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

non isochronicity of the center at the origin in polynomial hamiltonian systems with even degree nonlinearities abstract in 2002 x jarque and j villadelprat proved that no center in a planar polynomial hamiltonian system of degree 4 is isochronous and raised a question is there a planar polynomial hamiltonian system of even degree which has an isochronous center in this paper we give a criterion for non isochronicity of the center at the origin of planar polynomial hamiltonian systems moreover the orders of weak centers are determined our results answer a weak version of the question proving that there is no planar polynomial hamiltonian system with only even degree nonlinearities having an isochronous center at the origin

a tool for the analysis of quasi newton methods with application to unconstrained minimization the bfgs update formula is shown to have an important property that is inde pendent of the algorithmic context of the update and that is relevant to both constrained and unconstrained optimization the bfgs method for unconstrained optimization using a variety of line searches including backtracking is shown to be globally and superlinearly convergent on uniformly convex problems the analysis is particularly simple due to the use of some new tools introduced in this paper

distributed nonconvex constrained optimization over time varying digraphs this paper considers nonconvex distributed constrained optimization over networks modeled as directed possibly time varying graphs we introduce the first algorithmic framework for the minimization of the sum of a smooth nonconvex nonseparable function the agent s sum utility plus a difference of convex function with nonsmooth convex part this general formulation arises in many applications from statistical machine learning to engineering the proposed distributed method combines successive convex approximation techniques with a judiciously designed perturbed push sum consensus mechanism that aims to track locally the gradient of the smooth part of the sum utility sublinear convergence rate is proved when a fixed step size possibly different among the agents is employed whereas asymptotic convergence to stationary solutions is proved using a diminishing step size numerical results show that our algorithms compare favorably with current schemes on both convex and nonconvex problems

an optimal method for stochastic composite optimization this paper considers an important class of convex programming cp problems namely the stochastic composite optimization sco whose objective function is given by the summation of general nonsmooth and smooth stochastic components since sco covers non smooth smooth and stochastic cp as certain special cases a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming note however that the optimization algorithms that can achieve this lower bound had never been developed in this paper we show that the simple mirror descent stochastic approximation method exhibits the best known rate of convergence for solving these problems our major contribution is to introduce the accelerated stochastic approximation ac sa algorithm based on nesterov s optimal method for smooth cp nesterov in doklady an sssr 269 543 547 1983 nesterov in math program 103 127 152 2005 and show that the ac sa algorithm can achieve the aforementioned lower bound on the rate of convergence for sco to the best of our knowledge it is also the first universally optimal algorithm in the literature for solving non smooth smooth and stochastic cp problems we illustrate the significant advantages of the ac sa algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems

convergence of line search methods for unconstrained optimization line search methods are traditional and successful methods for solving unconstrained optimization problems its convergence has attracted more attention in recent years in this paper we analyze the general results on convergence of line search methods with seven line search rules it is clarified that the search direction plays a main role in these methods and that step size guarantees the global convergence in some cases it is also proved that many line search methods have same convergence property these convergence results can enable us to design powerful effective and stable algorithms in practice finally a class of special line search methods is investigated

steady state performance of spline adaptive filters recently a novel class of nonlinear adaptive filters called spline adaptive filters safs has been introduced and demonstrated to be very effective in many practical applications the learning rules of these architectures are based on the least mean square lms algorithm in order to provide theoretical foundation to the saf in this paper we provide a steady state performance evaluation in particular after the stochastic analysis of the mean behavior of the saf approach under the gaussian assumption the analytical derivation of the theoretical excess mean square error emse and the normalized misadjustment are derived and discussed the proposed analysis of emse and misadjustment is based on the energy conservation approach that has been extended to saf architecture the derived theoretical analysis allows to accurately predict the steady state performance therefore some properties for the correct choice of filter parameters are also provided experimental results demonstrate the effectiveness of the analysis results

convergence of approximate and incremental subgradient methods for convex optimization we present a unified convergence framework for approximate subgradient methods that covers various stepsize rules including both diminishing and nonvanishing stepsizes convergence in objective values and convergence to a neighborhood of the optimal set we discuss ways of ensuring the boundedness of the iterates and give efficiency estimates our results are extended to incremental subgradient methods for minimizing a sum of convex functions which have recently been shown to be promising for various large scale problems including those arising from lagrangian relaxation

