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This article proposes a robust method for the synthesis and design of optimal linear group delay filters with prescribed bandwidth and dispersion properties. The proposed filters are based on a cascade of second-order all-pass filter cells, for which all the parameters are optimized in order to reach the desired wideband linear group delay characteristic with positive slope. Compared to the previously published synthesis method, the proposed approach offers appropriate initialization enabling convergence and aiming for the optimal solution. The method’s convergence and robustness are proved through a 21-cells synthesis, which is the highest number of cascaded cells reported to date, due to the usual convergence problem. Then the electronic feasibility is illustrated through the measurement results of a lumped-elements linear group-delay filter prototype dimensioned with this method. The measured group delay presents a positive slope of 8 ns GHz−1 in the frequency range [100–725] MHz. It is, to the best of our knowledge, the highest positive slope reported to date with an electronic implementation (i.e except SAW and waveguide devices). [ABSTRACT FROM AUTHOR]

The limited memory version of the Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm is the most popular quasi-Newton algorithm in machine learning and optimization. Recently, it was shown that the stochastic L-BFGS (sL-BFGS) algorithm with the variance-reduced stochastic gradient converges linearly. In this paper, we propose a new sL-BFGS algorithm by importing a proper momentum. We prove an accelerated linear convergence rate under mild conditions. The experimental results on different data sets also verify this acceleration advantage. [ABSTRACT FROM AUTHOR]

This paper considers an unconstrained collaborative optimization of a sum of convex functions, where agents make decisions using local information in the presence of random interconnection topologies. We recast the problem as minimization of the sum of convex functions over a constraint set defined as the set of fixed-value points of a random operator derived from weighted matrices of random graphs. We show that the derived random operator has nonexpansivity property; therefore, this formulation does not need the distribution of random communication topologies. Hence, it includes random networks with/without asynchronous protocols. As an extension of the problem, we define a novel optimization problem, namely minimization of a convex function over the fixed-value point set of a nonexpansive random operator. We propose a discrete-time algorithm using diminishing step size for converging almost surely and in mean square to the global solution of the optimization problem under suitable assumptions. Consequently, as a special case, it reduces to a totally asynchronous algorithm for the distributed optimization problem. We show that fixed-value point is a bridge from deterministic analysis to random analysis of the algorithm. Finally, a numerical example illustrates the convergence of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Conjugate gradient (CG) methods are a class of important methods for solving linear equations and nonlinear optimization problems. In this paper, we propose a new stochastic CG algorithm with variance reduction and we prove its linear convergence with the Fletcher and Reeves method for strongly convex and smooth functions. We experimentally demonstrate that the CG with variance reduction algorithm converges faster than its counterparts for four learning models, which may be convex, nonconvex or nonsmooth. In addition, its area under the curve performance on six large-scale data sets is comparable to that of the LIBLINEAR solver for the $L2$ -regularized $L2$ -loss but with a significant improvement in computational efficiency. 1 CGVR algorithm is available on github: https://github.com/xbjin/cgvr [ABSTRACT FROM AUTHOR]

Many problems of interest for cyber-physical network systems can be formulated as mixed-integer linear programs in which the constraints are distributed among the agents. In this paper, we propose a distributed algorithmic framework to solve this class of optimization problems in a peer-to-peer network with no coordinator and with limited computation and communication capabilities. At each communication round, agents locally solve a small linear program, generate suitable cutting planes, and communicate a fixed number of active constraints. Within the distributed framework, we first propose an algorithm that, under the assumption of integer-valued optimal cost, guarantees finite-time convergence to an optimal solution. Second, we propose an algorithm for general problems that provides a suboptimal solution up to a given tolerance in a finite number of communication rounds. Both algorithms work under asynchronous, directed, unreliable networks. Finally, through numerical computations, we analyze the algorithm scalability in terms of the network size. Moreover, for a multi-agent multi-task assignment problem, we show, consistently with the theory, its robustness to packet loss. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of achieving average consensus over a random time-varying sequence of directed graphs by extending the class of so-called push-sum algorithms to such random scenarios. Provided that an ergodicity notion, which we term the directed infinite flow property, holds and the auxiliary states of agents are uniformly bounded away from zero infinitely often, we prove the almost sure convergence of the evolutions of this class of algorithms to the average of initial states. Moreover, for a random sequence of graphs generated using a so-called time-varying $B$ -irreducible probability matrix, we establish convergence rates for the proposed push-sum algorithm. [ABSTRACT FROM AUTHOR]

