我们研究了一个具有局部耦合的时间依赖性均场博弈（MFG）系统的数值近似。我们考虑的离散化方法源于[Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017]中针对稳态问题描述的变分方法，导致了Achdou和Capuzzo-Dolcetta在[SIAM J. Numer. Anal., 48(3):1136-1162, 2010]中引入的有限差分方案。为了求解有限维变分问题，在[Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017]中，作者实现了Chambolle和Pock在[J. Math. Imaging Vision, 40(1):120-145, 2011]中引入的对偶算法，其核心在于迭代求解线性系统并应用近邻算子。我们将该方法应用于时间依赖性MFG，并针对大粘性参数，通过使用合适的预条件迭代算法改进线性系统求解方法，以替代[ Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017]中使用的直接方法。
英文摘要：
We study a numerical approximation of a time-dependent Mean Field Game (MFG) system with local couplings. The discretization we consider stems from a variational approach described in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] for the stationary problem and leads to the finite difference scheme introduced by Achdou and Capuzzo-Dolcetta in [SIAM J. Numer. Anal., 48(3):1136-1162, 2010]. In order to solve the finite dimensional variational problems, in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] the authors implement the primal-dual algorithm introduced by Chambolle and Pock in [J. Math. Imaging Vision, 40(1):120-145, 2011], whose core consists in iteratively solving linear systems and applying a proximity operator. We apply that method to time-dependent MFG and, for large viscosity parameters, we improve the linear system solution by replacing the direct approach used in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] by suitable preconditioned iterative algorithms.
We consider a primal-dual algorithm for minimizing $f(x)+h\square l(Ax)$ with Fréchet differentiable $f$ and $l^*$. This primal-dual algorithm has two names in literature: Primal-Dual Fixed-Point algorithm based on the Proximity Operator (PDFP$^2$O) and Proximal Alternating Predictor-Corrector (PAPC). In this paper, we prove its convergence under a weaker condition on the stepsizes than existing ones. With additional assumptions, we show its linear convergence. In addition, we show that this condition (the upper bound of the stepsize) is tight and can not be weakened. This result also recovers…
Dual first-order methods are essential techniques for large-scale constrained convex optimization. However, when recovering the primal solutions, we need $T(ε^{-2})$ iterations to achieve an $ε$-optimal primal solution when we apply an algorithm to the non-strongly convex dual problem with $T(ε^{-1})$ iterations to achieve an $ε$-optimal dual solution, where $T(x)$ can be $x$ or $\sqrt{x}$. In this paper, we prove that the iteration complexity of the primal solutions and dual solutions have the same $O\left(\frac{1}{\sqrtε}\right)$ order of magnitude for the accelerated randomized dual coordin…
The primal-dual method of Chambolle and Pock is a widely used algorithm to solve various optimization problems written as convex-concave saddle point problems. Each update step involves the application of both the forward linear operator and its adjoint. However, in practical applications like computerized tomography, it is often computationally favourable to replace the adjoint operator by a computationally more efficient approximation. This leads to an adjoint mismatch in the algorithm. In this paper, we analyze the convergence of Chambolle-Pock's primal-dual method under the presence of a m…
We develop block structure adapted primal-dual algorithms for non-convex non-smooth optimisation problems whose objectives can be written as compositions $G(x)+F(K(x))$ of non-smooth block-separable convex functions $G$ and $F$ with a non-linear Lipschitz-differentiable operator $K$. Our methods are refinements of the non-linear primal-dual proximal splitting method for such problems without the block structure, which itself is based on the primal-dual proximal splitting method of Chambolle and Pock for convex problems. We propose individual step length parameters and acceleration rules for ea…
The Stochastic Primal-Dual Hybrid Gradient (SPDHG) was proposed by Chambolle et al. (2018) and is an efficient algorithm to solve some nonsmooth large-scale optimization problems. In this paper we prove its almost sure convergence for convex but not necessarily strongly convex functionals. We also look into its application to parallel Magnetic Resonance Imaging reconstruction in order to test performance of SPDHG. Our numerical results show that for a range of settings SPDHG converges significantly faster than its deterministic counterpart.