brief paper an adaptive optimization scheme with satisfactory transient performance adaptive optimization ao schemes based on stochastic approximation principles such as the random directions kiefer wolfowitz rdkw the simultaneous perturbation stochastic approximation spsa and the adaptive fine tuning aft algorithms possess the serious disadvantage of not guaranteeing satisfactory transient behavior due to their requirement for using random or random like perturbations of the parameter vector the use of random or random like perturbations may lead to particularly large values of the objective function which may result to severe poor performance or stability problems when these methods are applied to closed loop controller optimization applications in this paper we introduce and analyze a new algorithm for alleviating this problem mathematical analysis establishes satisfactory transient performance and convergence of the proposed scheme under a general set of assumptions application of the proposed scheme to the adaptive optimization of a large scale complex control system demonstrates the efficiency of the proposed scheme

a new class of incremental gradient methods for least squares problems the least mean squares lms method for linear least squares problems differs from the steepest descent method in that it processes data blocks one by one with intermediate adjustment of the parameter vector under optimization this mode of operation often leads to faster convergence when far from the eventual limit and to slower sublinear convergence when close to the optimal solution we embed both lms and steepest descent as well as other intermediate methods within a one parameter class of algorithms and we propose a hybrid class of methods that combine the faster early convergence rate of lms with the faster ultimate linear convergence rate of steepest descent these methods are well suited for neural network training problems with large data sets furthermore these methods allow the effective use of scaling based for example on diagonal or other approximations of the hessian matrix

a policy gradient method for semi markov decision processes with application to call admission control solving a semi markov decision process smdp using value or policy iteration requires precise knowledge of the probabilistic model and suffers from the curse of dimensionality to overcome these limitations we present a reinforcement learning approach where one optimizes the smdp performance criterion with respect to a family of parameterised policies we propose an online algorithm that simultaneously estimates the gradient of the performance criterion and optimises it using stochastic approximation we apply our algorithm to call admission control our proposed policy gradient smdp algorithm and its application to admission control is novel

consistent normalized least mean square filtering with noisy data matrix when the ordinary least squares method is applied to the parameter estimation problem with noisy data matrix it is well known that the estimates turn out to be biased while this bias term can be somewhat reduced by the use of models of higher order or by requiring a high signal to noise ratio snr it can never be completely removed consistent estimates can be obtained by means of the instrumental variable method ivm or the total data least squares method tls dls in the adaptive setting for the such problem a variety of least mean squares lms type algorithms have been researched rather than their recursive versions of ivm or tls dls that cost considerable computations motivated by these observations we propose a consistent lms type algorithm for the data least square estimation problem this novel approach is based on the geometry of the mean squared error mse function rendering the step size normalization and the heuristic filtered estimation of the noise variance respectively for fast convergence and robustness to stochastic noise monte carlo simulations of a zero forcing adaptive finite impulse response fir channel equalizer demonstrate the efficacy of our algorithm

general results on the convergence of stochastic algorithms a deterministic approach is proposed for proving the convergence of stochastic algorithms of the most general form under necessary conditions on the input noise and reasonable conditions on the nonnecessarily continuous mean field emphasis is placed on the case where more than one stationary point exists we also use this approach to prove the convergence of a stochastic algorithm with markovian dynamics

analysis of an lms algorithm for unbiased impulse response estimation in this correspondence a least mean squares lms based algorithm is devised for unbiased system identification in the presence of white input and output noise assuming that the ratio of the noise powers is known the proposed approach aims to minimize the mean square value of the equation error function under a constant norm constraint and is equivalent to minimizing a modified mean square error mse function an analysis of the algorithm shows that the estimates will converge to the true values in the mean sense the variances of the parameter estimates are also available computer simulations are included to corroborate the theoretical development and to evaluate the impulse response estimation performance of the lms algorithm under different conditions

decentralized delay optimal control for interference networks with limited renewable energy storage in this paper we consider delay minimization for interference networks with renewable energy source where the transmission power of a node comes from both the conventional utility power ac power and the renewable energy source we assume the transmission power of each node is a function of the local channel state local data queue state and local energy queue state only we consider two delay optimization formulations namely the decentralized partially observable markov decision process dec pomdp and noncooperative partially observable stochastic game posg in dec pomdp formulation we derive a decentralized online learning algorithm to determine the control actions and lagrangian multipliers lms simultaneously based on the policy gradient approach under some mild technical conditions the proposed decentralized policy gradient algorithm converges almost surely to a local optimal solution in the noncooperative posg formulation the transmitter nodes are noncooperative we extend the decentralized policy gradient solution and establish the technical proof for almost sure convergence of the learning algorithms in both cases the solutions are very robust to model variations

