si admm a stochastic inexact admm framework for stochastic convex programs we consider the structured stochastic convex program requiring the minimization of mathbb e xi tilde f x xi mathbb e xi tilde g y xi subject to the constraint ax by b motivated by the need for decentralized schemes we propose a stochastic inexact alternating direction method of multiplier si admm framework where subproblems are solved inexactly via stochastic approximation schemes we propose a stochastic inexact alternating direction method of multiplier si admm framework where subproblems are solved inexactly via stochastic approximation schemes based on this framework we prove the following 1 under suitable assumptions on the associated batch size of samples utilized at each iteration the si admm scheme produces a sequence that converges to the unique solution almost surely 2 if the number of gradient steps or equivalently the number of sampled gradients utilized for solving the subproblems in each iteration increases at a geometric rate the mean squared error diminishes to zero at a prescribed geometric rate and 3 the overall iteration complexity in terms of gradient steps or equivalently samples is found to be consistent with the canonical level of mathcal o 1 epsilon preliminary applications on lasso and distributed regression suggest that the scheme performs well compared to its competitors

multidimensional gains for stochastic approximation this paper deals with iterative jacobian based recursion technique for the root finding problem of the vector valued function whose evaluations are contaminated by noise instead of a scalar step size we use an iterate dependent matrix gain to effectively weigh the different elements associated with the noisy observations the analytical development of the matrix gain is built on an iterative dependent linear function interfered by additive zero mean white noise where the dimension of the function is m ge 1 and the dimension of the unknown variable is n ge 1 necessary and sufficient conditions for m ge n algorithms are presented pertaining to algorithm stability and convergence of the estimate error covariance matrix two algorithms are proposed one for the case where m ge n and the second one for the antithesis the two algorithms assume full knowledge of the jacobian the recursive algorithms are proposed for generating the optimal iterative dependent matrix gain the proposed algorithms here aim for per iteration minimization of the mean square estimate error we show that the proposed algorithm satisfies the presented conditions for stability and convergence of the covariance in addition the convergence rate of the estimation error covariance is shown to be inversely proportional to the number of iterations for the antithesis m contraction of the error covariance is guaranteed this underdetermined system of equations can be helpful in training neural networks numerical examples are presented to illustrate the performance capabilities of the proposed multidimensional gain while considering nonlinear functions

a smoothing method for mathematical programs with equilibrium constraints the mathematical program with equilibrium constraints mpec is an optimization problem with variational inequality constraints mpec problems include bilevel programming problems as a particular case and have a wide range of applications mpec problems with strongly monotone variational inequalities are considered in this paper they are transformed into an equivalent one level nonsmooth optimization problem then a sequence of smooth regular problems that progressively approximate the nonsmooth problem and that can be solved by standard available software for constrained optimization is introduced it is shown that the solutions stationary points of the approximate problems converge to a solution stationary point of the original mpec problem numerical results showing viability of the approach are reported

incremental stochastic subgradient algorithms for convex optimization this paper studies the effect of stochastic errors on two constrained incremental subgradient algorithms the incremental subgradient algorithms are viewed as decentralized network optimization algorithms as applied to minimize a sum of functions when each component function is known only to a particular agent of a distributed network first the standard cyclic incremental subgradient algorithm is studied in this the agents form a ring structure and pass the iterate in a cycle when there are stochastic errors in the subgradient evaluations sufficient conditions on the moments of the stochastic errors are obtained that guarantee almost sure convergence when a diminishing step size is used in addition almost sure bounds on the algorithm s performance with a constant step size are also obtained next the markov randomized incremental subgradient method is studied this is a noncyclic version of the incremental algorithm where the sequence of computing agents is modeled as a time nonhomogeneous markov chain such a model is appropriate for mobile networks as the network topology changes across time in these networks convergence results and error bounds for the markov randomized method in the presence of stochastic errors for diminishing and constant step sizes are obtained

