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Recently, several convergence rate results for Douglas-Rachford splitting and the alternating direction method of multipliers (ADMM) have been presented in the literature. In this paper, we show global linear convergence rate bounds for Douglas-Rachford splitting and ADMM under strong convexity and smoothness assumptions. We further show that the rate bounds are tight for the class of problems under consideration for all feasible algorithm parameters. For problems that satisfy the assumptions, we show how to select step-size and metric for the algorithm that optimize the derived convergence rate bounds. For problems with a similar structure that do not satisfy the assumptions, we present heuristic step-size and metric selection methods. [ABSTRACT FROM PUBLISHER]

Purpose – To improve the performance of the krill herd (KH) algorithm, in this paper, a series of chaotic particle-swarm krill herd (CPKH) algorithms are proposed for solving optimization tasks within limited time requirements. The paper aims to discuss these issues. Design/methodology/approach – In CPKH, chaos sequence is introduced into the KH algorithm so as to further enhance its global search ability. Findings – This new method can accelerate the global convergence speed while preserving the strong robustness of the basic KH. Originality/value – Here, 32 different benchmarks and a gear train design problem are applied to tune the three main movements of the krill in CPKH method. It has been demonstrated that, in most cases, CPKH with an appropriate chaotic map performs superiorly to, or at least highly competitively with, the standard KH and other population-based optimization methods. [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]

This paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problem à la Byrnes/Georgiou/Lindquist. The approximation is pursued with respect to the Kullback–Leibler pseudo-distance, which gives rise to a convex optimization problem. After developing the variational analysis, we discuss the properties of an efficient algorithm for the solution of the corresponding dual problem, based on the iteration of a nonlinear map in a bounded subset of the dual space. Our main result is the proof of local convergence of the latter, established as a consequence of the central manifold theorem. Supported by numerical evidence, we conjecture that, in the mentioned bounded set, the convergence is actually global. [ABSTRACT FROM AUTHOR]

Higher order sliding mode (HOSM) control design is considered for systems with a known permanent relative degree. In this paper, we introduce the Robust Fuller's Problem that is a robust generalization of the Fuller's problem, a standard optimal control problem for a chain of integrators with bounded control. By solving the Robust Fuller's Problem it is possible to obtain feed. back laws that are HOSM algorithms of generic order and, in addition, provide optimal finite-time reaching of the sliding manifold. A common difficulty in the use of existing HOSM algorithms is the tuning of design parameters: our methodology proves useful for the tuning of HOSM controller parameters in order to assure desired performances and prevent instabilities. The convergence and stability properties of the proposed family of controllers are theoretically analyzed. Simulation evidence demonstrates their effectiveness. [ABSTRACT FROM AUTHOR]

Abstract: Most real world multi-objective problems (MOPs) have a complicated solution space. Facing such problems, a direction vectors based co-evolutionary multi-objective optimization algorithm (DVCMOA) that introduces the decomposition idea from MOEA/D to co-evolutionary algorithms is proposed in this paper. It is novel in the sense that DVCMOA applies the concept of direction vectors to co-evolutionary algorithms. DVCMOA first divides the entire population into several subpopulations on the basis of the initial direction vectors in the objective space. Then, it solves MOPs through the co-evolutionary interaction among the subpopulations in which individuals are classified according to their direction vectors. Finally, it explores the less developed regions to maintain the relatively uniform distribution of the solution space. In this way, DVCMOA has advantages in convergence, diversity and uniform distribution of the non-dominated solution set, which are explained through comparison with other state-of-the-art multi-objective optimization evolutionary algorithms (MOEAs) in this paper. DVCMOA is shown to be effective on 6 multi-objective 0–1 knapsack problems. [Copyright &y& Elsevier]

