium seeking for multi-cluster games under partial-decision information

This paper considers the distributed strategy design for Nash equilibrium (NE) seeking in multi-cluster games under a partial-decision information scenario. In the considered game, there are multiple clusters and each cluster consists of a group of agents. A cluster is viewed as a virtual noncooperative player that aims to minimize its local payoff function and the agents in a cluster are the actual players that cooperate within the cluster to optimize the payoff function of the cluster through communication via a connected graph. In our setting, agents have only partial-decision information, that is, they only know local information and cannot have full access to opponents' decisions. To solve the NE seeking problem of this formulated game, a discrete-time distributed algorithm, called distributed gradient tracking algorithm (DGT), is devised based on the inter- and intra-communication of clusters. In the designed algorithm, each agent is equipped with strategy variables including its own strategy and estimates of other clusters' strategies. With the help of a weighted Fronbenius norm and a weighted Euclidean norm, theoretical analysis is presented to rigorously show the linear convergence of the algorithm. Finally, a numerical example is given to illustrate the proposed algorithm

Localization of Control Synthesis Problem for Large-Scale Interconnected System Using IQC and Dissipativity Theories

The synthesis problem for the compositional performance certification of interconnected systems is considered. A fairly unified description of control synthesis problem is given using integral quadratic constraints (IQC) and dissipativity. Starting with a given large-scale interconnected system and a global performance objective, an optimization problem is formulated to search for admissible dissipativity properties of each subsystems. Local control laws are then synthesized to certify the relevant dissipativity properties. Moreover, the term localization is introduced to describe a finite collection of syntheses problems, for the local subsystems, which are a feasibility certificate for the global synthesis problem. Consequently, the problem of localizing the global problem to a smaller collection of disjointed sets of subsystems, called groups, is considered. This works looks promising as another way of looking at decentralized control and also as a way of doing performance specifications for components in a large-scale system

Decay Properties Of Multilinear Oscillatory Integrals

In this thesis, we study the following multilinear oscillatory integral introduced by Christ, Li, Tao and Thiele \cite{CLTT} \begin{equation} I_{\lambda}(f_1,...f_n)=\int_{\mathbb{R}^m}e^{i\lambda P(x)}\prod_{j=1}^n f_j(\pi_j(x))\eta(x)dx, \end{equation} where $P:\mathbb{R}^m\to\mathbb{R}$ is a real-valued measurable function, $\eta$ is a compactly supported smooth cutoff function. Each $\pi_j$ is a surjective linear transformation from $\mathbb{R}^m$ to $\mathbb{R}^{k_j}$, where $1\le k_j\le m-1$. Each $f_j:\mathbb{R}^{k_j}\rightarrow \mathbb{C}$ is a locally integrable function with respect to Lebesgue measure on $\mathbb{R}^{k_j}$. In Chapter 2, we first introduce the nondegeneracy degree along with the nondegeneracy norm defined in \cite{CLTT} to characterize the nondegeneracy condition of the phase function. In the same chapter, we will summarize some powerful tools that can help to simplify the problem and introduce the idea of a special geometric structure called ``separation . There are three results in this thesis. The first proves trilinear oscillatory integrals with nondegenerate polynomial phase always have the decay property. The second one extends the one-dimensional case whose phase function has large nondegeneracy degree. The third result deals with the case where every linear mapping preserves the direct sum decomposition

Krull-Schmidt decompositions for thick subcategories

Following Krause \cite{Kr}, we prove Krull-Schmidt type decomposition theorems for thick subcategories of various triangulated categories including the derived categories of rings, Noetherian stable homotopy categories, stable module categories over Hopf algebras, and the stable homotopy category of spectra. In all these categories, it is shown that the thick ideals of small objects decompose uniquely into indecomposable thick ideals. We also discuss some consequences of these decomposition results. In particular, it is shown that all these decompositions respect K-theory.Comment: Added more references, fixed some typos, to appear in Journal of Pure and Applied Algebra, 22 pages, 1 figur

A Total Fractional-Order Variation Model for Image Restoration with Non-homogeneous Boundary Conditions and its Numerical Solution

To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total $\alpha$-order variation model, which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-order variations for image restoration; however first no analysis is done yet and second all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while non-zero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total $\alpha$-order variational model rigorously. It then develops four algorithms for solving the variational problem, one based on the variational Split-Bregman idea and three based on direct solution of the discretise-optimization problem. Numerical experiments show that, in terms of restoration quality and solution efficiency, the proposed model can produce highly competitive results, for smooth images, to two established high order models: the mean curvature and the total generalized variation.Comment: 26 page