convex optimization over fixed point sets of quasi nonexpansive and nonexpansive mappings in utility based bandwidth allocation problems with operational constraints network bandwidth allocation is a central issue in modern communication networks the main objective of the bandwidth allocation is to allocate an optimal bandwidth for maximizing a predefined utility over the capacity constraints to traffic sources when a centralized operator which manages all the bandwidth allocations in the network has a certain operational policy the bandwidth allocation reflecting the operational policy should result in the network being more stable and reliable accordingly we need to solve a network bandwidth allocation problem under both capacity constraints and operational constraints to develop a novel algorithm for solving the problem we translate the network bandwidth allocation problem into one of minimizing a convex objective function over the intersection of the fixed point sets of certain quasi nonexpansive and nonexpansive mappings and propose a fixed point optimization algorithm for solving it we numerically compare the proposed algorithm with the existing algorithm for solving a concrete bandwidth allocation problem and show its effectiveness

a branch and bound algorithm for quadratically constrained sparse filter design this paper presents an exact algorithm for sparse filter design under a quadratic constraint on filter performance the algorithm is based on branch and bound a combinatorial optimization procedure that can either guarantee an optimal solution or produce a sparse solution with a bound on its deviation from optimality to reduce the complexity of branch and bound several methods are developed for bounding the optimal filter cost bounds based on infeasibility yield incrementally accumulating improvements with minimal computation while two convex relaxations referred to as linear and diagonal relaxations are derived to provide stronger bounds the approximation properties of the two relaxations are characterized analytically as well as numerically design examples involving wireless channel equalization and minimum variance distortionless response beamforming show that the complexity of obtaining certifiably optimal solutions can often be significantly reduced by incorporating diagonal relaxations especially in more difficult instances in the case of early termination due to computational constraints diagonal relaxations strengthen the bound on the proximity of the final solution to the optimum

peak error constrained sparse fir filter design using iterative socp in this paper a novel algorithm is proposed to design sparse fir filters it is known that this design problem is highly nonconvex due to the existence of norm of a filter coefficient vector in its objective function to tackle this difficulty an iterative procedure is developed to search a potential sparsity pattern which is then used to compute the final solution by solving a convex optimization problem in each iterative step the original sparse filter design problem is successively transformed to a simpler subproblem it can be proved that under a weak condition globally optimal solutions of these subproblems can be attained by solving their dual problems in this case the overall iterative procedure converges to a locally optimal solution of the original design problem the design procedure described above can be repeated for several times to further improve the sparsity of design results the output of the previous stage can be used as the initial point of the subsequent design the performance of the proposed algorithm is evaluated by two sets of design examples and compared to other sparse fir filter design algorithms

tensorisation of vectors and their efficient convolution in recent papers the tensorisation of vectors has been discussed in principle this is the isomorphic representation of an mathbb r n vector as a tensor black box tensor approximation methods can be used to reduce the data size of the tensor representation in particular if the vector corresponds to a grid function the resulting data size can become much smaller than n e g o log n ll n in this article we discuss the convolution of two vectors which are given via a sparse tensor representation we want to obtain the result again in the tensor representation furthermore the cost of the convolution algorithm should be related to the operands data sizes while mathbb r n vectors can be considered as grid values of function we also apply the corresponding procedure to univariate functions

superfast solution of linear convolutional volterra equations using qtt approximation we address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular toeplitz matrices and the recently proposed qtt format the inverses of such matrices can be computed by the divide and conquer and modified bini s algorithms for which we present the versions with the qtt approximation we also present an efficient formula for the shift of vectors given in qtt format which is used in the divide and conquer algorithm as a result we reduce the complexity of inversion from the fast fourier level o n log n to the speed of superfast fourier transform i e o log 2 n the results of the paper are illustrated by numerical examples

robust stochastic approximation approach to stochastic programming in this paper we consider optimization problems where the objective function is given in a form of the expectation a basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy the aim of this paper is to compare two computational approaches based on monte carlo sampling techniques namely the stochastic approximation sa and the sample average approximation saa methods both approaches the sa and saa methods have a long history current opinion is that the saa method can efficiently use a specific say linear structure of the considered problem while the sa approach is a crude subgradient method which often performs poorly in practice we intend to demonstrate that a properly modified sa approach can be competitive and even significantly outperform the saa method for a certain class of convex stochastic problems we extend the analysis to the case of convex concave stochastic saddle point problems and present in our opinion highly encouraging results of numerical experiments

a stopping rule for stochastic approximation a stochastic approximation algorithm is a recursive procedure to find the solution to an unknown nonlinear equation via noisy measurements in this paper we present a stopping rule for a stochastic approximation we show that there is a high probability that the distance between the exact solution and the candidate solution is less than a specified tolerance level when the stochastic approximation stops according to our stopping rule furthermore the number of recursions required by the stopping rule is a polynomial function of the problem size