Abstract: The research on optimal design of infinite-impulse response (IIR) filters based on optimization techniques has gained much attention in recent years. However, due to the limited performance of the applied optimization techniques, the orders of the filters, which can be obtained, are very low in the previous research. Memetic algorithms (MAs) are widely recognized to have better convergence capability than their conventional counterparts. However, the universality of the MAs, e.g. the ability of solving diverse kinds of digital IIR filter designs, is still limited. In this paper, we design a Two-Stage ensemble Memetic Algorithm (TSMA) framework to more appropriately synthesize the strengths of the evolutionary global search and local search techniques. In the first optimization stage, a competition is held among the candidate local search techniques. Its major idea is to choose the best local search technique and to obtain good initial state. Inheriting the good information of the first stage, the second optimization stage is to implement effective adaptive MA to pursue high-quality solution. The experimental studies presented in this paper contain three aspects: (1) the benefits of the TSMA framework are experimentally investigated by comparing TSMA with its sub-optimizers and recent effective evolutionary algorithms (EAs) on 26 test functions; then (2) TSMA is compared with 4 MAs on the CEC05 functions to comprehensively show the advantages of TSMA; and (3) the TSMA and 6 state-of-the-art algorithms are applied to design high-order digital infinite-impulse response (IIR) filters. The experimental results definitely demonstrate the excellent effectiveness, efficiency and reliability of TSMA on both function optimization and digital IIR filter design tasks. [Copyright &y& Elsevier]

Abstract This paper develops a novel algorithm to solve real-world constrained optimization problems, which hybridizes multiobjective optimization techniques with an ϵ-constrained method. First, a constrained optimization problem at hand is transformed into a bi-objective optimization problem. By the transformation, the advantage of multiobjective optimization techniques can be utilized in the constrained optimization area to balance population diversity and convergence. Meanwhile, the ϵ-constrained method is applied, which keeps the population evolving toward feasible region of the constrained optimization problem. In our proposed algorithm, the differential evolution is employed as a search engine to create offspring at each generation. Further, different combinations of mutation operators have been developed to improve the search ability and the population convergence at different stages. The performance of our approach is evaluated on 64 benchmark test functions from three popular test suits. Experimental results demonstrate that our proposed approach is capable of obtaining high-quality solutions on the majority of benchmark test functions, when compared with some other state-of-the-art constrained optimization algorithms. [ABSTRACT FROM AUTHOR]

Abstract: Motivated by the recent success of diverse approaches based on differential evolution (DE) to solve constrained numerical optimization problems, in this paper, the performance of this novel evolutionary algorithm is evaluated. Three experiments are designed to study the behavior of different DE variants on a set of benchmark problems by using different performance measures proposed in the specialized literature. The first experiment analyzes the behavior of four DE variants in 24 test functions considering dimensionality and the type of constraints of the problem. The second experiment presents a more in-depth analysis on two DE variants by varying two parameters (the scale factor F and the population size NP), which control the convergence of the algorithm. From the results obtained, a simple but competitive combination of two DE variants is proposed and compared against state-of-the-art DE-based algorithms for constrained optimization in the third experiment. The study in this paper shows (1) important information about the behavior of DE in constrained search spaces and (2) the role of this knowledge in the correct combination of variants, based on their capabilities, to generate simple but competitive approaches. [ABSTRACT FROM AUTHOR]

Abstract-This paper studies state quantization schemes for feedback stabilization of control systems with limited information. The focus is on designing the least destabilizing quantizer subject to a given information constraint. We explore several ways of mea- suring the destabilizing effect of a quantizer on the closed-loop system, including (but not limited to) the worst-case quantization error. In each case, we show how quantizer design can be naturally reduced to a version of the so-called multicenter problem from locational optimization. Algorithms for obtaining solutions to such problems, all in terms of suitable Voronoi quantizers, are discussed. In particular, an iterative solver is developed for a novel weighted multicenter problem which most accurately represents the least destabilizing quantizer design. A simulation study is also presented. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an analysis of an adaptive random search (ARS) algorithm, a global minimization method. A probability model is introduced to characterize the statistical properties of the number of iterations required to find an acceptable solution. Moreover, based on this probability model, a new stopping criterion is introduced to predict the maximum number of iterations required to find an acceptable solution with a pre-specified level of confidence. Finally, this paper presents a systematic procedure for choosing the user-specified parameters in the ARS algorithm for fastest convergence. The results, which are valid for search spaces of arbitrary dimensions, are illustrated on a simple three-dimensional example. [Copyright &y& Elsevier]