a time domain feedback analysis of filtered error adaptive gradient algorithms this paper provides a time domain feedback analysis of gradient based adaptive schemes a key emphasis is on the robustness performance of the adaptive filters in the presence of disturbances and modeling uncertainties along the lines of h sup spl infin theory and robust filtering the analysis is carried out in a purely deterministic framework and assumes no prior statistical information or independence conditions it is shown that an intrinsic feedback structure can be associated with the varied adaptive schemes the feedback structure is motivated via energy arguments and is shown to consist of two major blocks a time variant lossless i e energy preserving feedforward path and a time variant feedback path the configuration is further shown to lend itself to analysis via a so called small gain theorem thus leading to stability and robustness conditions that require the contractivity of certain operators choices for the step size parameter in order to guarantee faster rates of convergence are also derived and simulation results are included to demonstrate the theoretical findings in addition the time domain analysis provided in this paper is shown to extend the so called transfer function approach to a general time variant scenario without any approximations

an efficient recursive total least squares algorithm for fir adaptive filtering an algorithm for recursively computing the total least squares tls solution to the adaptive filtering problem is described this algorithm requires o n multiplications per iteration to effectively track the n dimensional eigenvector associated with the minimum eigenvalue of an augmented sample covariance matrix it is shown that the recursive least squares rls algorithm generates biased adaptive filter coefficients when the filter input vector contains additive noise the tls solution on the other hand is seen to produce unbiased solutions examples of standard adaptive filtering applications that result in noise being added to the adaptive filter input vector are cited computer simulations comparing the relative performance of rls and recursive tls are described

minima of quasisuperminimizers abstract we show that unlike minima of superharmonic functions which are again superharmonic the same property fails for q quasisuperminimizers more precisely if u i is a q i quasisuperminimizer i 1 2 where 1 q 1 q 2 then u min u 1 u 2 is a q quasisuperminimizer but there is an increase in the optimal quasisuperminimizing constant q we provide the first examples of this phenomenon i e that q q 2 in addition to lower bounds for the optimal quasisuperminimizing constant of u we also improve upon the earlier upper bounds due to kinnunen and martio moreover our lower and upper bounds turn out to be quite close we also study a similar phenomenon in pasting lemmas for quasisuperminimizers where q q 1 q 2 turns out to be optimal and provide results on exact quasiminimizing constants of piecewise linear functions on the real line which can serve as approximations of more general quasiminimizers

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

sharp exponents and a wiener type condition for boundary regularity of quasiminimizers abstract we obtain a sufficient condition for boundary regularity of quasiminimizers of the p energy integral in terms of a wiener type sum the exponent in the sum is independent of the dimension and is explicitly expressed in terms of p and the quasiminimizing constant we also show by an example that the exponent is sharp in a certain sense

on the nonergodic convergence rate of an inexact augmented lagrangian framework for composite convex programming in this paper we consider the linearly constrained composite convex optimization problem whose objective is a sum of a smooth function and a possibly nonsmooth function we propose an inexact aug

power control with imperfect exchanges and applications to spectrum sharing in various applications the effect of errors in gradient based iterations is of particular importance when seeking saddle points of the lagrangian function associated with constrained convex optimization problems of particular interest here are problems arising in power control applications where network utility is maximized subject to minimum signal to interference plus noise ratio sinr constraints maximum interference constraints maximum received power constraints or simultaneous minimum and maximum sinr constraints especially when the gradient iterations are executed in a distributed fashion imperfect exchanges among the link nodes may result in erroneous gradient vectors in order to assess and cope with such errors two running averages ergodic sequences are formed from the iterates generated by the perturbed saddle point method each with complementary strengths under the assumptions of problem convexity and error boundedness bounds on the constraint violation and the suboptimality per iteration index are derived the two types of running averages are tested on a spectrum sharing problem with minimum and maximum sinr constraints as well as maximum interference constraints

