This paper concerns the generalized Nash equilibrium problem of polynomials (GNEPP). We apply the Gauss-Seidel method and Lasserre type Moment-SOS relaxations to solve GNEPPs. The convergence of the Gauss-Seidel method is known for some special GNEPPs, such as generalized potential games (GPGs). We give a sufficient condition for GPGs and propose a numerical certificate, based on Putinar's Positivstellensatz. Numerical examples for both convex and nonconvex GNEPPs are given for demonstrating the efficiency of the proposed method.
For multi-block alternating direction method of multipliers(ADMM), where the objective function can be decomposed into multiple block components, we show that with block symmetric Gauss-Seidel iteration, the algorithm will converge quickly. The method will apply a block symmetric Gauss-Seidel iteration in the primal update and a linear correction that can be derived in view of Richard iteration. We also establish the linear convergence rate for linear systems.
This paper aims to present a fairly accessible generalization of several symmetric Gauss-Seidel decomposition based multi-block proximal alternating direction methods of multipliers (ADMMs) for convex composite optimization problems. The proposed method unifies and refines many constructive techniques that were separately developed for the computational efficiency of multi-block ADMM-type algorithms. Specifically, the majorized augmented Lagrangian functions, the indefinite proximal terms, the inexact symmetric Gauss-Seidel decomposition theorem, the tolerance criteria of approximately solving…
In this paper, we propose a novel decomposition approach for mixed-integer stochastic programming (SMIP) problems that is inspired by the combination of penalty-based Lagrangian and block Gauss-Seidel methods (PBGS). In this sense, PBGS is developed such that the inherent decomposable structure that SMIPs present can be exploited in a computationally efficient manner. The performance of the proposed method is compared with the Progressive Hedging method (PH), which also can be viewed as a Lagrangian-based method for obtaining solutions for SMIP. Numerical experiments performed using instances …
Transform Invariant Low-Rank Textures, referred to as TILT, can accurately and robustly extract textural or geometric information in a 3D from user-specified windows in 2D in spite of significant corruptions and warping. It was discovered that the task can be characterized, both theoretically and numerically, by solving a sequence of matrix nuclear-norm and $\ell_1$-norm involved convex minimization problems. For solving this problem, the direct extension of Alternating Direction Method of Multipliers (ADMM) in an usual Gauss-Seidel manner often performs numerically well in practice but there …
Consider the classical problem of solving a general linear system of equations $Ax=b$. It is well known that the (successively over relaxed) Gauss-Seidel scheme and many of its variants may not converge when $A$ is neither diagonally dominant nor symmetric positive definite. Can we have a linearly convergent G-S type algorithm that works for {\it any} $A$? In this paper we answer this question affirmatively by proposing a doubly stochastic G-S algorithm that is provably linearly convergent (in the mean square error sense) for any feasible linear system of equations. The key in the algorithm de…
In this paper, we propose an inexact multi-block ADMM-type first-order method for solving a class of high-dimensional convex composite conic optimization problems to moderate accuracy. The design of this method combines an inexact 2-block majorized semi-proximal ADMM and the recent advances in the inexact symmetric Gauss-Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block-variable. One distinctive feature of our proposed method (the sGS-imsPADMM) is that it only needs one cycle of an in…
Large-scale linear complementarity problems (LCPs) are repeatedly solved in interactive rigid-body simulations. The projected Gauss-Seidel method is often employed for LCPs, since it has advantages in computation time, numerical robustness, and memory use. Zheng and Yin (2013) proposed modulus-based matrix splitting iteration methods and showed their effectiveness for large problems, but a simple application of their approach to large-scale LCPs in interactive rigid-body simulations is not effective since such a simple application demands large matrix multiplications. In this paper, we propose…
The H2 guaranteed cost decentralized control problem is investigated in this work. More specifically, on the basis of an appropriate H2 re-formulation that we put in place, the optimal control problem in the presence of parameter uncertainties is then suitably characterized by convex restriction and solved in parameter space. It is shown that a set of stabilizing decentralized controller gains for the uncertain system is parameterized in a convex set through appropriate convex restriction, and then an approximated conic optimization problem is constructed. This facilitates the use of the symme…
This paper involves the innovative development of a symmetric Gauss-Seidel ADMM algorithm to solve the H-infinity guaranteed cost control problem. In the presence of parametric uncertainties, the H-infinity guaranteed cost control problem generally leads to the large-scale optimization. This is due to the exponential growth of the number of the extreme systems involved with respect to the number of parametric uncertainties. In this work, through a variant of the Youla-Kucera parameterization, the stabilizing controllers are parameterized in a convex set; yielding the outcome that the H-infinit…
This paper investigates a class of mixed stochastic linear-quadratic-Gaussian (LQG) social optimization and Nash game in the context of a large scale system. Two types of interactive agents are involved: a major agent and a large number of weakly-coupled minor agents. All minor agents are cooperative to minimize the social cost as the sum of their individual costs, whereas such social cost are conflictive to that of major agent. Thus, major agent and all minor agents are further competitive to reach some non-zero Nash equilibrium. The control processes enter both diffusion and drift terms of a…
The Bellman operator constitutes the foundation of dynamic programming (DP). An alternative is presented by the Gauss-Seidel operator, whose evaluation, differently from that of the Bellman operator where the states are all processed at once, updates one state at a time, while incorporating into the computation the interim results. The provably better convergence rate of DP methods based on the Gauss-Seidel operator comes at the price of an inherent sequentiality, which prevents the exploitation of modern multi-core systems. In this work we propose a new operator for dynamic programming, namel…
We show that an almost cyclic (or generalized Gauss- Seidel) Dykstra's algorithm which incorporates the SHQP (supporting halfspace- quadratic programming) strategy can achieve nonasymptotic and asymptotic linear convergence for polyhedral problems.
