Distributed Bregman-Distance Algorithms for Min-Max Optimization

This article studies the distributed convex optimization problem with a common decision variable, a global inequality constraint, and local constraint sets over a time-varying multiagent network, the objective function of which is a sum of agents’ local convex cost functions. To solve such problem, a penalty-based distributed continuous-time subgradient algorithm with time-varying gain is developed for each agent to seek the saddle point of the penalty Lagrangian function. It is shown that an exact primal optimal solution can be obtained with certain assumption on time-varying gain. Moreover, the proposed algorithm adopts fixed-time projection scheme to ensure that for any initial state value, each local state estimate converges to its convex constraint set within fixed time. Finally, numerical examples are provided to show the effectiveness of the theoretical results.

The paper mainly talks about the distributed optimal issue under the continuous-time high-order multi-agent system with event-triggered communication. Every agent possesses its local convex cost function only known by itself. and our aim is to minimize the sum of all the local cost functions and make the agents convergent to the optimal point through the communication between the agents. Utilizing the knowledge of convex analysis and graph theory, we propose a distributed protocol with event-triggered communication under a undirected graph for the high-order multi-agent system. Through Lyapunov method, the agents gradually converging to the optimal point is testified. In addition, we provide an example to validate the proposed protocol.

Since the initial proposal in the late 80s, spectral gradient methods continue to receive significant attention, especially due to their excellent numerical performance on various large scale applications. However, to date, they have not been sufficiently explored in the context of distributed optimization. In this paper, we consider unconstrained distributed optimization problems where n nodes constitute an arbitrary connected network and collaboratively minimize the sum of their local convex cost functions. In this setting, building from existing exact distributed gradient methods, we propose a novel exact distributed gradient method wherein nodes’ step-sizes are designed according to the novel rules akin to those in spectral gradient methods. We refer to the proposed method as Distributed Spectral Gradient method. The method exhibits R-linear convergence under standard assumptions for the nodes’ local costs and safeguarding on the algorithm step-sizes. We illustrate the method’s performance through simulation examples.

SummaryIn this paper, we study a distributed constrained optimal consensus problem for discrete‐time first‐order integrator systems under bounded input. Each agent is assigned with a local convex cost function, and all agents are required to achieve consensus at the minimum of the aggregate cost over a common convex constraint set, which is only accessible by part of the agents. A 2‐step control protocol is designed to solve the problem under bounded input. Firstly, at each time step, each agent moves a bounded step of the subgradient descent from an individual cost and another one along the projection direction if it can access the constraint. The second movement is then given by a bounded average of the relative positions to neighbors. Specifically, to coordinate the subgradient step within the network without using global information, we introduce an estimate of the upper bound of all agents' subgradients, which is updated by a unilateral consensus mechanism. Under the given control protocol, we obtain the necessary and sufficient conditions to achieve the constrained optimal consensus for a fixed topology. Under similar conditions, we also solve the problem for switching topologies and conduct a convergence rate analysis for strongly convex costs.

The distributed optimal coordinated control problem of heterogeneous linear multi-agent systems is studied in this paper. Specifically, each agent in the system under consideration has a local private cost function, which is assumed to be strongly convex with a Lipschitz continuous gradient. The agents aim to achieve output consensus on the optimal solution of a global cost function, which is the sum of local cost functions. The communication topology among agents is assumed to be a weight-unbalanced and strongly connected directed graph. To solve such a control problem, a tracking-based approach is suggested and utilized, where an auxiliary system is designed to seek the optimal solution. Based on the output regulation technique, distributed tracking control inputs are proposed for the agents, which ensure that the outputs of the agents track the trajectories generated by the auxiliary system. Under the assumptions on the local cost functions, the communication topology and system matrices, it is proved that with the proposed control scheme, the outputs of all agents can achieve consensus on the optimal solution. Finally, two numerical simulation examples are presented to illustrate the effectiveness of the proposed tracking control scheme.