This study addresses primal-dual dynamics for a stochastic programming problem for capacity network design. It is proven that consensus can be achieved on the \textit{here and now} variables which represent the capacity of the network. The main contribution is a heuristic approach which involves the formulation of the problem as a mean-field game. Every agent in the mean-field game has control over its own primal-dual dynamics and seeks consensus with neighboring agents according to a communication topology. We obtain theoretical results concerning the existence of a mean-field equilibrium. Mo…
Population games can be regarded as a tool to study the strategic interaction of a population of players. Although several attention has been given to such field, most of the available works have focused only on the unconstrained case. That is, the allowed equilibrium of the game is not constrained. To further extend the capabilities of population games, in this paper we propose a novel class of primal-dual evolutionary dynamics that allow the consideration of constraints that must be satisfied at the equilibrium of the game. Using duality theory and Lyapunov stability theory, we provide suffi…
In this paper, we consider a large class of hierarchical congestion population games. One can show that the equilibrium in a game of such type can be described as a minimum point in a properly constructed multi-level convex optimization problem. We propose a fast primal-dual composite gradient method and apply it to the problem, which is dual to the problem describing the equilibrium in the considered class of games. We prove that this method allows to find an approximate solution of the initial problem without increasing the complexity.
We study structured nonsmooth convex finite-sum optimization that appears widely in machine learning applications, including support vector machines and least absolute deviation. For the primal-dual formulation of this problem, we propose a novel algorithm called \emph{Variance Reduction via Primal-Dual Accelerated Dual Averaging (\vrpda)}. In the nonsmooth and general convex setting, \vrpda~has the overall complexity $O(nd\log\min \{1/ε, n\} + d/ε)$ in terms of the primal-dual gap, where $n$ denotes the number of samples, $d$ the dimension of the primal variables, and $ε$ the desired accuracy…
Given two couplings between "primal" and "dual" sets, we prove a general implication that relates an inequality involving "primal" sets to a reverse inequality involving the "dual" sets.% More precisely, let be given two "primal" sets $\PRIMAL$, $\PRIMALBIS$and two "dual" sets $\DUAL$, $\DUALBIS$, together with two {coupling} functions \(\PRIMAL \overset{\coupling}{\leftrightarrow} \DUAL \) and \(\PRIMALBIS \overset{\couplingbis}{\leftrightarrow} \DUALBIS \). We define a new coupling \(\SumCoupling{\coupling}{\couplingbis} \) between the "primal" product set~$\PRIMAL \times \PRIMALBIS$ and the…
In this work we study a constrained monotone inclusion involving the normal cone to a closed vector subspace and a priori information on primal solutions. We model this information by imposing that solutions belongs to the fixed point set of an averaged nonexpansive mapping. We characterize the solutions using an auxiliary inclusion that involves the partial inverse operator. Then, we propose the primal-dual partial inverse splitting and we prove its weak convergence to a solution of the inclusion, generalizing several methods in the literature. The efficiency of the proposed method is illustr…
In this article we develop a new primal dual variational formulation suitable for a large class of non-convex problems in the calculus of variations. The results are obtained through basic tools of convex analysis, duality theory, the Legendre transform concept and the respective relations between the primal and dual variables. The novelty here is that the dual formulation is established also for the primal variables, however with a large domain region of concavity about a critical point. Finally, we formally prove there is no duality gap between the primal and dual formulations in a local ext…
In this work, we show that for linearly constrained optimization problems the primal-dual hybrid gradient algorithm, analyzed by Chambolle and Pock [3], can be written as an entirely primal algorithm. This allows us to prove convergence of the iterates even in the degenerate cases when the linear system is inconsistent or when the strong duality does not hold. We also obtain new convergence rates which seem to improve existing ones in the literature. For a decentralized distributed optimization we show that the new scheme is much more efficient than the original one.