This paper is concerned with a new class of mean-field games which involve a finite number of agents. Necessary and sufficient conditions are obtained for the existence of the decentralized open-loop Nash equilibrium in terms of non-standard forward-backward stochastic differential equations (FBSDEs). By solving the FBSDEs, we design a set of decentralized strategies by virtue of two differential Riccati equations. Instead of the $\varepsilon$-Nash equilibrium in classical mean-field games, the set of decentralized strategies is shown to be a Nash equilibrium. For the infinite-horizon problem,…
In this paper, we propose a novel symmetric alternating minimization algorithm to solve a broad class of total variation (TV) regularization problems. Unlike the usual $z^k\to x^k$ Gauss-Seidel cycle, the proposed algorithm performs the special $\overline{x}^{k}\to z^k\to x^k$ cycle. The main idea for our setting is the recent symmetric Gauss-Seidel (sGS) technique which is developed for solving the multi-block convex composite problem. This idea also enables us to build the equivalence between the proposed method and the well-known accelerated proximal gradient (APG) method. The faster conver…
This work investigates a problem of simultaneous global cost minimization and Nash equilibrium seeking, which commonly exists in $N$-cluster non-cooperative games. Specifically, the agents in the same cluster collaborate to minimize a global cost function, being a summation of their individual cost functions, and jointly play a non-cooperative game with other clusters as players. For the problem settings, we suppose that the explicit analytical expressions of the agents' local cost functions are unknown, but the function values can be measured. We propose a gradient-free Nash equilibrium seeki…
This paper studies an $N$-coalition non-cooperative game problem, where the players in the same coalition cooperatively minimize the sum of their local cost functions under a directed communication graph, while collectively acting as a virtual player to play a non-cooperative game with other coalitions. Moreover, it is assumed that the players have no access to the explicit functional form but only the function value of their local costs. To solve the problem, a discrete-time gradient-free Nash equilibrium seeking strategy, based on the gradient tracking method, is proposed. Specifically, a gr…
We investigate a privacy-signaling game problem in which a sender with privacy concerns observes a pair of correlated random vectors which are modeled as jointly Gaussian. The sender aims to hide one of these random vectors and convey the other one whereas the objective of the receiver is to accurately estimate both of the random vectors. We analyze these conflicting objectives in a game theoretic framework with quadratic costs where depending on the commitment conditions (of the sender), we consider Nash or Stackelberg (Bayesian persuasion) equilibria. We show that a payoff dominant Nash equi…
We show that the moving arithmetic average is closely connected to a Gauss-Seidel type fixed point method studied by Bauschke, Wang and Wylie, and which was observed to converge only numerically. Our analysis establishes a rigorous proof of convergence of their algorithm in a special case; moreover, limit is explicitly identified. Moving averages in Banach spaces and Kolmogorov means are also studied. Furthermore, we consider moving proximal averages and epi-averages of convex functions.
This paper studies the decentralized quadratic cheap talk and signaling game problems when an encoder and a decoder, viewed as two decision makers, have misaligned objective functions. The main contributions of this study are the extension of Crawford and Sobel's cheap talk formulation to multi-dimensional sources and to noisy channel setups. We consider both (simultaneous) Nash equilibria and (sequential) Stackelberg equilibria. We show that for arbitrary scalar sources, in the presence of misalignment, the quantized nature of all equilibrium policies holds for Nash equilibria in the sense th…
This paper studies a two-person linear-quadratic-Gaussian pursuit-evasion differential game with costly but controlled information. One player can decide when to observe the other player's state. However, one observation of another player's state comes with two costs: the direct cost of observing and the implicit cost of exposing his state. We call games of this type a Pursuit-Evasion-Exposure-Concealment (PEEC) game. The PEEC game constitutes two types of strategies: The control strategies and the observation strategies. We fully characterize the Nash control strategies of the PEEC game using…