This paper concerns a spectral estimation problem in which we want to find a spectral density function that is consistent with estimated second-order statistics. It is an inverse problem admitting multiple solutions, and selection of a solution can be based on prior functions. We show that the problem is well-posed when formulated in a parametric fashion, and that the solution parameter depends continuously on the prior function. In this way, we are able to obtain a smooth parametrization of admissible spectral densities. Based on this result, the problem is reparametrized via a bijective change of variables out of a numerical consideration, and then a continuation method is used to compute the unique solution parameter. Numerical aspects, such as convergence of the proposed algorithm and certain computational procedures are addressed. A simple example is provided to show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a distributed Newton method for decenteralized optimization of large sums of convex functions. Our proposed method is based on creating a set of separable finite sum minimization problems by utilizing a decomposition technique known as Global Consensus that distributes the computation across nodes of a graph and enforces a consensus constraint among the separated variables. The key idea is to exploit the sparsity of the dual Hessian and recast the computation of the Newton step as one of the efficiently solving symmetric diagonally dominant linear equations. We show that our method outperforms the state-of-the-art algorithms, including ADMM. We validate our algorithm both theoretically and empirically. On the theory side, we demonstrate that our algorithm exhibits superlinear convergence within a neighborhood of optimality. Empirically, we show the superiority of this new method on a variety of large-scale optimization problems. The proposed approach is scalable to large problems and has a low communication overhead. [ABSTRACT FROM AUTHOR]

This article develops a communication-efficient algorithm to solve the stochastic optimization problem defined over a distributed network, aiming at reducing the burdensome communication in applications, such as distributed machine learning. Different from the existing works based on quantization and sparsification, we introduce a communication-censoring technique to reduce the transmissions of variables, which leads to our communication-censored distributed stochastic gradient descent (CSGD) algorithm. Specifically, in CSGD, the latest minibatch stochastic gradient at a worker will be transmitted to the server if and only if it is sufficiently informative. When the latest gradient is not available, the stale one will be reused at the server. To implement this communication-censoring strategy, the batch size is increasing in order to alleviate the effect of stochastic gradient noise. Theoretically, CSGD enjoys the same order of convergence rate as that of SGD but effectively reduces communication. Numerical experiments demonstrate the sizable communication saving of CSGD. [ABSTRACT FROM AUTHOR]

In this article, distributed algorithms are proposed for training a group of neural networks with private data sets. Stochastic gradients are utilized in order to eliminate the requirement for true gradients. To obtain a universal model of the distributed neural networks trained using local data sets only, consensus tools are introduced to derive the model toward the optimum. Most of the existing works employ diminishing learning rates, which are often slow and impracticable for online learning, while constant learning rates are studied in some recent works, but the principle for choosing the rates is not well established. In this article, constant learning rates are adopted to empower the proposed algorithms with tracking ability. Under mild conditions, the convergence of the proposed algorithms is established by exploring the error dynamics of the connected agents, which provides an upper bound for selecting the constant learning rates. Performances of the proposed algorithms are analyzed with and without gradient noises, in the sense of mean square error (MSE). It is proved that the MSE converges with bounded errors determined by the gradient noises, and the MSE converges to zero if the gradient noises are absent. Simulation results are provided to validate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a family of robust set-membership normalized subband adaptive filtering (RSM-NSAF) algorithms for acoustic echo cancellation (AEC). By using a new robust set-membership error bound, the RSM-NSAF algorithm obtains improved robustness against impulsive noises and decreased steady-state misalignment relative to the conventional set-membership NSAF (SM-NSAF) algorithm. To exploit the sparsity of the impulse response, the L0 norm constraint robust set-membership NSAF ( L0 -RSM-NSAF), robust set-membership improved proportionate NSAF (RSM-IPNSAF), and L0 norm constraint robust set-membership improved proportionate NSAF ( L0 -RSM-IPNSAF) algorithms are derived by minimizing a differentiable cost function that utilizes the Riemannian distance between the updated and previous weight vectors as well as the L0 norm of the weighted updated weight vector. Simulations in AEC application confirm the improvements of the proposed algorithms in performance. [ABSTRACT FROM PUBLISHER]