the monotonicity of period function for codimension four quadratic system q4 abstract in this paper we study the codimension four quadratic system q 4 z iz 4 z 2 2 z 2 b ic z 2 where b and c are real constants i 2 1 z x iy b ic 2 it is proved that the period function of periodic trajectories for q 4 is monotone the content of this paper can be viewed as a contribution to the proof of chicone s conjecture cf mr 94h 58072

bifurcation of limit cycles from quadratic isochrones abstract for a one parameter family of plane quadratic vector fields x e depending analytically on a small real parameter e we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous or linearizable center techniques are developed for treating the bifurcations of all orders and these are applied to prove the following results for the linear isochrone the maximum number of continuous families of limit cycles which can emerge is three for one class of nonlinear isochrones at most one continuous family of limit cycles can emerge whereas for all other nonlinear isochrones at most two continuous families of limit cycles can emerge moreover for each isochrone in one of these classes there are small perturbations such that the indicated maximum number of continuous families of limit cycles can be made to emerge from a corresponding number of arbitrarily prescribed periodic orbits within the period annulus of the isochronous center

convergence property of gradient type methods with non monotone line search in the presence of perturbations in this paper two new kinds of methods which are called gradient type method and hybrid projection method with perturbations are proposed and non monotone line search technique is employed at the same time global convergence of these methods is proved only in the case where the gradient function is uniformly continuous on an open convex set containing the iteration sequence numerical examples are given at the end of this paper

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

the dirichlet problem for p harmonic functions on metric spaces we study the dirichlet problem for p harmonic functions and p energy minimizers in bounded domains in proper pathconnected metric measure spaces equipped with a doubling measure and supporting a poincare inequality the dirichlet problem has previously been solved for sobolev type boundary data and we extend this result and solve the problem for all continuous boundary data we study the regularity of boundary points and prove the kellogg property i e that the set of irregular boundary points has zero p capacity we also construct p capacitary p singular and p harmonic measures on the boundary we show that they are all absolutely continuous with respect to the p capacity for p 2 we show that all the boundary measures are comparable and that the singular and harmonic measures coincide we give an integral representation for the solution to the dirichlet problem when p 2 enabling us to extend the solvability of the problem to l 1 boundary data in this case moreover we give a trace result for newtonian functions when p 2 finally we give an estimate for the hausdorff dimension of the boundary of a bounded domain in ahlfors q regular spaces less

on the convergence of the lms algorithm with adaptive learning rate for linear feedforward networks we consider the problem of training a linear feedforward neural network by using a gradient descent like lms learning algorithm the objective is to find a weight matrix for the network by repeatedly presenting to it a finite set of examples so that the sum of the squares of the errors is minimized kohonen showed that with a small but fixed learning rate or stepsize some subsequences of the weight matrices generated by the algorithm will converge to certain matrices close to the optimal weight matrix in this paper we show that by dynamically decreasing the learning rate during each training cycle the sequence of matrices generated by the algorithm will converge to the optimal weight matrix we also show that for any given 0 the lms algorithm with decreasing learning rates will generate an optimal weight matrix i e a matrix of distance at most away from the optimal matrix after o 1 training cycles this is in contrast to 1 log 1 training cycles needed to generate an optima

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

a sufficient condition in order that the real jacobian conjecture in r 2 holds let f f g r2 r2 be a polynomial map such that det df x y is different from zero for all x y r2 and f 0 0 0 0 we prove that for the injectivity of f it is sufficient to assume that the higher homogeneous terms of the polynomials ffx ggx and ffy ggy do not have real linear factors in common the proofs are based on qualitative theory of dynamical systems

fast approximate inverse power iteration algorithm for adaptive total least squares fir filtering the presence of contaminating noises at both the input and the output of an finite impulse response fir system constitutes a major impediment to unbiased parameter estimation the total least squares tls method is known to be effective in achieving unbiased estimation in this correspondence we develop a fast recursive algorithm with a view to finding the tls solution for adaptive fir filtering given the fact that the tls solution is obtainable via inverse power iteration we introduce a novel but approximate inverse power iteration in combination with galerkin method so that the tls solution can be updated adaptively at a lower computational cost we also take advantage of the regular form of the tls solution to constrain the last element of the filter parameter vector to the negative one we further reduce the computational complexity of the developed algorithm by making efficient computation of the fast gain vector defined in and using rank one update of the augmented autocorrelation matrix the developed algorithm saves seven m mad s number of multiplies divides and square roots when compared with the recursive tls algorithm in moreover unlike the algorithms given in and the developed algorithm does not deal with the solution to a one variable quadratic equation and it avoids square root operation therefore it has the simpler structure and may be more easily implemented we then make a careful investigation into global convergence of the developed algorithm simulation results are provided that clearly illustrate appealing performance of the developed algorithm including its good long term numerical stability

