Distributed Control over Networks Using Smoothing Techniques

In this paper we propose a distributed dual gradient algorithm for minimizing linearly constrained separable convex problems and analyze its rate of convergence. In particular, we show that under the assumption that the Hessian of the primal objective function is bounded we have a global error bound type property for the dual problem. Using this error bound property we devise a fully distributed dual gradient scheme for which we derive global linear rate of convergence. The proposed dual gradient method is fully distributed, requiring only local information, since is based on a weighted stepsize. Our method can be applied in many applications, e.g. distributed model predictive control, network utility maximization or optimal power flow.

BackgroundContraceptive use among HIV-positive women prevents unwanted pregnancy and reduces STI and HIV transmission. Recent studies link the use of hormonal contraceptive methods to increased risk of HIV transmission, consequently...

It is well-known that primal first-order algorithms achieve sublinear (linear) convergence for smooth convex (smooth strongly convex) constrained minimization. However, these methods encounter numerical difficulties when the primal feasible set is complicated, since they require exact projection onto this set. Algorithmic alternatives to convex problems with complicated feasible set are the dual first-order methods. Dual methods are able to handle easily complicated constraints, but they have difficulties in convergence when the norm of the optimal Lagrange multiplier is large, since this norm appears linearly in the convergence estimates of these methods. Moreover, they have typically sublinear convergence rate in an average primal sequence, even when the primal problem has smooth and strongly convex objective function. Motivated by these issues, in this paper, we analyze the convergence of primal first-order methods with inexact projections for solving constrained convex problems with smooth and then strongly convex objective function. In particular, we consider the inexact variants of Projected Gradient and Projected Fast Gradient methods, where instead of computing an exact projection onto the complicated primal feasible set, an approximate projection, not necessarily feasible, is used. We prove that we can still achieve similar convergence rates for these inexact projection first-order algorithms with those given in the exact projection settings, provided that the approximate projection is sufficiently accurate. Our convergence analysis allows us to derive explicitly the accuracy of the inexact projection and the number of iterations we need to perform in order to obtain an approximate solution for our convex problem. Finally, practical performance on random quadratic problems show encouraging results.

A Local Dual Method for Structured Nonconvex Optimisation Problems

Integer Linear Programming (ILP) is among the most popular optimization techniques found in practical applications, however, it often faces computational issues in modeling real-world problems. Computation can easily outgrow the computing power of standalone computers as the size of problem increases. The modern distributed computing releases the computing power constraints by providing scalable computing resources to match application needs, which boosts large-scale optimization. This chapter presents a paradigm that leverages Hadoop, an open-source distributed computing framework, to solve a large-scale ILP problem that is abstracted from real-world air traffic flow management. The ILP involves millions of decision variables, which is intractable even with existing state-of-the-art optimization software package. Dual decomposition method is used to separate variables into a set of dual subproblems that are smaller ILPs with lower dimensions, the computation complexity is downsized. As a result, the subproblems are solvable with optimization tools. It is shown that the iterative update on Lagrangian multipliers in dual decomposition method can fit into the Hadoop’s MapReduce programming model, which is designed to allocate computations to cluster for parallel processing and collect results from each node to report aggregate results. Thanks to the scalability of the distributed computing, parallelism can be improved by assigning more working nodes to the Hadoop cluster. As a result, the computational efficiency for solving the whole ILP problem is not impacted by the input size.

Increasing accuracy of two-class pattern recognition with enhanced fuzzy functions

This paper considers the primary user activity or the subchannel availability in optimally distributing the available resources for an orthogonal frequency-division multiple-access (OFDMA) cognitive radio multicast network. For this purpose, a risk-return model is presented, and a general rate-loss function, which gives a reduction in the attainable throughput whenever primary users reoccupy the temporarily accessible subchannels, is introduced. Taking the maximization of the expected sum rate of secondary multicast groups as the design objective, an efficient joint subcarrier and power-allocation scheme is proposed. Specifically, the design problem is solved via a dual optimization method under constraints on the tolerable interference thresholds at individual primary user's frequency bands. It is shown that as the number of subcarriers gets large (which is often the case in practice), the dual-domain solution becomes globally optimum with regard to the primal problem. More attractively, the ?practically optimal? performance of this approach is achieved with a substantially lower complexity, which is only linear in the total number of subcarriers as opposed to exponential complexity typically required by a direct search method. Our proposed design is valid for unicast and multicast transmissions and is applicable for a wide range of rate-loss functions, among which, the linear function is a special case. The superiority of the dual scheme is thoroughly verified by numerical examples.

By taking both control and state vectors as decision variables, the subproblems of model predictive control scheme can be considered as a class of separable convex optimisation problems with coupling linear constraints. A Lagrangian dual method is introduced to deal with the optimisation problem, in which, the primal problem is solved by a parallel coordinate descent method, and a fast dual ascend method is adopted to solve the dual problem iteratively. The proposed approach is applied to the well-known hierarchical and distributed model predictive control four-tank benchmark. Experimental results have testified the effectiveness of the proposed approach and shown that the benchmark problem can be well stabilised.