Consider a connected network of agents endowed with local cost functions representing private objectives. Agents seek to find an agreement on some minimizer of the aggregate cost, by means of repeated communications between neighbors. Consensus on the average over the network, usually addressed by gossip algorithms, is a special instance of this problem, corresponding to quadratic private objectives. Consensus on the median, or more generally, consensus on a given quantile, is also a special instance of this problem. In this paper we show that optimizing the aggregate cost function regularized by a total variation (TV) term has appealing properties. First, it can be done very naturally in a distributed way, yielding algorithms that are efficient on numerical simulations. Secondly, the optimum for the regularized cost is shown to be also the optimum for the initial aggregate cost function under assumptions that are simple to state. Finally, these algorithms are robust to unreliable agents that keep injecting some false value in the network. This is remarkable enough, and is not the case, for instance, of gossip algorithms that are entirely ruled by unreliable agents as detailed in the paper. [ABSTRACT FROM PUBLISHER]

This paper considers the problem of a leader that seeks to optimally influence the opinions of agents in a directed network through connecting with a limited number of the agents (“direct followers”), possibly in the presence of a fixed competing leader. The settings involving a single leader and two competing leaders are unified into a general combinatoric optimization problem, for which two heuristic approaches are developed. The first approach is based on a convex relaxation scheme, possibly in combination with the $\ell _1$ -norm regularization technique, and the second is based on a greedy selection strategy. The main technical novelties of this work are in the establishment of supermodularity of the objective function and convexity of its continuous relaxation. The greedy approach is guaranteed to have a lower bound on the approximation ratio sharper than $(1-1/e)$ , while the convex approach can benefit from efficient (customized) numerical solvers to have practically comparable solutions possibly with faster computation times. The two approaches can be combined to provide improved results. In numerical examples, the approximation ratio can be made to reach ${\text{90}\%}$ or higher depending on the number of direct followers. [ABSTRACT FROM AUTHOR]

This paper proposes a distributed model predictive control (MPC) approach for a family of discrete-time linear systems with local (uncoupled) and global (coupled) constraints. The proposed approach is based on the dual problem of an overall MPC optimization problem involving all systems, which is then solved distributively using a modified distributed Nesterov-accelerated-gradient algorithm. To further reduce the computational requirement, this approach allows for early termination of the distributed gradient algorithm. This is made possible via a consensus algorithm that determines the satisfaction of the termination condition and by appropriate tightening of the coupled constraints. Under reasonable assumptions, the approach is able to produce a suboptimal solution as long as the network of the systems is connected while ensuring recursive feasibility and exponential stability of the closed-loop system. The performance of the proposed approach is demonstrated by a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we consider the learning problem of multilayer perceptrons (MLPs) formulated as the problem of minimizing a smooth error function. As well known, the learning problem of MLPs can be a difficult nonlinear nonconvex optimization problem. Typical difficulties can be the presence of extensive flat regions and steep sided valleys in the error surface, and the possible large number of training data and of free network parameters. We define a wide class of batch learning algorithms for MLP, based on the use of block decomposition techniques in the minimization of the error function. The learning problem is decomposed into a sequence of smaller and structured minimization problems in order to advantageously exploit the structure of the objective function. Theoretical convergence results are established, and a specific algorithm is constructed and evaluated through an extensive numerical experimentation. The comparisons with the state-of-the-art learning algorithms show the effectiveness of the proposed techniques. [ABSTRACT FROM AUTHOR]

In many real-world applications, data are represented by matrices or high-order tensors. Despite the promising performance, the existing 2-D discriminant analysis algorithms employ a single projection model to exploit the discriminant information for projection, making the model less flexible. In this paper, we propose a novel compound rank- $k$ projection (CRP) algorithm for bilinear analysis. The CRP deals with matrices directly without transforming them into vectors, and it, therefore, preserves the correlations within the matrix and decreases the computation complexity. Different from the existing 2-D discriminant analysis algorithms, objective function values of CRP increase monotonically. In addition, the CRP utilizes multiple rank- $k$ projection models to enable a larger search space in which the optimal solution can be found. In this way, the discriminant ability is enhanced. We have tested our approach on five data sets, including UUIm, CVL, Pointing’04, USPS, and Coil20. Experimental results show that the performance of our proposed CRP performs better than other algorithms in terms of classification accuracy. [ABSTRACT FROM PUBLISHER]

The proximal inertial gradient descent (PIGD) is efficient for the composite minimization and applicable for broad of machine learning problems. In this article, we revisit the computational complexity of this algorithm and present other novel results, especially on the convergence rates of the objective function values. The nonergodic $O(1/k)$ rate is proved for PIGD with constant step size when the objective function is coercive. When the objective function fails to promise coercivity, we prove the sublinear rate with diminishing inertial parameters. In the case that the objective function satisfies the Polyak– Lojasiewicz (PŁ) property, the linear convergence is proved with much larger and general step size than the previous literature. We also extend our results to the multiblock version and present the computational complexity. Both cyclic and stochastic index selection strategies are considered. [ABSTRACT FROM AUTHOR]