decomposition optimization algorithms for distributed radar systems distributed radar systems are capable of enhancing the detection performance by using multiple widely spaced distributed antennas with prior statistic information of targets resource allocation is of critical importance for further improving the system s achievable performance in this paper the total transmitted power is minimized at a given mean square target estimation error we derive two iterative decomposition algorithms for solving this nonconvex constrained optimization problem namely the optimality condition decomposition ocd based and the alternating direction method of multipliers admm based algorithms both the convergence performance and the computational complexity of our algorithms are analyzed theoretically which are then confirmed by our simulation results the ocd method imposes a much lower computational burden per iteration while the admm method exhibits a higher per iteration complexity but as a benefit of its significantly faster convergence speed it requires less iterations therefore the admm imposes a lower total complexity than the ocd the results also show that both of our schemes outperform the state of the art benchmark scheme for the multiple target case in terms of the total power allocated at the cost of some degradation in localization accuracy for the single target case all the three algorithms achieve similar performance our admm algorithm has similar total computational complexity per iteration and convergence speed to those of the benchmark

qtt rank one vectors with qtt rank one and full rank fourier images abstract quantics tensor train qtt a new data sparse format for one and multi dimensional vectors is based on a bit representation of mode indices followed by a separation of variables a radix 2 recursion that lays behind the famous fft algorithm can be efficiently applied to vectors in the qtt format if input and all intermediate vectors of the fft algorithm have moderate qtt ranks the resulted qtt fft algorithm outperforms the fft for large vectors and has asymptotically the same complexity as the superfast quantum fourier transform it is instructive to describe a class of such vectors explicitly we identify all vectors that have qtt ranks one on input intermediate steps and output of the fft algorithm we also give an example of qtt rank one vector that has the fourier image with full qtt ranks we show by numerical experiments that for certain rank one vectors with full rank fourier images the practical ranks remain moderate for large mode sizes

randomized algorithms for probabilistic robustness with real and complex structured uncertainty there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems unlike classical worst case methods these algorithms provide probabilistic estimates assessing for instance if a certain design specification is met with a given probability one of the advantages of this approach is that the robustness margins can be often increased by a considerable amount at the expense of a small risk in this sense randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information the applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres sampling in more general uncertainty sets is generally performed through overbounding at the expense of an exponential rejection rate in the paper randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general m spl delta configuration are studied in particular efficient polynomial time algorithms for uncertainty structures spl delta consisting of an arbitrary number of full complex blocks and uncertain parameters are developed

results on nonrecursive digital filters with nonequidistant taps a new method in nonrecursive digial filter design with nonequidistant taps is presented with characteristic examples the method consists of increasing the number of taps progressively and optimizing their positions at every addition of a new tap the efficiency of the method manifests itself in the drastic reduction of the number of taps and in the reduction dynamic range of tap weights as compared to the equally spaced case for the same filter performance

interpolated finite impulse response filters a new approach to implement computationally efficient finite impulse response fir digital filters is presented the filter structure is a cascade of two sections the first section generates a sparse set of impulse response samples and the other section generates the remaining samples by using interpolation the method can be used to implement most practical fir filters with significant savings in the number of arithmetic operations typically 1 2 to 1 8 of the number of multipliers and adders of conventional fir filters are required in the implementation the saving is achieved both in the linear phase and the non linear phase cases in addition the new implementation gives smaller coefficient sensitivities and better roundoff noise properties than conventional implementations

stochastic approximation method with gradient averaging for unconstrained problems a stochastic approximation algorithm is studied in which random gradient estimates are averaged by an auxiliary filter convergence of the algorithm is proved under a rather general noise condition a practical version of the method is described and numerical results are given

accelerated stochastic approximation a technique to accelerate convergence of stochastic approximation algorithms is studied it is based on kesten s idea of equalization of the gain coefficient for the robbins monro algorithm convergence with probability 1 is proved for the multidimensional analog of the kesten accelerated stochastic approximation algorithm asymptotic normality of the delivered estimates is also shown results of numerical simulations are presented that demonstrate the efficiency of the acceleration procedure