“ Fuzzy Theoretic Approach to Signals and Systems ” assumes all system variables and parameters as uncertain (i.e. being characterized by membership functions), develops a mathematical theory for analytically determining the membership functions on system variables and parameters, derives algorithms for estimating the parameters of membership functions, and establishes robustness and convergence properties of the estimation algorithms. The membership functions are analytically determined via solving a variational optimization problem that maximizes the “ over-uncertainties-averaged-log-membership ” of the observed data around an initial guess. This paper develops the analytical fuzzy theory for the particular case of a multi-input single-output static system affected by noises. The theory facilitates designing an adaptive filtering algorithm. The robustness of the adaptive filtering algorithm is proved theoretically via a mathematical analysis. Numerical experiments further demonstrate the robustness of the filtering algorithm. A comparison of the algorithm with the state-of-art methods is made using the practical biomedical applications related to the modeling and analysis of heart rate signals for assessing the physiological state of an individual. [ABSTRACT FROM AUTHOR]

Recently, krill herd algorithm (KHA) is proposed for the solution of global optimization problem. Various chaotic maps are considered in the proposed chaotic KHA (CKHA) of the present work to improve the performance of basic KHA method. It is observed that Logistic map based CKHA offers better results as compared other chaotic maps. The performance of the proposed CKHA is tested on standard 26-bus and IEEE 57-bus test power systems for the solution of optimal power flow of power system with different objectives that reflect minimization of fuel cost or active power loss or sum of total voltage deviation. Results obtained by using CKHA are compared to other evolutionary optimization techniques surfaced in the recent state-of-the-art literature. The results presented in this paper show that the proposed CKHA algorithm outperforms the other techniques in terms of convergence rate and global search ability. [ABSTRACT FROM AUTHOR]

Abstract: Recently, Gandomi and Alavi proposed a meta-heuristic optimization algorithm, called Krill Herd (KH). This paper introduces the chaos theory into the KH optimization process with the aim of accelerating its global convergence speed. Various chaotic maps are considered in the proposed chaotic KH (CKH) method to adjust the three main movements of the krill in the optimization process. Several test problems are utilized to evaluate the performance of CKH. The results show that the performance of CKH, with an appropriate chaotic map, is better than or comparable with the KH and other robust optimization approaches. [Copyright &y& Elsevier]

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]

Abstract: This paper presents a new technique for designing IIR filters that have minimum deviation from equiripple response. The algorithm is also able to find transfer functions with unequal numerator and denominator orders, which are suitable for both digital and analog IIR sampled-data realizations. Elliptic filters are produced as a particular case, when equal numerator and denominator orders are specified. Pole-zero mapping is used for scalar update of optimization parameters, thereby reducing the algorithm complexity. Zeros are structurally allocated on the unit circumference for efficient stopband shaping. Moreover, filter stability is easily enforced by restricting the radii of the poles to be lower than 1. A Taylor series expansion is employed to determine the step size of the parameter updates. The proposed approach is based on evaluations of partial cost functions to avoid local minima, and hence increase the robustness and the convergence rate of the optimization process. Design examples are shown to illustrate the efficacy of the proposed design technique, compared to alternative design techniques. [Copyright &y& Elsevier]

Abstract: Load balancers distribute workload across multiple nodes based on a variation of the round robin algorithm, or a more complex algorithm that optimizes a specified objective or allows for horizontal scalability and higher availability. In this paper, we investigate whether robust load balancing can be achieved using a local co-operative mechanism between the resources (nodes). The local aspect of the mechanism implies that each node interacts with a small subset of the nodes that define its neighborhood. The co-operative aspect of the mechanism implies that a node may offload some of load to its neighbor nodes that have lesser load or accept jobs from neighbor nodes that have higher load. Each node is thus only aware of the state of its neighboring nodes and there is no central entity that has the knowledge of the state of all the nodes. We model the overall mechanism of load balancing based on local interactions as a congestion game and show that convergence to the Nash equilibrium is possible using only local interactions. We derive worst case bounds on the number of transfers (time) required to achieve global load balancing under this setup. We also include simulation results to demonstrate emergent global load balancing based only on local interactions and local information. [Copyright &y& Elsevier]