In sparse empirical risk minimization (ERM) models, when sensitive personal data are used, e.g., genetic, healthcare, and financial data, it is crucial to preserve the differential privacy (DP) in training. In many applications, the information (i.e., features) of an individual is held by different organizations, which give rise to the prevalent yet challenging setting of the featurewise distributed multiparty model training. Such a setting is also beneficial to the scalability when the number of features exceeds the computation and storage capacity of a single node. However, existing private sparse optimizations are limited to centralized and samplewise distributed datasets only. In this article, we develop a differentially private algorithm for the sparse ERM model training under the featurewise distributed datasets setting. Our algorithm comes with guaranteed DP, nearly optimal utility, and reduced uplink communication complexity. Accordingly, we present a more generalized convergence analysis for block-coordinate Frank–Wolfe (BCFW) under arbitrary sampling (denoted as BCFW-AS in short), which significantly extends the known convergence results that apply to two specific sampling distributions only. To further reduce the uplink communication cost, we design an active private feature sharing scheme, which is new in both design and analysis of BCFW, to guarantee the convergence of communicating Johnson–Lindenstrauss transformed features. Empirical studies justify the new convergence as well as the nearly optimal utility theoretical results. [ABSTRACT FROM AUTHOR]

The recent literature indicates that preserving global pairwise sample similarity is of great importance for feature selection and that many existing selection criteria essentially work in this way. In this paper, we argue that besides global pairwise sample similarity, the local geometric structure of data is also critical and that these two factors play different roles in different learning scenarios. In order to show this, we propose a global and local structure preservation framework for feature selection (GLSPFS) which integrates both global pairwise sample similarity and local geometric data structure to conduct feature selection. To demonstrate the generality of our framework, we employ methods that are well known in the literature to model the local geometric data structure and develop three specific GLSPFS-based feature selection algorithms. Also, we develop an efficient optimization algorithm with proven global convergence to solve the resulting feature selection problem. A comprehensive experimental study is then conducted in order to compare our feature selection algorithms with many state-of-the-art ones in supervised, unsupervised, and semisupervised learning scenarios. The result indicates that: 1) our framework consistently achieves statistically significant improvement in selection performance when compared with the currently used algorithms; 2) in supervised and semisupervised learning scenarios, preserving global pairwise similarity is more important than preserving local geometric data structure; 3) in the unsupervised scenario, preserving local geometric data structure becomes clearly more important; and 4) the best feature selection performance is always obtained when the two factors are appropriately integrated. In summary, this paper not only validates the advantages of the proposed GLSPFS framework but also gains more insight into the information to be preserved in different feature selection tasks. [ABSTRACT FROM AUTHOR]

Nonnegative matrix factorization (NMF) has recently become a very popular unsupervised learning method because of its representational properties of factors and simple multiplicative update algorithms for solving the NMF. However, for the common NMF approach of minimizing the Euclidean distance between approximate and true values, the convergence of multiplicative update algorithms has not been well resolved. This paper first discusses the convergence of existing multiplicative update algorithms. We then propose a new multiplicative update algorithm for minimizing the Euclidean distance between approximate and true values. Based on the optimization principle and the auxiliary function method, we prove that our new algorithm not only converges to a stationary point, but also does faster than existing ones. To verify our theoretical results, the experiments on three data sets have been conducted by comparing our proposed algorithm with other existing methods. [ABSTRACT FROM AUTHOR]

We study the robustness of accelerated first-order algorithms to stochastic uncertainties in gradient evaluation. Specifically, for unconstrained, smooth, strongly convex optimization problems, we examine the mean-squared error in the optimization variable when the iterates are perturbed by additive white noise. This type of uncertainty may arise in situations where an approximation of the gradient is sought through measurements of a real system or in a distributed computation over a network. Even though the underlying dynamics of first-order algorithms for this class of problems are nonlinear, we establish upper bounds on the mean-squared deviation from the optimal solution that are tight up to constant factors. Our analysis quantifies fundamental tradeoffs between noise amplification and convergence rates obtained via any acceleration scheme similar to Nesterov's or heavy-ball methods. To gain additional analytical insight, for strongly convex quadratic problems, we explicitly evaluate the steady-state variance of the optimization variable in terms of the eigenvalues of the Hessian of the objective function. We demonstrate that the entire spectrum of the Hessian, rather than just the extreme eigenvalues, influences robustness of noisy algorithms. We specialize this result to the problem of distributed averaging over undirected networks and examine the role of network size and topology on the robustness of noisy accelerated algorithms. [ABSTRACT FROM AUTHOR]