gradient convergence in gradient methods with errors we consider the gradient method x t 1 x t g t s t w t where s t is a descent direction of a function f rn to re and w t is a deterministic or stochastic error we assume that gr f is lipschitz continuous that the stepsize g t diminishes to 0 and that s t and w t satisfy standard conditions we show that either f x t to infty or f x t converges to a finite value and gr f x t to0 with probability 1 in the stochastic case and in doing so we remove various boundedness conditions that are assumed in existing results such as boundedness from below of f boundedness of gr f x t or boundedness of xt

linear programming algorithms for sparse filter design in designing discrete time filters the length of the impulse response is often used as an indication of computational cost in systems where the complexity is dominated by arithmetic operations the number of nonzero coefficients in the impulse response may be a more appropriate metric to consider instead and computational savings are realized by omitting arithmetic operations associated with zero valued coefficients this metric is particularly relevant to the design of sensor arrays where a set of array weights with many zero valued entries allows for the elimination of physical array elements resulting in a reduction of data acquisition and communication costs however designing a filter with the fewest number of nonzero coefficients subject to a set of frequency domain constraints is a computationally difficult optimization problem this paper describes several approximate polynomial time algorithms that use linear programming to design filters having a small number of nonzero coefficients i e filters that are sparse specifically we present two approaches that have different computational complexities in terms of the number of required linear programs the first technique iteratively thins the impulse response of a non sparse filter until frequency domain constraints are violated the second minimizes the 1 norm of the impulse response of the filter using the resulting design to determine the coefficients that are constrained to zero in a subsequent re optimization stage the algorithms are evaluated within the contexts of array design and acoustic equalization

a stochastic successive minimization method for nonsmooth nonconvex optimization with applications to transceiver design in wireless communication networks consider the problem of minimizing the expected value of a cost function parameterized by a random variable the classical sample average approximation method for solving this problem requires minimization of an ensemble average of the objective at each step which can be expensive in this paper we propose a stochastic successive upper bound minimization method ssum which minimizes an approximate ensemble average at each iteration to ensure convergence and to facilitate computation we require the approximate ensemble average to be a locally tight upper bound of the expected cost function and be easily optimized the main contributions of this work include the development and analysis of the ssum method as well as its applications in linear transceiver design for wireless communication networks and online dictionary learning moreover using the ssum framework we extend the classical stochastic sub gradient method to the case of minimizing a nonsmooth nonconvex objective function and establish its convergence

learning algorithms for separable approximations of discrete stochastic optimization problems we propose the use of sequences of separable piecewise linear approximations for solving nondifferentiable stochastic optimization problems the approximations are constructed adaptively using a combination of stochastic subgradient information and possibly sample information on the objective function itself we prove the convergence of several versions of such methods when the objective function is separable and has integer break points and we illustrate their behavior on numerical examples we then demonstrate the performance on nonseparable problems that arise in the context of two stage stochastic programming problems and demonstrate that these techniques provide near optimal solutions with a very fast rate of convergence compared with other solution techniques

the o d e method for convergence of stochastic approximation and reinforcement learning it is shown here that stability of the stochastic approximation algorithm is implied by the asymptotic stability of the origin for an associated ode this in turn implies convergence of the algorithm several specific classes of algorithms are considered as applications it is found that the results provide i a simpler derivation of known results for reinforcement learning algorithms ii a proof for the first time that a class of asynchronous stochastic approximation algorithms are convergent without using any a priori assumption of stability iii a proof for the first time that asynchronous adaptive critic and q learning algorithms are convergent for the average cost optimal control problem

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

multilevel toeplitz matrices generated by tensor structured vectors and convolution with logarithmic complexity we study the tensor structure of two operations the transformation of a given multidimensional vector into a multilevel toeplitz matrix and the convolution of two given multidimensional vectors we show that the low rank tensor structure of the input is preserved in the output and propose efficient algorithms for these operations in the newly introduced quantized tensor train qtt format consider a d dimensional 2n times cdots times 2n vector boldsymbol x if it is represented elementwise the number of parameters is 2n d however if we assume that boldsymbol x is given in a qtt representation with ranks bounded by p the number of parameters is reduced to mathcal o left dp 2 log n right under this assumption we show how the multilevel toeplitz matrix generated by boldsymbol x can be obtained in the qtt format with ranks bounded by 2p in mathcal o left dp 2 log n right operations we also describe how the convolution boldsymbol x star boldsymbol y of

randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems in this paper we study two general semi infinite programming problems by means of a randomized strategy based on statistical learning theory the sample size results obtained with this approach are generally considered to be very conservative by the control community the first main contribution of this paper is to demonstrate that this is not necessarily the case utilizing as a starting point one sided results from statistical learning theory we obtain bounds on the number of required samples that are manageable for ldquoreasonablerdquo values of probabilistic confidence and accuracy in particular we show that the number of required samples grows with the accuracy parameter epsiv as 1 epsivln 1 epsiv and this is a significant improvement when compared to the existing bounds which depend on 1 epsiv2ln 1 epsiv2 secondly we present new results for optimization and feasibility problems involving boolean expressions consisting of polynomials in this case when the accuracy parameter is sufficiently small an explicit bound that only depends on the number of decision variables and on the confidence and accuracy parameters is presented for convex optimization problems we also prove that the required sample size is inversely proportional to the accuracy for fixed confidence thirdly we propose a randomized algorithm that provides a probabilistic solution circumventing the potential conservatism of the bounds previously derived

a stochastic quasi newton method for large scale optimization the question of how to incorporate curvature information into stochastic approximation methods is challenging the direct application of classical quasi newton updating techniques for deterministic optimization leads to noisy curvature estimates that have harmful effects on the robustness of the iteration in this paper we propose a stochastic quasi newton method that is efficient robust and scalable it employs the classical bfgs update formula in its limited memory form and is based on the observation that it is beneficial to collect curvature information pointwise and at spaced intervals one way to do this is through subsampled hessian vector products this technique differs from the classical approach that would compute differences of gradients at every iteration and where controlling the quality of the curvature estimates can be difficult we present numerical results on problems arising in machine learning that suggest that the proposed method shows much promise

convergence of simultaneous perturbation stochastic approximation for nondifferentiable optimization we consider simultaneous perturbation stochastic approximation for function minimization the standard assumption for convergence is that the function be three times differentiable although weaker assumptions have been used for special cases however all work that we are aware of at least requires differentiability we relax the differentiability requirement and prove convergence using convex analysis

a parallel decomposition method for nonconvex stochastic multi agent optimization problems this paper considers the problem of minimizing the expected value of a possibly nonconvex cost function parameterized by a random vector variable when the expectation cannot be computed accurately e g because the statistics of the random variables are unknown and or the computational complexity is prohibitive classical stochastic gradient methods for solving this problem may suffer from slow convergence in this paper we propose a stochastic parallel successive convex approximation based best response algorithm for general nonconvex stochastic sum utility optimization problems which arise naturally in the design of multi agent networks the proposed novel decomposition approach enables all users to update their optimization variables in parallel by solving a sequence of strongly convex subproblems one for each user almost sure convergence to stationary points is proved we then customize the algorithmic framework to solve the stochastic sum rate maximization problem over single input single output siso frequency selective interference channels multiple input multiple output mimo interference channels and mimo multiple access channels numerical results corroborate that the proposed algorithms can converge faster than state of the art stochastic gradient schemes while achieving the same or better sum rates

a distributed algorithm for convex constrained optimization under noise we present a novel distributed algorithm for convex constrained optimization problems that are subject to noise corruption and uncertainties the proposed scheme can be classified as a distributed stochastic approximation method where a unique feature here is that we allow for multiple noise terms to appear in both the computation and communication stages of the distributed iterative process specifically we consider problems that involve multiple agents optimizing a separable convex objective function subject to convex local constraints and linear coupling constraints this is a richer class of problems compared to those that can be handled by existing distributed stochastic approximation methods which consider only consensus constraints and fewer sources of noise the proposed algorithm utilizes the augmented lagrangian al framework which has been widely used recently to solve deterministic optimization problems in a distributed way we show that the proposed method generates sequences of primal and dual variables that converge to their respective optimal sets almost surely