Irrigation canals are large-scale systems, consisting of many interacting components, and spanning vast geographical areas. For safe and efficient operation of these canals, maintaining the levels of the water flows close to pre-specified reference values is crucial, both under normal operating conditions as well as in extreme situations.    Irrigation canals are equipped with local controllers, to control the flow of water by adjusting the settings of control structures such as gates and pumps. Traditionally, the local controllers operate in a decentralized way in the sense that they use local information only, that they are not explicitly aware of the presence of other controllers or subsystems, and that no communication among them takes place. Hence, an obvious drawback of such a decentralized control scheme is that adequate performance at a system-wide level may be jeopardized, due to the unexpected and unanticipated interactions among the subsystems and the actions of the local controllers.    In this paper we survey the state-of-the-art literature on distributed control of water systems in general, and irrigation canals in particular. We focus on the model predictive control (MPC) strategy, which is a model-based control strategy in which prediction models are used in an optimization to determine optimal control inputs over a given horizon. We discuss how communication among local MPC controllers can be included to improve the performance of the overall system. We present a distributed control scheme in which each controller employs MPC to determine those actions that maintain water levels after disturbances close to pre-specified reference values. Using the presented scheme the local controllers cooperatively strive for obtaining the best system-wide performance. A simulation study on an irrigation canal with seven reaches illustrates the potential of the approach.

There are two important approaches based on linear ranking functions for solving linear programming problems with cost coefficients as an auxiliary problem to obtain a fuzzy solution of fuzzy variable linear programming problem. The first approach uses the primal simplex method that assumes an initial primal feasible basic solution is at hand. The second approach is based on dual simplex method that begins with a basic dual feasible basic solution and proceeds by pivoting through a series of dual basic solutions until the associated complementary primal basic fuzzy solution is feasible. In this paper, we propose a new method called the primal-dual algorithm, which is similar to the dual simplex method and begins with dual feasibility and proceeds to obtain primal feasibility while maintaining complementary slackness. An important difference between the dual simplex method and the primal-dual method is that the primal-dual algorithm does not require a dual feasible solution to be basic. This algorithm is useful specially for solving minimum fuzzy cost flow problem in which finding an initial dual feasible solution turns out to be a trivial task.

An iterative substructuring method with Lagrange multipliers is considered for second order elliptic problems, which is a variant of the FETI-DP method. The standard FETI-DP formulation is associated with the saddle-point problem which is induced from the minimization problem with a constraint for imposing the continuity across the interface. Starting from the slightly changed saddle-point problem by addition of a penalty term with a positive penalization parameter η, we propose a dual substructuring method which is implemented iteratively by the conjugate gradient method. In spite of the absence of any preconditioners, it is shown that the proposed method is numerically scalable in the sense that for a large value of η, the condition number of the resultant dual problem is bounded by a constant independent of both the subdomain size H and the mesh size h. Computational issues and numerical results are presented.

An ε-sensitivity analysis in the primal–dual interior point method

An interior-point algorithm for linear optimization based on a new barrier function

Motivated by large-scale signal processing and machine learning applications, this paper considers the distributed multi-agent optimization problem for a linearly constrained non-convex problem. Each of the agents owns a local cost function and local variable, but are coupled with each other due to the linear constraint. Most of the existing methods are either applicable for convex problems only or are developed under the non-convex setting subject to a specific type of linear constraint. There still lacks a distributed method for solving the linear constrained problem under the general and non-convex setting. In this paper, we propose such a method, called the proximal dual consensus (PDC) method, that combines a proximal technique and the dual consensus method. Theoretical analysis shows that the proposed PDC method can yield a Karush-Kuhn-Tucker solution of the linearly constrained non-convex problem and it has an O(1/e) iteration complexity, where e is a solution accuracy. The practical behavior of the proposed method is examined by numerical results.

The idea of network protocol design based on optimization theory has been proposed and used practically in Internet for about 15 years. However, for large-scale wireless ad hoc network, although protocol could be viewed as a recursive solving of a global optimization problem, protocol design is still facing huge challenge because an effective distributed algorithm for solving global optimization problem is still lacking. We solve the problem by putting forward a systematic design method based on optimization decomposition. The systematic method includes primal decomposition method and dual decomposition method, with which a complex optimization problem can be decomposed into several smaller and independent optimization subproblems. By using subgradient method, each of these subproblems can be solved distributively. Further, the above two methods can be combined in different sequences or used recursively to solve more complex optimization problems. Two examples of wireless protocol design, the transmission control protocol and the joint congestion control and power control protocol, are given to demonstrate its validity.