a stochastic gradient type algorithm for closed loop problems we focus on the numerical solution of closed loop stochastic problems and propose a perturbed gradient algorithm to achieve this goal the main hurdle in such problems is the fact that the control variables are infinite dimensional due to e g the information constraints alternatively said control variables are feedbacks i e functions such controls have hence to be represented in a finite way in order to solve the problem numerically in the same way the gradient of the criterion is itself an infinite dimensional object our algorithm replaces this exact and unknown gradient by a perturbed one which consists of the product of the true gradient evaluated at a random point and a kernel function which extends this gradient to the neighbourhood of the random point proceeding this way we explore the whole space iteration after iteration through random points since each kernel function is perfectly known by a small number of parameters say n the control at iteration k is perfectly known as an infinite dimensional object by at most n k parameters the main strength of this method is that it avoids any discretization of the underlying space provided that we can sample as many points as needed in this space moreover our algorithm can take into account the possible measurability constraints of the problem in a new way finally the randomized strategy implemented by the algorithm causes the most probable parts of the space to be the most explored ones which is a priori an interesting feature in this paper we first prove two convergence results of this algorithm in the strongly convex and convex cases and then give some numerical examples showing the interest of this method for practical stochastic optimization problems

an online gradient based parameter identification algorithm for the neuro fuzzy systems abstract online gradient descent method has been widely applied for parameter learning in neuro fuzzy systems the success of the application relies on the convergence of the learning procedure however there barely have been convergence analyses on the online learning procedure for neuro fuzzy systems in this paper an online gradient learning algorithm with adaptive learning rate is proposed to identify the parameters of the neuro fuzzy systems representing the mamdani fuzzy model with gaussian fuzzy sets we take the reciprocals of the variances of the gaussian membership functions rather than the variances themselves as independent variables when computing the gradient with respect to the variance parameters subsequently oscillation of the gradient value in the learning process can be avoided furthermore some convergence results for this online learning scheme are studied finally three numerical examples are provided to illustrate the performance of the proposed algorithm

isochronicity for several classes of hamiltonian systems abstract in this paper we study isochronous centers of analytic hamiltonian systems giving special attention to the polynomial case we first revisit the potential systems and we show the connection between isochronicity and involutions we then study a more general system namely the ones associated to hamiltonians of the form h x y a x b x y c x y 2 as an application we classify the cubic polynomial hamiltonian isochronous centers and we give examples of nontrivial and nonglobal polynomial hamiltonian isochronous centers

darboux linearization and isochronous centers with a rational first integral abstract in this paper we study isochronous centers of polynomial systems it is known that a center is isochronous if and only if it is linearizable we introduce the notion of darboux linearizability of a center and give an effective criterion for verifying darboux linearizability if a center is darboux linearizable the method produces a linearizing change of coordinates most of the known polynomial isochronous centers are darboux linearizable moreover using this criterion we find a new two parameter family of cubic isochronous centers and give the linearizing changes of coordinates for centers belonging to that family we also determine all hamiltonian cubic systems which are darboux linearizable in the second part of this work we restrict to the study of isochronous centers having a rational first integral we prove that under certain conditions the cycle vanishing at the isochronous center is either zero homologous in the closure of a generic fiber or the function obtained from the first integral by eliminating the indeterminacy points has several critical points in the singular fiber passing through the isochronous center

steered sequential projections for the inconsistent convex feasibility problem we study a steered sequential gradient algorithm which minimizes the sum of convex functions by proceeding cyclically in the directions of the negative gradients of the functions and using steered step sizes this algorithm is applied to the convex feasibility problem by minimizing a proximity function which measures the sum of the bregman distances to the members of the family of convex sets the resulting algorithm is a new steered sequential bregman projection method

asynchronous stochastic approximations the asymptotic behavior of a distributed asynchronous stochastic approximation scheme is analyzed in terms of a limiting nonautonomous differential equation the relation between the latter and the relative values of suitably rescaled relative frequencies of updates of different components is underscored