This paper discusses dynamically weighted model predictive control based on two-timescale neurodynamic optimization. Minimax optimization problems with dynamic weights in objective functions are used in the model predictive control. The minimax optimization problems are solved by using a two-timescale neurodynamic optimization approach. Examples on controlling HVAC (heating, ventilation, and air-conditioning) and CSTR (cooling continuous stirred tank reactor) systems are elaborated to substantiate the efficacy of the control approach.

In this chapter, we focus on introducing a distributed online optimization problem for a set of nodes communicating on a time-varying unbalanced directed network, while considering the problem of how to preserve the privacy of their local cost functions. The main goal of this set of nodes is to cooperatively minimize the sum of all locally known convex cost functions (global cost function). We propose a differentially private distributed stochastic subgradient-push algorithm, named DP-DSSP, to solve such an optimization problem in a collaborative and distributed manner. The algorithm ensures that nodes interact with their in-neighbors and collectively optimize the global cost function. Unlike most existing distributed algorithms that do not consider privacy issues, DP-DSSP successfully protects the privacy of participating nodes through a differential privacy strategy, which is more practical in applications involving sensitive information, such as military affairs or medical treatment. An important feature of DP-DSSP is that it handles distributed online optimization problems in the context of time-varying unbalanced directed networks. Theoretical analysis shows that DP-DSSP can effectively protect differential privacy and can achieve sublinear regret. The tradeoff between the level of privacy and accuracy of DP-DSSP is also revealed. Moreover, DP-DSSP is able to handle arbitrarily large but uniformly bounded delays in the communication link. Finally, simulation experiments confirm the usefulness of DP-DSSP and the findings of this chapter.KeywordsDistributed online optimizationDifferential privacyStochastic subgradient-push algorithmTime-varying directed networksCommunication delays

In this paper the method of design the robust MPC controller for nonzero targets tracking is presented. The proposed method is based on dual mode paradigm in which the inner unconstrained controller is designed together with the robust feasible and invariant region. This robust feasible and invariant set unsure that the unconstrained controller does not violate the process constraints. So, the theory of robust invariant sets is investigated and the algorithm for finding needed robust invariant and feasible is presented. The outer controller is based on the solution of a min-max finite horizon optimisation problem. The mentioned min-max formulation of MPC involves minimisation of maximum of disturbance influence which range over the given polytope of possible disturbance realisations In the formulation of the robust MPC is a new linear cost function used that allow computation of the min-max optimisation problem using only single linear program. At the end, the illustrative real-time experiment is given.

The computational complexity and convergence speed of adaptive feedback cancellation algorithms depend on the number of adaptive parameters used to model the acoustic feedback path. To reduce the number of adaptive parameters it has been proposed to decompose the acoustic feedback path as the convolution of a (time-invariant) common part and a (time-varying) variable part. Typically the problem of estimating all the required coefficients has been formulated as a least-squares optimization problem. In contrast, in this paper we propose to formulate the estimation problem as a minmax optimization problem and show how this is associated with the maximum stable gain of a hearing aid. Experimental results using measured acoustic feedback paths from a two-microphone behind-the-ear hearing aid show that the proposed minmax optimization outperforms the least-squares optimization in terms of maximum stable gain. Furthermore, the robustness of proposed common part decomposition for different feedback paths is evaluated.

Sharing content over a mobile network through opportunistic contacts has recently received considerable attention. In proposed scenarios, users store content they download in a local cache and share it with other users they meet, e.g., via Bluetooth or WiFi. The storage capacity of mobile devices is typically limited; therefore, identifying which content a user should store in her cache is a fundamental problem in the operation of any such content distribution system. In this work, we propose Psephos, a novel mechanism for determining the caching policy of each mobile user. Psephos is fully distributed: users compute their own policies individually, in the absence of a central authority. Moreover, it is designed for a heterogeneous environment, in which demand for content, access to resources, and mobility characteristics may vary across different users. Most importantly, the caching policies computed by our mechanism are optimal: we rigorously show that Psephos maximizes the system's social welfare. Our results are derived formally using techniques from stochastic approximation and convex optimization; to the best of our knowledge, our work is the first to address caching with heterogeneity in a fully distributed manner.