The Adwords problem has always been an interesting internet advertising problem. There are many ways to solve the Adwords problem with the adversarial order model, including the Greedy Algorithm, the Balance Algorithm, and the Scale-bid Algorithm, which is also known as MSVV. In this survey, I will review the Adwords problem with different input models under a primal-dual setting, and with the small-bid assumption. In the first section, I will focus on Adwords with adversarial order model, and use duality to prove the efficiency of the Greedy Algorithm and the MSVV algorithm. Next, I will look…
This work proposes an accelerated primal-dual dynamical system for affine constrained convex optimization and presents a class of primal-dual methods with nonergodic convergence rates. In continuous level, exponential decay of a novel Lyapunov function is established and in discrete level, implicit, semi-implicit and explicit numerical discretizations for the continuous model are considered sequentially and lead to new accelerated primal-dual methods for solving linearly constrained optimization problems. Special structures of the subproblems in those schemes are utilized to develop efficient …
This article develops a new primal dual formulation for the Kirchhoff-Love non-linear plate model. At first we establish a duality principle which includes sufficient conditions of global optimality through the dual formulation. At this point we highlight this first duality principle is specially suitable for the case in which the membrane stress tensor is negative definite. In a second step, from such a general principle, we develop a primal dual variational formulation which also includes the corresponding sufficient conditions for global optimality. The results are based on standard tools o…
In this paper, we consider the decentralized dynamic optimization problem defined over a multi-agent network. Each agent possesses a time-varying local objective function, and all agents aim to collaboratively track the drifting global optimal solution that minimizes the summation of all local objective functions. The decentralized dynamic optimization problem can be solved by primal or primal-dual methods, and when the problem degenerates to be static, it has been proved in literature that primal-dual methods are superior to primal ones. This motivates us to ask: are primal-dual methods neces…
This paper studies the asymptotic convergence properties of the primal-dual dynamics designed for solving constrained concave optimization problems using classical notions from stability analysis. We motivate the need for this study by providing an example that rules out the possibility of employing the invariance principle for hybrid automata to study asymptotic convergence. We understand the solutions of the primal-dual dynamics in the Caratheodory sense and characterize their existence, uniqueness, and continuity with respect to the initial condition. We use the invariance principle for dis…
This paper presents a simple primal dual method named DPD which is a flexible framework for a class of saddle point problem with or without strongly convex component. The presented method has linearized version named LDPD and exact version EDPD. Each iteration of DPD updates sequentially the dual and primal variable via simple proximal mapping and refines the dual variable via extrapolation. Convergence analysis with smooth or strongly convex primal component recovers previous state-of-the-art results, and that with strongly convex dual component attains full acceleration $O(1/k^2)$ in terms o…
We provide an overview of primal-dual algorithms for nonsmooth and non-convex-concave saddle-point problems. This flows around a new analysis of such methods, using Bregman divergences to formulate simplified conditions for convergence.
We propose primal-dual stochastic mirror descent for the convex optimization problems with functional constraints. We obtain the rate of convergence in terms of probability of large deviations.
This paper investigates the asymptotic and non-asymptotic behavior of the quantized primal dual algorithm in network utility maximization problems, in which a group of agents maximize the sum of their individual concave objective functions under linear constraints. In the asymptotic scenario, we use the information theoretic notion of differential entropy power to establish universal lower bounds on the exponential convergence rates of joint primal dual, primal and dual variables under optimum achieving quantization schemes. These results provide trade offs between the speed of exponential con…
This article develops a primal dual formulation for a primal proximal approach suitable for a large class of non-convex models in the calculus of variations. The results are established through standard tools of functional analysis, convex analysis and duality theory and are applied to a Ginzburg-Landau type model. Finally, in the last two sections, we present concerning optimal conditions and another related duality principle for the model in question.