a characterization of centres via first integrals on considere l existence d une integrale premiere lisse et l existence d une mesure invariante equivalente a la mesure de lebesgue pour des systemes differentiels dans le plan

p harmonic functions with boundary data having jump discontinuities and baernstein s problem for p harmonic functions on unweighted r 2 with 1 andlt p andlt infinity we show that if the boundary values f has a jump at an asymptotic corner point zo then the perron solution pf is asymptotically a b arg z z 0 o vertical bar z z 0 vertical bar we use this to obtain a positive answer to baernsteins problem on the equality of the p harmonic measure of a union g of open arcs on the boundary of the unit disc and the p harmonic measure of g over bar we also obtain various invariance results for functions with jumps and perturbations on small sets for p andgt 2 these results are new also for continuous functions finally we look at generalizations to r n and metric spaces

a convergent incremental gradient method with a constant step size an incremental aggregated gradient method for minimizing a sum of continuously differentiable functions is presented the method requires a single gradient evaluation per iteration and uses a constant step size for the case that the gradient is bounded and lipschitz continuous we show that the method visits infinitely often regions in which the gradient is small under certain unimodality assumptions global convergence is established in the quadratic case a global linear rate of convergence is shown the method is applied to distributed optimization problems arising in wireless sensor networks and numerical experiments compare the new method with other incremental gradient methods

distributed optimization in an energy constrained network analog versus digital communication schemes we consider a distributed optimization problem whereby a network of n nodes sl l 1 n wishes to minimize a common strongly convex function f x x x1 xn t under the constraint that node sl controls variable xl only the nodes locally update their respective variables and periodically exchange their values with their neighbors over a set of predefined communication channels previous studies of this problem have focused mainly on the convergence issue and the analysis of convergence rate in this study we consider noisy communication channels and study the impact of communication energy on convergence in particular we study the minimum amount of communication energy required for nodes to obtain an e minimizer of f x in the mean square sense for linear analog communication schemes we prove that the communication energy to obtain an e minimizer of f x must grow at least at the rate of 1 e and this bound is tight when f is convex quadratic furthermore we show that the same energy requirement can be reduced to o log21 e if a suitable digital communication scheme is used

the period function for quadratic integrable systems with cubic orbits this paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves it is proved that each such system has a period function with at most one critical point

sphericalization and p harmonic functions on unbounded domains in ahlfors regular spaces abstract we use sphericalization to study the dirichlet problem perron solutions and boundary regularity for p harmonic functions on unbounded sets in ahlfors regular metric spaces boundary regularity for the point at infinity is given special attention in particular we allow for several approach directions towards infinity and take into account the massiveness of their complements in 2005 llorente manfredi wu showed that the p harmonic measure on the upper half space r n n 2 is not subadditive on null sets when p 2 using their result and spherical inversion we create similar bounded examples in the unit ball b r n showing that the n harmonic measure is not subadditive on null sets when n 3 and neither are the p harmonic measures in b generated by certain weights depending on p 2 and n 2

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

analysis of the gradient descent total least squares adaptive filtering algorithm the gradient descent total least squares gd tls algorithm is a stochastic gradient adaptive filtering algorithm that compensates for error in both input and output data we study the local convergence of the gd tls algoritlun and find bounds for its step size that ensure its stability we also analyze the steady state performance of the gd tls algorithm and calculate its steady state mean square deviation our steady state analysis is inspired by the energy conservation based approach to the performance analysis of adaptive filters the results predicted by the analysis show good agreement with the simulation experiments

when does online bp training converge the backpropogation bp neural networks have been widely applied in scientific research and engineering the success of the application however relies upon the convergence of the training procedure involved in the neural network learning we settle down the convergence analysis issue through proving two fundamental theorems on the convergence of the online bp training procedure one theorem claims that under mild conditions the gradient sequence of the error function will converge to zero the weak convergence and another theorem concludes the convergence of the weight sequence defined by the procedure to a fixed value at which the error function attains its minimum the strong convergence the weak convergence theorem sharpens and generalizes the existing convergence analysis conducted before while the strong convergence theorem provides new analysis results on convergence of the online bp training procedure the results obtained reveal that with any analytic sigmoid activation function the online bp training procedure is always convergent which then underlies successful application of the bp neural networks

the perron method for p harmonic functions in metric spaces we use the perron method to construct and study solutions of the dirichlet problem for p harmonic functions in proper metric measure spaces endowed with a doubling borel measure supporting a weak 1 q poincare inequality for some 1 q p the upper and lower perron solutions are constructed for functions defined on the boundary of a bounded domain and it is shown that these solutions are p harmonic in the domain it is also shown that newtonian sobolev functions and continuous functions are resolutive i e that their upper and lower perron solutions coincide and that their perron solutions are invariant under perturbations of the function on a set of capacity zero we further study the problem of resolutivity and invariance under perturbations for semicontinuous functions we also characterize removable sets for bounded p super harmonic functions