In this article, we provide a distributed optimization algorithm, termed as TV- AB, that minimizes a sum of convex functions over time-varying, random directed graphs. Contrary to the existing work, the algorithm we propose does not require eigenvector estimation to estimate the (non-1) Perron eigenvector of a stochastic matrix. Instead, the proposed approach relies on a novel information mixing approach that exploits both row- and column-stochastic weights to achieve agreement toward the optimal solution when the underlying graph is directed. We show that TV-AB converges linearly to the optimal solution when the global objective is smooth and strongly convex, and the underlying time-varying graphs exhibit bounded connectivity, i.e., a union of every C consecutive graphs is strongly connected. We derive the convergence results based on the stability analysis of a linear system of inequalities along with a matrix perturbation argument. Simulations confirm the findings in this article. [ABSTRACT FROM AUTHOR]

Inspired by the subgradient push method developed recently by Nedić et al. we present a distributed dual averaging push algorithm for constrained nonsmooth convex optimization over time-varying directed graph. Our algorithm combines the dual averaging method with the push-sum technique and achieves an $O(1/ \sqrt{k})$ convergence rate. Compared with the subgradient push algorithm, our algorithm, first, addresses the constrained problems, and, second, has a faster convergence rate, and, third, simplifies the convergence analysis. We also generalize the proposed algorithm so that input variables of subgradient oracles have guaranteed convergence. [ABSTRACT FROM AUTHOR]

This note develops a distributed algorithm to solve a convex optimization problem with coupled constraints. Both coupled equality and inequality constraints are considered, where functions in the equality constraints are affine and functions in the inequality constraints are convex. Different from primal-dual subgradient methods with decreasing stepsizes for nonsmooth optimizations, our algorithm focuses on smooth problems and uses a fixed stepsize to find the exact optimal solution. Convergence analysis is derived with rigorous proofs. Our result is also illustrated by simulations. [ABSTRACT FROM AUTHOR]

Operator splitting techniques have recently gained popularity in convex optimization problems arising in various control fields. Being fixed-point iterations of nonexpansive (NE) operators, such methods suffer many well known downsides, which include high sensitivity to ill conditioning and parameter selection, and consequent low accuracy and robustness. As universal solution we propose SuperMann, a Newton-type algorithm for finding fixed points of NE operators. It generalizes the classical Krasnosel'skii–Mann scheme, enjoys its favorable global convergence properties and requires exactly the same oracle. It is based on a novel separating hyperplane projection tailored for nonexpansive mappings which makes it possible to include steps along any direction. In particular, when the directions satisfy a Dennis–Moré condition we show that SuperMann converges superlinearly under mild assumptions, which, surprisingly, do not entail nonsingularity of the Jacobian at the solution but merely metric subregularity. As a result, SuperMann enhances and robustifies all operator splitting schemes for structured convex optimization, overcoming their well known sensitivity to ill conditioning. [ABSTRACT FROM AUTHOR]

This technical note considers the dynamic average consensus problem, where a group of networked agents are required to estimate the average of their time-varying reference signals. Almost all existing solutions to this problem require a specific initialization of the estimator states, and such constraints render the algorithms vulnerable to network disruptions. Here, we present three robust algorithms that do not entail any initialization criteria. Furthermore, the proposed algorithms do not rely on the full knowledge of the dynamics generating the reference signals nor assume access to its time derivatives. Two of the proposed algorithms focus on undirected networks and make use of an adaptive scheme that removes the explicit dependence of the algorithm on any upper bounds on the reference signals or its time derivatives. The third algorithm presented here provides a robust solution to the dynamic average consensus problem on directed networks. Compared to the existing algorithms for directed networks, the proposed algorithm guarantees an arbitrarily small steady-state error bound that is independent of any bounds on the reference signals or its time derivatives. The current formulation allows each agent to select its own performance criteria, and the algorithm parameters are distributedly selected such that the most stringent requirement among them is satisfied. A performance comparison of the proposed approach to existing algorithms is presented. [ABSTRACT FROM AUTHOR]