We consider empirical risk minimization of linear predictors with convex loss functions. Such problems can be reformulated as convex-concave saddle point problems, and thus are well suitable for primal-dual first-order algorithms. However, primal-dual algorithms often require explicit strongly convex regularization in order to obtain fast linear convergence, and the required dual proximal mapping may not admit closed-form or efficient solution. In this paper, we develop both batch and randomized primal-dual algorithms that can exploit strong convexity from data adaptively and are capable of ac…
In this paper, we propose an inertial accelerated primal-dual method for the linear equality constrained convex optimization problem. When the objective function has a ``nonsmooth + smooth'' composite structure, we further propose an inexact inertial primal-dual method by linearizing the smooth individual function and solving the subproblem inexactly. Assuming merely convexity, we prove that the proposed methods enjoy $\mathcal{O}(1/k^2)$ convergence rate on the objective residual and the feasibility violation in the primal model. Numerical results are reported to demonstrate the validity of t…
We present a parallelized primal-dual algorithm for solving constrained convex optimization problems. The algorithm is "block-based," in that vectors of primal and dual variables are partitioned into blocks, each of which is updated only by a single processor. We consider four possible forms of asynchrony: in updates to primal variables, updates to dual variables, communications of primal variables, and communications of dual variables. We construct a family of explicit counterexamples to show the need to eliminate asynchronous communication of dual variables, though the other forms of asynchr…
By time discretization of a second-order primal-dual dynamical system with damping $α/t$ where an inertial construction in the sense of Nesterov is needed only for the primal variable, we propose a fast primal-dual algorithm for a linear equality constrained convex optimization problem. Under a suitable scaling condition, we show that the proposed algorithm enjoys a fast convergence rate for the objective residual and the feasibility violation, and the decay rate can reach $\mathcal{O}(1/k^{α-1})$ at the most. We also study convergence properties of the corresponding primal-dual dynamical syst…
We introduce a novel primal-dual flow for affine constrained convex optimization problems. As a modification of the standard saddle-point system, our primal-dual flow is proved to possess the exponential decay property, in terms of a tailored Lyapunov function. Then two primal-dual methods are obtained from numerical discretizations of the continuous model, and global nonergodic linear convergence rate is established via a discrete Lyapunov function. Instead of solving the subproblem of the primal variable, we apply the semi-smooth Newton iteration to the subproblem with respect to the multipl…
We consider mixed model of traffic flow distribution in large networks (BMW model, 1954 & Stable Dynamic model, 1999). We build dual problem and consider primal-dual mirror descent method for the dual problem. There are two ways to recover the solution of the primal problem. First technique is based on "models" approach and second one is based on Lagrange multiplyers technique. Both approaches have its own advantages and drawbacks. We discuss them.
Computational methods are proposed for solving a convex quadratic program (QP). Active-set methods are defined for a particular primal and dual formulation of a QP with general equality constraints and simple lower bounds on the variables. In the first part of the paper, two methods are proposed, one primal and one dual. These methods generate a sequence of iterates that are feasible with respect to the equality constraints associated with the optimality conditions of the primal-dual form. The primal method maintains feasibility of the primal inequalities while driving the infeasibilities of t…
In this paper we propose distributed dual gradient algorithms for linearly constrained separable convex problems and analyze their rate of convergence under different assumptions. Under the strong convexity assumption on the primal objective function we propose two distributed dual fast gradient schemes for which we prove sublinear rate of convergence for dual suboptimality but also primal suboptimality and feasibility violation for an average primal sequence or for the last generated primal iterate. Under the additional assumption of Lipshitz continuity of the gradient of the primal objective…
Linear constrained convex programming has many practical applications, including support vector machine and machine learning portfolio problems. We propose the randomized primal-dual coordinate (RPDC) method, a randomized coordinate extension of the first-order primal-dual method by Cohen and Zhu, 1984 and Zhao and Zhu, 2019, to solve linear constrained convex programming. We randomly choose a block of variables based on a uniform distribution, linearize, and apply a Bregman-like function (core function) to the selected block to obtain simple parallel primal-dual decomposition. We then establi…
Spingarn's method of partial inverses has found many applications in nonlinear analysis and in optimization. We show that it can be employed to solve composite monotone inclusions in duality, thus opening a new range of applications for the partial inverse formalism. The versatility of the resulting primal-dual splitting algorithm is illustrated through applications to structured monotone inclusions and optimization.