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

dgm a deep learning algorithm for solving partial differential equations abstract high dimensional pdes have been a longstanding computational challenge we propose to solve high dimensional pdes by approximating the solution with a deep neural network which is trained to satisfy the differential operator initial condition and boundary conditions our algorithm is meshfree which is key since meshes become infeasible in higher dimensions instead of forming a mesh the neural network is trained on batches of randomly sampled time and space points the algorithm is tested on a class of high dimensional free boundary pdes which we are able to accurately solve in up to 200 dimensions the algorithm is also tested on a high dimensional hamilton jacobi bellman pde and burgers equation the deep learning algorithm approximates the general solution to the burgers equation for a continuum of different boundary conditions and physical conditions which can be viewed as a high dimensional space we call the algorithm a deep galerkin method dgm since it is similar in spirit to galerkin methods with the solution approximated by a neural network instead of a linear combination of basis functions in addition we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic pdes

a regularity classification of boundary points for p harmonic functions and quasiminimizers a regularity classification of boundary points for p harmonic funktions and quasiminimizers

linearization of isochronous centers abstract in this paper we study isochronous centers of polynomial systems we first discuss isochronous centers of quadratic systems cubic symmetric systems and reduced kukles system all these systems have rational first integrals we give a unified proof of the isochronicity of these systems by constructing algebraic linearizing changes of coordinates we then study two other classes of systems with isochronous centers namely the class of complex systems z ip z and the class of cubic systems symmetric with respect to a line and satisfying 1 both classes consist of darboux integrable systems we discuss their geometric properties and construct the linearizing changes of coordinates we show that the class of polynomial isochronous systems carries a very rich geometry finally we discuss the geometry of the linearizing changes of coordinates in the complex plane

a unified approach to the steady state and tracking analyses of adaptive filters most adaptive filters are inherently nonlinear and time variant systems the nonlinearities in the update equations tend to lead to difficulties in the study of their steady state performance as a limiting case of their transient performance this paper develops a unified approach to the steady state and tracking analyses of adaptive algorithms that bypasses many of these difficulties the approach is based on the study of the energy flow through each iteration of an adaptive filter and it relies on a fundamental error variance relation

against the convergence of the minor component analysis neurons we investigate the dynamical behavior of some linear neurons devised for the minor component analysis contrary to their convergence theorems our analysis reveals the existence of a divergence in a finite time

robust inversion dimensionality reduction and randomized sampling we consider a class of inverse problems in which the forward model is the solution operator to linear odes or pdes this class admits several dimensionality reduction techniques based on data averaging or sampling which are especially useful for large scale problems we survey these approaches and their connection to stochastic optimization the data averaging approach is only viable however for a least squares misfit which is sensitive to outliers in the data and artifacts unexplained by the forward model this motivates us to propose a robust formulation based on the student s t distribution of the error we demonstrate how the corresponding penalty function together with the sampling approach can obtain good results for a large scale seismic inverse problem with 50 corrupted data

the dirichlet problem for p harmonic functions on the topologist s comb in this paper we study the perron method for solving the p harmonic dirichlet problem on the topologist s comb for functions which are bounded and continuous at the accessible points we obtain invariance of the perron solutions under arbitrary perturbations on the set of inaccessible points we also obtain some results allowing for jumps and perturbations at a countable set of points

isochronicity and linearizability of planar polynomial hamiltonian systems in this paper we study isochronicity and linearizability of planar polynomial hamiltonian systems first we prove a theorem which supports a negative answer to the following open question stated by jarque and villadelprat in 15 do there exist planar polynomial hamiltonian systems of even degree having an isochronous center additionally we obtain some conditions for linearizability of complex cubic hamiltonian systems