Recently, low-rank representation (LRR) has shown promising performance in many real-world applications such as face clustering. However, LRR may not achieve satisfactory results when dealing with the data from nonlinear subspaces, since it is originally designed to handle the data from linear subspaces in the input space. Meanwhile, the kernel-based methods deal with the nonlinear data by mapping it from the original input space to a new feature space through a kernel-induced mapping. To effectively cope with the nonlinear data, we first propose the kernelized version of LRR in the clean data case. We also present a closed-form solution for the resultant optimization problem. Moreover, to handle corrupted data, we propose the robust kernel LRR (RKLRR) approach, and develop an efficient optimization algorithm to solve it based on the alternating direction method. In particular, we show that both the subproblems in our optimization algorithm can be efficiently and exactly solved, and it is guaranteed to obtain a globally optimal solution. Besides, our proposed algorithm can also solve the original LRR problem, which is a special case of our RKLRR when using the linear kernel. In addition, based on our new optimization technique, the kernelization of some variants of LRR can be similarly achieved. Comprehensive experiments on synthetic data sets and real-world data sets clearly demonstrate the efficiency of our algorithm, as well as the effectiveness of RKLRR and the kernelization of two variants of LRR. [ABSTRACT FROM AUTHOR]

The push-sum algorithm allows distributed computing of the average on a directed graph, and is particularly relevant when one is restricted to one-way and/or asynchronous communications. We investigate its behavior in the presence of unreliable communication channels where messages can be lost. We show that exponential convergence still holds and deduce fundamental properties that implicitly describe the distribution of the final value obtained. We analyze the error of the final common value we get for the essential case of two nodes, both theoretically and numerically. We provide performance comparison with a standard consensus algorithm. [ABSTRACT FROM AUTHOR]

This technical note deals with switched linear system identification and more particularly aims at solving switched linear regression problems in a large-scale setting with both numerous data and many parameters to learn. We consider the recent minimum-of-error framework with a quadratic loss function, in which an objective function based on a sum of minimum errors with respect to multiple submodels is to be minimized. The technical note proposes a new approach to the optimization of this nonsmooth and nonconvex objective function, which relies on Difference of Convex (DC) functions programming. In particular, we formulate a proper DC decomposition of the objective function, which allows us to derive a computationally efficient DC algorithm. Numerical experiments show that the method can efficiently and accurately learn switching models in large dimensions and from many data points. [ABSTRACT FROM AUTHOR]

We present a fast distributed solution to the convex network flow optimization problem. Our approach uses a family of dual descent algorithms that approximate the Newton direction to achieve faster convergence rates than existing distributed methods. The approximate Newton directions are obtained through matrix splitting techniques and sparse Taylor approximations of the inverse Hessian. We couple this descent direction with a distributed line search algorithm which requires the same information as our descent direction to compute. We show that, similarly to conventional Newton methods, the proposed algorithm exhibits super-linear convergence within a neighborhood of the optimal value. Numerical experiments corroborate that convergence times are between one to two orders of magnitude faster than existing distributed optimization methods. A connection with recent developments that use consensus to compute approximate Newton directions is also presented. [ABSTRACT FROM PUBLISHER]

Cumulative prospect theory (CPT) is a popular approach for modeling human preferences. It is based on probabilistic distortions and generalizes the expected utility theory. We bring the CPT to a stochastic optimization framework and propose algorithms for both estimation and optimization of CPT-value objectives. We propose an empirical distribution function-based scheme to estimate the CPT value, and then, use this scheme in the inner loop of a CPT-value optimization procedure. We propose both gradient based as well as gradient-free CPT-value optimization algorithms that are based on two well-known simulation optimization ideas: simultaneous perturbation stochastic approximation and model-based parameter search, respectively. We provide theoretical convergence guarantees for all the proposed algorithms and also illustrate the potential of CPT-based criteria in a traffic signal control application. [ABSTRACT FROM AUTHOR]

We develop algorithms that find and track the optimal solution trajectory of time-varying convex optimization problems that consist of local and network-related objectives. The algorithms are derived from the prediction-correction methodology, which corresponds to a strategy where the time-varying problem is sampled at discrete time instances, and then, a sequence is generated via alternatively executing predictions on how the optimizers at the next time sample are changing and corrections on how they actually have changed. Prediction is based on how the optimality conditions evolve in time, while correction is based on a gradient or Newton method, leading to decentralized prediction-correction gradient and decentralized prediction-correction Newton. We extend these methods to cases where the knowledge on how the optimization programs are changing in time is only approximate and propose decentralized approximate prediction-correction gradient and decentralized approximate prediction-correction Newton. Convergence properties of all the proposed methods are studied and empirical performance is shown on an application of a resource allocation problem in a wireless network. We observe that the proposed methods outperform existing running algorithms by orders of magnitude. The numerical results showcase a tradeoff between convergence accuracy, sampling period, and network communications. [ABSTRACT FROM PUBLISHER]