Abstract: In this paper, a ship lock scheduling problem is investigated. Ships arrive randomly over time, and the instantaneous arrival rates are allowed to vary both temporally and stochastically in an arbitrary manner. A data-driven approach is applied to a single ship lock scheduling, which is a typical optimizing and decision-making problem. The objective is to minimize the operation costs and other costs(e.g. water cost, electricity cost, and staff welfare cost) by selecting an appropriate slot number during a planned period. The convergence of data-driven approach is discussed from three aspects: the convergence of ant colony optimization algorithm, the convergence of the proposed algorithm, and the error between the historical ship data and the current arrival ship data. The research findings are beneficial for the convergence analysis of data-driven theory and the management of waterway transportation. [Copyright &y& Elsevier]

Abstract: Motivated by concepts in quantum mechanics and particle swarm optimization (PSO), quantum-behaved particle swarm optimization (QPSO) was proposed as a variant of PSO with better global search ability. Although it has been shown to perform well in finding optimal solutions for many optimization problems, there has so far been little theoretical analysis on its convergence and performance. This paper presents a convergence analysis and performance evaluation of the QPSO algorithm and it also proposes two variants of the QPSO algorithm. First, we investigate in detail the convergence of the QPSO algorithm on a probabilistic metric space and prove that the QPSO algorithm is a form of contraction mapping and can converge to the global optimum. This is the first time that the theory of probabilistic metric spaces has been employed to analyze a stochastic optimization algorithm. We provided a new definition for the convergence rate of a stochastic algorithm as well as definitions for three types of convergence according to the correlations between the convergence rate and the objective function values. With these definitions, the effectiveness of the QPSO is evaluated by computing and analyzing the time complexity and the convergence rate of the algorithm. Then, the QPSO with random mean best position (QPSO-RM) and the QPSO with ranking operator (QPSO-RO) are proposed as two improvements of the QPSO algorithm. Finally, some empirical studies on popular benchmark functions are performed in order to make a full performance evaluation and comparison between QPSO, QPSO-RM, QPSO-RO and other variants of PSO. [Copyright &y& Elsevier]

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]

Abstract: When the Newton–Raphson algorithm or the Fisher scoring algorithm does not work and the EM-type algorithms are not available, the quadratic lower-bound (QLB) algorithm may be a useful optimization tool. However, like all EM-type algorithms, the QLB algorithm may also suffer from slow convergence which can be viewed as the cost for having the ascent property. This paper proposes a novel ‘shrinkage parameter’ approach to accelerate the QLB algorithm while maintaining its simplicity and stability (i.e., monotonic increase in log-likelihood). The strategy is first to construct a class of quadratic surrogate functions that induces a class of QLB algorithms indexed by a ‘shrinkage parameter’ () and then to optimize over under some criterion of convergence. For three commonly used criteria (i.e., the smallest eigenvalue, the trace and the determinant), we derive a uniformly optimal shrinkage parameter and find an optimal QLB algorithm. Some theoretical justifications are also presented. Next, we generalize the optimal QLB algorithm to problems with penalizing function and then investigate the associated properties of convergence. The optimal QLB algorithm is applied to fit a logistic regression model and a Cox proportional hazards model. Two real datasets are analyzed to illustrate the proposed methods. [Copyright &y& Elsevier]

Abstract: In evolutionary multi-objective optimization (EMO), the convergence to the Pareto set of a multi-objective optimization problem (MOP) and the diversity of the final approximation of the Pareto front are two important issues. In the existing definitions and analyses of convergence in multi-objective evolutionary algorithms (MOEAs), convergence with probability is easily obtained because diversity is not considered. However, diversity cannot be guaranteed. By combining the convergence with diversity, this paper presents a new definition for the finite representation of a Pareto set, the B-Pareto set, and a convergence metric for MOEAs. Based on a new archive-updating strategy, the convergence of one such MOEA to the B-Pareto sets of MOPs is proved. Numerical results show that the obtained B-Pareto front is uniformly distributed along the Pareto front when, according to the new definition of convergence, the algorithm is convergent. [Copyright &y& Elsevier]