adaptive stochastic approximation by the simultaneous perturbation method stochastic approximation sa has long been applied for problems of minimizing loss functions or root finding with noisy input information as with all stochastic search algorithms there are adjustable algorithm coefficients that must be specified and that can have a profound effect on algorithm performance it is known that choosing these coefficients according to an sa analog of the deterministic newton raphson algorithm provides an optimal or near optimal form of the algorithm however directly determining the required hessian matrix or jacobian matrix for root finding to achieve this algorithm form has often been difficult or impossible in practice the paper presents a general adaptive sa algorithm that is based on a simple method for estimating the hessian matrix while concurrently estimating the primary parameters of interest the approach applies in both the gradient free optimization kiefer wolfowitz and root finding stochastic gradient based robbins monro settings and is based on the simultaneous perturbation sp idea introduced previously the algorithm requires only a small number of loss function or gradient measurements per iteration independent of the problem dimension to adaptively estimate the hessian and parameters of primary interest aside from introducing the adaptive sp approach the paper presents practical implementation guidance asymptotic theory and a nontrivial numerical evaluation also included is a discussion and numerical analysis comparing the adaptive sp approach with the iterate averaging approach to accelerated sa

on stochastic gradient and subgradient methods with adaptive steplength sequences traditionally stochastic approximation sa schemes have been popular choices for solving stochastic optimization problems however the performance of standard sa implementations can vary significantly based on the choice of the steplength sequence and in general little guidance is provided about good choices motivated by this gap we present two adaptive steplength schemes for strongly convex differentiable stochastic optimization problems equipped with convergence theory that aim to overcome some of the reliance on user specific parameters the first scheme referred to as a recursive steplength stochastic approximation rsa scheme optimizes the error bounds to derive a rule that expresses the steplength at a given iteration as a simple function of the steplength at the previous iteration and certain problem parameters the second scheme termed as a cascading steplength stochastic approximation csa scheme maintains the steplength sequence as a piecewise constant decreasing function with the reduction in the steplength occurring when a suitable error threshold is met then we allow for nondifferentiable objectives but with bounded subgradients over a certain domain in such a regime we propose a local smoothing technique based on random local perturbations of the objective function that leads to a differentiable approximation of the function assuming a uniform distribution on the local randomness we establish a lipschitzian property for the gradient of the approximation and prove that the obtained lipschitz bound grows at a modest rate with problem size this facilitates the development of an adaptive steplength stochastic approximation framework which now requires sampling in the product space of the original measure and the artificially introduced distribution

smooth sqp methods for mathematical programs with nonlinear complementarity constraints mathematical programs with nonlinear complementarity constraints are reformulated using better posed but nonsmooth constraints we introduce a class of functions parameterized by a real scalar to approximate these nonsmooth problems by smooth nonlinear programs this smoothing procedure has the extra benefits that it often improves the prospect of feasibility and stability of the constraints of the associated nonlinear programs and their quadratic approximations we present two globally convergent algorithms based on sequential quadratic programming sqp as applied in exact penalty methods for nonlinear programs global convergence of the implicit smooth sqp method depends on existence of a lower level nondegenerate strictly complementary limit point of the iteration sequence global convergence of the explicit smooth sqp method depends on a weaker property i e existence of a limit point at which a generalized constraint qualification holds we also discuss some practical matters relating to computer implementations

multivariate stochastic approximation using a simultaneous perturbation gradient approximation the problem of finding a root of the multivariate gradient equation that arises in function minimization is considered when only noisy measurements of the function are available a stochastic approximation sa algorithm for the general kiefer wolfowitz type is appropriate for estimating the root the paper presents an sa algorithm that is based on a simultaneous perturbation gradient approximation instead of the standard finite difference approximation of keifer wolfowitz type procedures theory and numerical experience indicate that the algorithm can be significantly more efficient than the standard algorithms in large dimensional problems

design of nonuniformly spaced linear phase fir filters using mixed integer linear programming an optimization problem for designing a nonuniformly spaced linear phase fir filter with minimal complexity is formulated and solved by mixed integer linear programming milp examples illustrate that the proposed method is useful for designing a wide range of filter types and can outperform subset selection based design methods

decentralized resource allocation in dynamic networks of agents we consider the problem of n agents that share m common resources the objective is to derive an optimal allocation that maximizes a global objective expressed as a separable concave objective function we propose a decentralized asynchronous gradient descent method that is suitable for implementation in the case where the communication between agents is described in terms of a dynamic network this communication model accommodates situations such as mobile agents and communication failures the method is shown to converge provided that the objective function has lipschitz continuous gradients we further consider a randomized version of the same algorithm for the case where the objective function is nondifferentiable but has bounded subgradients we show that both algorithms converge to near optimal solutions and derive convergence rates in terms of the magnitude of the gradient of the objective function we show how to accommodate nonnegativity constraints on the resources using the results derived ex