A novel optimization method is proposed to minimize a convex function subject to bilinear matrix inequality (BMI) constraints. The key idea is to decompose the bilinear mapping as a difference between two positive semidefinite convex mappings. At each iteration of the algorithm the concave part is linearized, leading to a convex subproblem. Applications to various output feedback controller synthesis problems are presented. In these applications, the subproblem in each iteration step can be turned into a convex optimization problem with linear matrix inequality (LMI) constraints. The performance of the algorithm has been benchmarked on the data from the COMPle\rm ib library. [ABSTRACT FROM AUTHOR]

The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multi-agent coordination, estimation in sensor networks, and large-scale machine learning. We develop and analyze distributed algorithms based on dual subgradient averaging, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our analysis allows us to clearly separate the convergence of the optimization algorithm itself and the effects of communication dependent on the network structure. We show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network, and confirm this prediction's sharpness both by theoretical lower bounds and simulations for various networks. Our approach includes the cases of deterministic optimization and communication, as well as problems with stochastic optimization and/or communication. [ABSTRACT FROM AUTHOR]

We develop a gradient algorithm for optimizing the performance of product-form networks through online adjustment of control parameters. The use of standard algorithms for finding optimal parameter settings is hampered by the prohibitive computational burden of calculating the gradient in terms of the stationary probabilities. The proposed approach instead relies on measuring empirical frequencies of the various states through simulation or online operation so as to obtain estimates for the gradient. Besides the reduction in computational effort, a further benefit of the online operation lies in the natural adaptation to slow variations in ambient parameters as commonly occurring in dynamic environments. On the downside, the measurements result in inherently noisy and biased estimates. We exploit mixing time results in order to overcome the impact of the bias and establish sufficient conditions for convergence to a globally optimal solution. We discuss our algorithm in the context of different systems, including queueing networks, loss networks, and wireless networks. We also illustrate how the algorithm can be used in such systems to optimize a service/cost trade-off, to map parameter regions that lead to systems meeting specified constraints, and to achieve target performance measures. For the latter application, we first identify which performance measures can be controlled depending on the set of configurable parameters. We then characterize the achievable region of performance measures in product-form networks, and finally we describe how our algorithm can be used to achieve the target performance in an online, distributed fashion, depending on the application context. [ABSTRACT FROM PUBLISHER]

A novel decomposition scheme to solve parametric nonconvex programs as they arise in Nonlinear Model Predictive Control (NMPC) is presented. It consists of a fixed number of alternating proximal gradient steps and a dual update per time step. Hence, the proposed approach is attractive in a real-time distributed context. Assuming that the Nonlinear Program (NLP) is semi-algebraic and that its critical points are strongly regular, contraction of the sequence of primal-dual iterates is proven, implying stability of the sub-optimality error, under some mild assumptions. Moreover, it is shown that the performance of the optimality-tracking scheme can be enhanced via a continuation technique. The efficacy of the proposed decomposition method is demonstrated by solving a centralized NMPC problem to control a dc motor and a distributed NMPC program for collaborative tracking of unicycles, both within a real-time framework. Furthermore, an analysis of the sub-optimality error as a function of the sampling period is proposed given a fixed computational power. [ABSTRACT FROM PUBLISHER]

In principle component pursuit (PCP), the essential idea is to replace the original non-convex optimization problem of the matrix rank and the count of non-zero entries by a convex optimization problem of the nuclear and \ell1 norms. In the PCP literature, it is rigorously proved that the validity of this idea depends on the coherence of the uncontaminated data. Specifically, the lower the coherence is, the equivalence of the convex optimization problem to the original non-convex one will hold by a larger probability. Although the coherence index is fixed for a given data set, it is possible to adjust this index by introducing different scalings to the data. The target of this work is thus to find the optimal scaling of the data such that the coherence index is minimized. Based on the analysis of the PCP problem structure, a non-convex optimization problem with implicit dependence on the scaling parameters is firstly formulated. To solve this problem, a coordinate descent algorithm is proposed. Under mild conditions on the structure of the data matrix, the convergence of the algorithm to a global optimal point is rigorously proved by treating the algorithm as a discrete-time dynamic system and utilizing a Lyapunov-type approach. Monte Carlo simulation experiments are performed to verify the effectiveness of the developed results. [ABSTRACT FROM PUBLISHER]