Abstract: An adaptation of a parametric ant colony optimization (ACO) to multi-objective optimization (MOO) is presented in this paper. In this algorithm (here onwards called MACO) the concept of MOO is achieved using the reference point (or goal vector) optimization strategy by applying scalarization. This method translates the multi-objective optimization problem to a single objective optimization problem. The ranking is done using ϵ-dominance with modified L p metric strategy. The minimization of the maximum distance from the goal vector drives the solution close to the goal vector. A few validation test cases with multi-objectives have been demonstrated. MACO was found to out perform R-NSGA-II for the test cases considered. This algorithm was then integrated with a meshless computational fluid dynamics (CFD) solver to perform aerodynamic shape optimization of an airfoil. The algorithm was successful in reaching the optimum solutions near to the goal vector on one hand. On the other hand the algorithm converged to an optimum outside the boundary specified by the user for the control variables. These make MACO a good contender for multi-objective shape optimization problems. [Copyright &y& Elsevier]

We present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time-varying connectivity. Our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem, where the global objective function is a combination of local agent objective functions. Our main focus is on constrained problems where the estimates of each agent are restricted to lie in different convex sets. To highlight the effects of constraints, we first consider a constrained consensus problem and present a distributed "projected consensus algorithm" in which agents combine their local averaging operation with projection on their individual constraint sets. This algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents. We establish convergence and convergence rate results for the projected consensus algorithm.We next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets. We present a distributed "projected subgradient algorithm" which involves each agent performing a local averaging operation, taking a subgradient step to minimize its own objective function, and projecting on its constraint set. We show that, with an appropriately selected stepsize rule, the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal, and when the weights are time-varying but all agents have the same constraint set. [ABSTRACT FROM AUTHOR]

We develop a receding horizon control approach to stochastic linear systems with control and state multiplicative noise that also contain constraints. Our receding horizon formulation is based upon an on-line optimization that utilizes open-loop plus linear feedback and is solved as a semi-definite programming problem. We also provide a characterization of stability, performance, and constraint satisfaction properties of the receding horizon controlled system under a specific choice of terminal weight and terminal constraint. A simple numerical example is used to illustrate the approach. [ABSTRACT FROM AUTHOR]

A hybrid evolutionary programming (EP) method is presented for global optimization of complex circuits. The conventional EP is integrated with a clustering algorithm to improve the robustness of the algorithm for complex multimodal circuit optimization problems. The EP generates populations around the regions of the search space which can potentially contain a minimum but may be overlooked. The clustering algorithm is used to identify these regions dynamically. In order to improve the speed of optimization, the EP is combined with a gradient-based search method in an efficient fashion. The local search is performed from the center of each identified cluster in order to find the minimum in the region very fast. The hybrid algorithm can also reduce the search space by avoiding the search in the areas that were previously investigated. This feature greatly improves the speed of optimization and prevents the premature convergence as well. The algorithm performed very well in several benchmark problems including a test function minimization and global optimization of a complex RF diplexer circuit. [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]

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]

We consider an optimal control problem for the hybrid model of a deterministic flow shop system, in which the jobs are processed in the order they arrive at the system. The problem is decomposed into a higher-level discrete-event system control problem of determining the optimal service times, and a set of lower-level classical control problems of determining the optimal control inputs for given service times. We focus on the higher-level problem which is nonconvex and nondifferentiable. The arrival times are known and the decision variables are the service times that are controllable within constraints. We present an equivalent convex optimization problem with linear constraints. Under some cost assumptions, we show that no waiting is observed on the optimal sample path. This property allows us to simplify the convex optimization problem by eliminating variables and constraints. We also prove, under an additional strict convexity assumption, the uniqueness of the optimal solution and propose two algorithms to decompose the simplified convex optimization problem into a set of smaller convex optimization problems. The effects of the simplification and the decomposition on the solution times are shown on an example problem. [ABSTRACT FROM AUTHOR]

We develop nonsmooth optimization techniques to solve H∞ synthesis problems under additional structural constraints on the controller. Our approach avoids the use of Lyapunov variables and therefore leads to moderate size optimization programs even for very large systems. The proposed framework is versatile and can accommodate a number of challenging design problems including static, fixed-order, fixed-structure, decentralized control, design of PID controllers and simultaneous design and stabilization problems. Our algorithmic strategy uses generalized gradients and bundling techniques suited for the H∞ norm and other nonsmooth performance criteria. We compute descent directions by solving quadratic programs and generate steps via line search. Convergence to a critical point from an arbitrary starting point is proved and numerical tests are included to validate our methods. The proposed approach proves to be efficient even for systems with several hundreds of states. [ABSTRACT FROM AUTHOR]