acceleration of stochastic approximation by averaging a new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated convergence with probability one is proved for a variety of classical optimization and identification problems it is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence

feedback and weighting mechanisms for improving jacobian estimates in the adaptive simultaneous perturbation algorithm it is known that a stochastic approximation sa analogue of the deterministic newton raphson algorithm provides an asymptotically optimal or near optimal form of stochastic search however directly determining the required jacobian matrix or hessian matrix for optimization has often been difficult or impossible in practice this paper presents a general adaptive sa algorithm that is based on a simple method for estimating the jacobian matrix while concurrently estimating the primary parameters of interest relative to prior methods for adaptively estimating the jacobian matrix the paper introduces two enhancements that generally improve the quality of the estimates for underlying jacobian hessian matrices thereby improving the quality of the estimates for the primary parameters of interest the first enhancement rests on a feedback process that uses previous jacobian estimates to reduce the error in the current estimate the second enhancement is based on an optimal weighting of per iteration jacobian estimates from the use of simultaneous perturbations the algorithm requires only a small number of loss function or gradient measurements per iteration independent of the problem dimension to adaptively estimate the jacobian matrix and parameters of primary interest

randomized smoothing for stochastic optimization we analyze convergence rates of stochastic optimization algorithms for nonsmooth convex optimization problems by combining randomized smoothing techniques with accelerated gradient methods we obtain convergence rates of stochastic optimization procedures both in expectation and with high probability that have optimal dependence on the variance of the gradient estimates to the best of our knowledge these are the first variance based rates for nonsmooth optimization we give several applications of our results to statistical estimation problems and provide experimental results that demonstrate the effectiveness of the proposed algorithms we also describe how a combination of our algorithm with recent work on decentralized optimization yields a distributed stochastic optimization algorithm that is order optimal

fir filter design over a discrete powers of two coefficient space fir digital filters with discrete coefficient values selected from the powers of two coefficient space are designed using the methods of integer programming the frequency responses obtained are shown to be superior to those obtained by simply rounding the coefficients both the weighted minimax and the weighted least square error criteria are considered using a weighted least square error criterion it is shown that it is possible to predict the improvement that can be expected when integer quadratic programming is used instead of simple coefficient rounding

on the use of cyclotomic polynomial prefilters for efficient fir filter design the authors present an efficient fir finite impulse response filter design algorithm that generalizes existing cascaded fir prefilter equalizer methods they propose using cyclotomic polynomial building blocks to form a multiplierless prefilter with impressive stopband performance and they provide a straightforward strategy for choosing the polynomials to match filter specification two options for design of the equalizer are provided a uniformly spaced equalizer can be optimally l sub infinity designed via a modified parks mcclellan algorithm a new algorithm based on complex basis function subset selection methods is also proposed for optimal design of a more efficient nonuniformly spaced equalizer the techniques which can be applied to a broad class of filter design problems typically provide a 35 85 reduction in the number of additions and multiplications required with a cost of 10 45 additional delays the methods also provide reduced coefficient quantization sensitivity and reduced roundoff noise

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

the rank of a random matrix this work is concerned with the numerical rank of matrix in the matrix computations we conclude that a real random matrix has full rank with probability 1 and a rational random matrix has full rank with probability 1 too finally the applications of the numerical matrix are given

multivariate adaptive stochastic approximation on considere une version multivariable de l approximation stochastique adaptative developpee recemment par lai et robbins 1978 1979

probabilistic robust design with linear quadratic regulators in this paper we study robust design of uncertain systems in a probabilistic setting by means of linear quadratic regulators lqr we consider systems affected by random bounded nonlinear uncertainty so that classical optimization methods based on linear matrix inequalities cannot be used without conservatism the approach followed here is a blend of randomization techniques for the uncertainty together with convex optimization for the controller parameters in particular we propose an iterative algorithm for designing a controller which is based upon subgradient iterations at each step of the sequence we first generate a random sample and then we perform a subgradient step for a convex constraint defined by the lqr problem the main result of the paper is to prove that this iterative algorithm provides a controller which quadratically stabilizes the uncertain system with probability one in a finite number of steps in addition at a fixed step we compute a lower bound of the probability that a quadratically stabilizing controller is found