This paper associates a dual problem to the minimization of an arbitrary linear perturbation of the robust sum function introduced in DOI 10.1007/s11228-019-00515-2. It provides an existence theorem for primal optimal solutions and, under suitable duality assumptions, characterizations of the primal-dual optimal set, the primal optimal set, and the dual optimal set, as well as a formula for the subdiffential of the robust sum function. The mentioned results are applied to get simple formulas for the robust sums of subaffine functions (a class of functions which contains the affine ones) and to…
In this paper, we propose two novel non-stationary first-order primal-dual algorithms to solve nonsmooth composite convex optimization problems. Unlike existing primal-dual schemes where the parameters are often fixed, our methods use pre-defined and dynamic sequences for parameters. We prove that our first algorithm can achieve $\mathcal{O}(1/k)$ convergence rate on the primal-dual gap, and primal and dual objective residuals, where $k$ is the iteration counter. Our rate is on the non-ergodic (i.e., the last iterate) sequence of the primal problem and on the ergodic (i.e., the averaging) sequ…
In this paper, we adapt proximal incremental aggregated gradient methods to saddle point problems, which is motivated by decoupling linear transformations in regularized empirical risk minimization models. First, the Primal-Dual Proximal Incremental Aggregated (PD-PIAG) methods with extrapolations were proposed. We proved that the primal-dual gap of the averaged iteration sequence sublinearly converges to 0, and the iteration sequence converges to some saddle point. Under the strong convexity of $f$ and $h^\ast$, we proved that the iteration sequence linearly converges to the saddle point. The…
Dual first-order methods are powerful techniques for large-scale convex optimization. Although an extensive research effort has been devoted to studying their convergence properties, explicit convergence rates for the primal iterates have only been established under global Lipschitz continuity of the dual gradient. This is a rather restrictive assumption that does not hold for several important classes of problems. In this paper, we demonstrate that primal convergence rate guarantees can also be obtained when the dual gradient is only locally Lipschitz. The class of problems that we analyze ad…
In this paper we study the relaxed primal-dual algorithm for solving composite monotone inclusions in real Hilbert spaces with critical preconditioners. Our approach is based in new results on the asymptotic behaviour of Krasnosel'ski\uı-Mann (KM) iterations defined in the range of monotone self-adjoint linear operators. These results generalize the convergence of classical KM iterations aiming at approximating fixed points. We prove that the relaxed primal-dual algorithm with critical preconditioners define KM iterations in the range of a particular monotone self-adjoint linear operator with …
In this paper, a primal-dual gradient flow algorithm for distributed support vector machines (DSVM) is proposed. A network of computing nodes, each carrying a subset of horizontally partitioned large dataset is considered. The nodes are represented as dynamical systems with Arrow-Hurwicz-Uzawa gradient flow dynamics, derived from the Lagrangian function of the DSVM problem. It is first proved that the nodes are passive dynamical systems. Then, by employing the Krasovskii type candidate Lyapunov functions, it is proved that the computing nodes asymptotically converge to the optimal primal-dual …
We present a parallelized primal-dual algorithm for solving constrained convex optimization problems. The algorithm is "block-based," in that vectors of primal and dual variables are partitioned into blocks, each of which is updated only by a single processor. We consider four possible forms of asynchrony: in updates to primal variables, updates to dual variables, communications of primal variables, and communications of dual variables. We explicitly construct a family of counterexamples to rule out permitting asynchronous communication of dual variables, though the other forms of asynchrony a…
We introduce the primal-dual quasi-Newton (PD-QN) method as an approximated second order method for solving decentralized optimization problems. The PD-QN method performs quasi-Newton updates on both the primal and dual variables of the consensus optimization problem to find the optimal point of the augmented Lagrangian. By optimizing the augmented Lagrangian, the PD-QN method is able to find the exact solution to the consensus problem with a linear rate of convergence. We derive fully decentralized quasi-Newton updates that approximate second order information to reduce the computational burd…
This paper investigates two accelerated primal-dual mirror dynamical approaches for smooth and nonsmooth convex optimization problems with affine and closed, convex set constraints. In the smooth case, an accelerated primal-dual mirror dynamical approach (APDMD) based on accelerated mirror descent and primal-dual framework is proposed and accelerated convergence properties of primal-dual gap, feasibility measure and the objective function value along with trajectories of APDMD are derived by the Lyapunov analysis method. Then, we extend APDMD into two distributed dynamical approaches to deal w…
We revisit the operator splitting schemes proposed in a recent work of [Some extensions of the operator splitting schemes based on Lagrangian and primal-dual: A unified proximal point analysis, Feng Xue, Optimization, 2022, doi: 10.1080/02331934.2022.2057309], and further analyze the convergence of the generalized Bregman distance and the primal-dual gap of these algorithms within a unified proximal point framework. The possibility of reduction to a simple resolvent is also discussed by exploiting the structure and possible degeneracy of the underlying metric.