For hybrid systems where the continuous state space is partitioned by switching manifolds, the discrete state changes autonomously if the continuous state hits a manifold. Recently, we introduced a version of the minimum principle for the optimal control of such systems, where necessary optimality conditions were provided for the case that a trajectory passes through the intersection of switching manifolds refid="ref1"/. Further, we introduced an algorithm based on these conditions for computing optimal controls by varying not only the continuous state on switching manifolds based on gradient information, but also the sequence of discrete states refid="ref2"/. It was shown that, the combinatorial complexity of former computational schemes based on the minimum principle can be avoided, since not all possible discrete state sequences need to be analyzed separately. In this note, theoretical aspects of the algorithm and practical considerations for a successful implementation are discussed in detail. Further, a comparison to existing algorithms for the solution of hybrid optimal control problems with autonomous switching is presented. The efficiency of the proposed algorithm is demonstrated by a novel numerical example. [ABSTRACT FROM AUTHOR]

This technical note develops a novel robust control algorithm for linear systems subject to additive and bounded disturbances. The approach is based on the constraint tightening method. While this problem can be tackled using existing robust model predictive control techniques, the proposed method has an advantage in that it is computationally efficient and avoids the need to solve repeatedly an online optimization problem, while the optimization problem solved at initialization is a simple linear programming problem. The algorithm elaborated in this technical note guarantees convergence to a minimal disturbance invariant set, and the terminal predicted state constraint set is allowed to be larger than the minimal disturbance invariant set. As an illustration, the developed algorithm is applied to constrained roll control of a ship operating in a wave field. Simulation results show that the proposed approach reduces the ship roll motion while the input and dynamic stall constraints are satisfied. [ABSTRACT FROM AUTHOR]

Motivated by the need for fast computations in wireless sensor networks, the new F-Lipschitz optimization theory is introduced for a novel class of optimization problems. These problems are defined by simple qualifying properties specified in terms of increasing objective function and contractive constraints. It is shown that feasible F-Lipschitz problems have always a unique optimal solution that satisfies the constraints at equality. The solution is obtained quickly by asynchronous algorithms of certified convergence. F-Lipschitz optimization can be applied to both centralized and distributed optimization. Compared to traditional Lagrangian methods, which often converge linearly, the convergence time of centralized F-Lipschitz problems is at least superlinear. Distributed F-Lipschitz algorithms converge fast, as opposed to traditional Lagrangian decomposition and parallelization methods, which generally converge slowly and at the price of many message passings. In both cases, the computational complexity is much lower than traditional Lagrangian methods. Examples of application of the new optimization method are given for distributed estimation and radio power control in wireless sensor networks. The drawback of the F-Lipschitz optimization is that it might be difficult to check the qualifying properties. For more general optimization problems, it is suggested that it is convenient to have conditions ensuring that the solution satisfies the constraints at equality. [ABSTRACT FROM AUTHOR]

Nonlinear model predictive control (NMPC) strategies based on linearization about predicted system trajectories enable the online NMPC optimization to be performed by a sequence of convex optimization problems. The approach relies on bounds on linearization errors in order to ensure constraint satisfaction and convergence of the performance index, both during the optimization at each sampling instant and along closed loop system trajectories. This technical note proposes bounds based on robust tubes constructed around predicted trajectories. To ensure local optimality, the bounds are non-conservative for the case of zero linearization error, which requires the tube cross sections to vary along predicted trajectories. The feasibility, stability and convergence properties of the algorithm are established without the need for predictions to satisfy local optimality criteria. The strategy is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

We consider a distributed multi-agent network system where each agent has its own convex objective function, which can be evaluated with stochastic errors. The problem consists of minimizing the sum of the agent functions over a commonly known constraint set, but without a central coordinator and without agents sharing the explicit form of their objectives. We propose an asynchronous broadcast-based algorithm where the communications over the network are subject to random link failures. We investigate the convergence properties of the algorithm for a diminishing (random) stepsize and a constant stepsize, where each agent chooses its own stepsize independently of the other agents. Under some standard conditions on the gradient errors, we establish almost sure convergence of the method to an optimal point for diminishing stepsize. For constant stepsize, we establish some error bounds on the expected distance from the optimal point and the expected function value. We also provide numerical results. [ABSTRACT FROM AUTHOR]