Random projection algorithm is an iterative gradient method with random projections. Such an algorithm is of interest for constrained optimization when the constraint set is not known in advance or the projection operation on the whole constraint set is computationally prohibitive. This paper presents a distributed random projection (DRP) algorithm for fully distributed constrained convex optimization problems that can be used by multiple agents connected over a time-varying network, where each agent has its own objective function and its own constrained set. With reasonable assumptions, we prove that the iterates of all agents converge to the same point in the optimal set almost surely. In addition, we consider a variant of the method that uses a mini-batch of consecutive random projections and establish its convergence in almost sure sense. Experiments on distributed support vector machines demonstrate fast convergence of the algorithm. It actually shows that the number of iteration required until convergence is much smaller than scanning over all training samples just once.

This letter deals with the robust joint design of radar transmit waveform and receive filter bank in a background of range unambiguous signal-dependent clutter. Assuming an unknown Doppler shift for the target, the worst-case Signal-to-Interference-plus-Noise-Ratio (SINR) at the output of the receive filter bank is considered as the figure of merit. The transceiver design is pursued considering a max-min optimization problem with some constraints on the transmit energy, similarity, and signal dynamic range. Hidden convexity is shown and a procedure to derive optimal waveform and filters is given. Simulation results highlight the effectiveness of the devised method.

Over the past few years the circuit and system designers have shown great interest in minimax algorithms. The purpose of this paper is to present a new algorithm to solve the nonlinear minimax problem. The minimax optimization problem can be stated as: \[\mathop {{\text{minimize}}}\limits_x M_f (x)\] where \[M_f (x) = \mathop {\max }\limits_{1 \leqq i \leqq m} f_i (x)\] and $x = [x_1 ,x_2 , \cdots ,x_n ]^T $. The above objective function has discontinuous first partial derivatives at points where two or more of the functions $f_i$ are equal to $M_f$ even if $f_i (x),1 \leqq i \leqq m$ have continuous first partial derivatives. Thus we cannot use directly the well known gradient methods to minimize $M_f (x)$. Unlike the work by Bandler and Charalambous where they tackle the minimax problem as a limiting case of the least pth problem (so as to overcome the difficulty of discontinuous first partial derivatives) our approach is direct. We use two distinct search directions in the algorithm. The first, the horizontal direction, attempts to reduce $M_f (x)$ whilst, at the same time, keeping those functions whose values are close to $M_f (x)$, approximately equal. The second, the vertical direction, amounts to attempting to decrease the error to within which those functions are equal to $M_f (x)$ by means of linearization. A linear search follows after the horizontal direction has been calculated. The linear search incorporates several simple features of the algorithm and numerical results to date suggest the resulting algorithm is very efficient.

This article investigates the distributed online optimization problem in a zero-sum game between two distinct time-varying multiagent networks. At each iteration, the agents not only communicate with their neighbors but also gather information about agents in the opposing network through a time-varying network, assigning weights accordingly. Moreover, we consider quantized communication and bandit feedback mechanisms, with agents transmitting quantized information and adopting one-point estimators. At each iteration, agents make and submit decisions and then receive the cost function values near their decision points rather than the full cost function information. To guarantee the payoff of each network, we design an algorithm named quantized distributed online bandit optimization in two-network (QDOBO-TN). We use dynamic Nash equilibrium regret to measure the positive payoff discrepancy between the decision sequence produced by Algorithm QDOBO-TN and the Nash equilibrium sequence. Furthermore, we propose a multiepoch version of Algorithm QDOBO-TN. The regret bounds for both algorithms are sublinear with respect to the iteration count T. Finally, we conduct a series of simulation experiments that further validate the effectiveness of the algorithms.

In this paper, we study the distributed constrained optimization problem where the objective function is the sum of local convex cost functions of distributed nodes in a network, subject to a global inequality constraint. To solve this problem, we propose a consensus-based distributed regularized primal-dual subgradient method. In contrast to the existing methods, most of which require projecting the estimates onto the constraint set at every iteration, only one projection at the last iteration is needed for our proposed method. We establish the convergence of the method by showing that it achieves an O ( K (-1/4) ) convergence rate for general distributed constrained optimization, where K is the iteration counter. Finally, a numerical example is provided to validate the convergence of the propose method.