analysis of gradient algorithms for tls based adaptive iir filters steepest descent gradient algorithms for unbiased equation error adaptive infinite impulse response iir filtering are analyzed collectively for both the total least squares and mixed least squares total least squares framework these algorithms have a monic normalization that allows for a direct filtering implementation we show that the algorithms converge to the desired filter coefficient vector we achieve the convergence result by analyzing the stability of the equilibrium points and demonstrate that only the desired solution is locally stable additionally we describe a region of initialization under which the algorithm converges to the desired solution we derive the results using interlacing relationships between the eigenvalues of the data correlation matrices and their respective schur complements finally we illustrate the performance of these new approaches through simulation

recursive total least squares algorithm based on inverse power method and dichotomous coordinate descent iterations we develop a recursive total least squares rtls algorithm for errors in variables system identification utilizing the inverse power method and the dichotomous coordinate descent dcd iterations the proposed algorithm called dcd rtls outperforms the previously proposed rtls algorithms which are based on the line search method with reduced computational complexity we perform a comprehensive analysis of the dcd rtls algorithm and show that it is asymptotically unbiased as well as being stable in the mean we also find a lower bound for the forgetting factor that ensures mean square stability of the algorithm and calculate the theoretical steady state mean square deviation msd we verify the effectiveness of the proposed algorithm and the accuracy of the predicted steady state msd via simulations

incremental proximal methods for large scale convex optimization we consider the minimization of a sum sum i 1 mf i x consisting of a large number of convex component functions f i for this problem incremental methods consisting of gradient or subgradient iterations applied to single components have proved very effective we propose new incremental methods consisting of proximal iterations applied to single components as well as combinations of gradient subgradient and proximal iterations we provide a convergence and rate of convergence analysis of a variety of such methods including some that involve randomization in the selection of components we also discuss applications in a few contexts including signal processing and inference machine learning

simulation based optimal sensor scheduling with application to observer trajectory planning the sensor scheduling problem can be formulated as a controlled hidden markov model and this paper solves the problem when the state observation and action spaces are continuous this general case is important as it is the natural framework for many applications the aim is to minimise the variance of the estimation error of the hidden state w r t the action sequence we present a novel simulation based method that uses a stochastic gradient algorithm to find optimal actions

the mca exin neuron for the minor component analysis the minor component analysis mca deals with the recovery of the eigenvector associated to the smallest eigenvalue of the autocorrelation matrix of the input data and is a very important tool for signal processing and data analysis it is almost exclusively solved by linear neurons this paper presents a linear neuron endowed with a novel learning law called mca exinn and analyzes its features the neural literature about mca is very poor in the sense that both a little theoretical basis is given almost always focusing on the ode asymptotic approximation and only experiments on toy problems at most four dimensional problems are presented without any numerical analysis this work addresses these problems and lays sound theoretical foundations for the neural mca theory in particular it classifies the mca neurons according to the riemannian metric and justifies from the analysis of the degeneracy of the error cost the different behavior in approaching convergence the cost landscape is studied and used as a basis for the analysis of the asymptotic behavior all the phases of the dynamics of the mca algorithms are investigated in detail and together with the numerical analysis lead to the identification of three possible kinds of divergence here called sudden dynamic and numerical the importance of the choice of low initial conditions is also explained a lot of importance is given to the experimental part where simulations on high dimensional problems are presented and analyzed the orthogonal regression or total least squares tls technique is also presented together with a real world application on the identification of the parameters of an electrical machine it can be concluded that mca exin is the best mca neuron in terms of stability no finite time divergence speed and accuracy

from structured data to evolution linear partial differential equations abstract this paper is devoted to the derivation of computational methods for constructing partial differential equations from data following some recent works 7 14 15 20 we propose a methodology based on symbolic calculus 8 9 13 pseudospectral methods 2 3 and stochastic processes 6 in order to determine non constant coefficients of linear evolution partial differential equations pdes from a set of structured data constituted by solutions at given times and positions of an unknown linear pde

nonexistence of isochronous centers in planar polynomial hamiltonian systems of degree four abstract in this paper we study centers of planar polynomial hamiltonian systems and we are interested in the isochronous ones we prove that every center of a polynomial hamiltonian system of degree four that is with its homogeneous part of degree four not identically zero is nonisochronous the proof uses the geometric properties of the period annulus and it requires the study of the hamiltonian systems associated to a hamiltonian function of the form h x y a x b x y c x y 2 d x y 3