an infeasible bundle method for nonsmooth convex constrained optimization without a penalty function or a filter global convergence in constrained optimization algorithms has traditionally been enforced by the use of parametrized penalty functions recently the filter strategy has been introduced as an alternative at least part of the motivation for using filter methods consists of avoiding the need for estimating a suitable penalty parameter which is often a delicate task in this paper we demonstrate that the use of a parametrized penalty function in nonsmooth convex optimization can be avoided without using the relatively complex filter methods we propose an approach which appears to be more direct and easier to implement in the sense that it is closer in spirit and structure to the well developed unconstrained bundle methods preliminary computational results are also reported

a trust region spectral bundle method for nonconvex eigenvalue optimization we present a nonsmooth optimization technique for nonconvex maximum eigenvalue functions and for nonsmooth functions which are infinite maxima of eigenvalue functions we prove global convergence of our method in the sense that for an arbitrary starting point every accumulation point of the sequence of iterates is critical the method is tested on several problems in feedback control synthesis

a doubly stabilized bundle method for nonsmooth convex optimization we propose a bundle method for minimizing nonsmooth convex functions that combines both the level and the proximal stabilizations most bundle algorithms use a cutting plane model of the objective function to formulate a subproblem whose solution gives the next iterate proximal bundle methods employ the model in the objective function of the subproblem while level methods put the model in the subproblem s constraints the proposed algorithm defines new iterates by solving a subproblem that employs the model in both the objective function and in the constraints one advantage when compared to the proximal approach is that the level set constraint provides a certain lagrange multiplier which is used to update the proximal parameter in a novel manner we also show that in the case of inexact function and subgradient evaluations no additional procedure needs to be performed by our variant to deal with inexactness as opposed to the proximal bundle methods that require special modifications numerical experiments on almost 1 000 instances of different types of problems are presented our experiments show that the doubly stabilized bundle method inherits useful features of the level and the proximal versions and compares favorably to both of them

an aggregate subgradient method for nonsmooth convex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

stochastic nonlinear minimax dynamic games with noisy measurements this note is concerned with nonlinear stochastic minimax dynamic games which are subject to noisy measurements the minimizing players are control inputs while the maximizing players are square integrable stochastic processes the minimax dynamic game is formulated using an information state which depends on the paths of the observed processes the information state satisfies a partial differential equation of the hamilton jacobi bellman hjb type the hjb equation is employed to characterize the dissipation properties of the system to derive a separation theorem between the design of the estimator and the controller and to introduce a certainty equivalence principle along the lines of whittle finally the separation theorem and the certainty equi valence principle are applied to solve the linear quadratic gaussian minimax game the results of this note generalize the l sup 2 gain of deterministic systems to stochastic analogs they are related to the controller design of stochastic systems which employ risk sensitive performance criteria and to the controller design of deterministic systems which employ minimax performance criteria

exact hausdorff measure on the boundary of a galton watson tree a necessary and sufficient condition for the almost sure existence of an absolutely continuous with respect to the branching measure exact hausdorff measure on the boundary of a galton watson tree is obtained in the case where the absolutely continuous exact hausdorff measure does not exist almost surely a criterion which classifies gauge functions 0 according to whether 0 hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given important examples are discussed in four additional theorems in particular hawkes s conjecture in 1981 is solved problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved

the exact hausdorff dimension of a branching set we obtain a critical function for which the hausdorff measure of a branching set generated by a simple galton watson process is positive and finite

certain nonlinear partially observable stochastic optimal control problems with explicit control laws equivalent to leqg lqg problems this paper is concerned with partially observed stochastic optimal control problems when nonlinearities enter the dynamics of the unobservable state and the observations as gradients of potential functions explicit representations for the information state are derived in terms of a finite number of sufficient statistics consequently the partially observed problem is recast as one of complete information with a new state generated by a modified version of the kalman filter when the terminal cost is quadratic in the unobservable state and includes the integral of the nonlinearities the optimal control laws are explicitly computed similar to linear exponential quadratic gaussian leqg and linear quadratic gaussian lqg tracking problems the results are applicable to filtering and control of hamiltonian systems

new variants of bundle methods in this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization convex concave games and variational inequalities we outline the ideas underlying these methods and present rate of convergence estimates

discriminative semi supervised feature selection via manifold regularization feature selection has attracted a huge amount of interest in both research and application communities of data mining we consider the problem of semi supervised feature selection where we are given a small amount of labeled examples and a large amount of unlabeled examples since a small number of labeled samples are usually insufficient for identifying the relevant features the critical problem arising from semi supervised feature selection is how to take advantage of the information underneath the unlabeled data to address this problem we propose a novel discriminative semi supervised feature selection method based on the idea of manifold regularization the proposed approach selects features through maximizing the classification margin between different classes and simultaneously exploiting the geometry of the probability distribution that generates both labeled and unlabeled data in comparison with previous semi supervised feature selection algorithms our proposed semi supervised feature selection method is an embedded feature selection method and is able to find more discriminative features we formulate the proposed feature selection method into a convex concave optimization problem where the saddle point corresponds to the optimal solution to find the optimal solution the level method a fairly recent optimization method is employed we also present a theoretic proof of the convergence rate for the application of the level method to our problem empirical evaluation on several benchmark data sets demonstrates the effectiveness of the proposed semi supervised feature selection method

a proximal cutting plane method using chebychev center for nonsmooth convex optimization an algorithm is developed for minimizing nonsmooth convex functions this algorithm extends elzinga moore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones thus removing compactness assumption our method is to elzinga moore s algorithm what a proximal bundle method is to kelley s algorithm instead of lower approximations used in proximal bundle methods the present approach is based on some objects regularizing translated functions of the objective function we propose some variants and using some academic test problems we conduct a numerical comparative study with elzinga moore algorithm and two other well known nonsmooth methods

hausdorff and packing spectra large deviations and free energy for branching random walks in mathbb r d consider an mathbb r d valued branching random walk brw on a supercritical galton watson tree without any assumption on the distribution of this brw we compute almost surely and simultaneously the hausdorff and packing dimensions of the level sets e k of infinite branches in the boundary of the tree endowed with its standard metric along which the averages of the brw have a given closed connected set of limit points k this goes beyond multifractal analysis which only considers those level sets when k ranges in the set of singletons alpha alpha in mathbb r d we also give a 0 law for the hausdorff and packing measures of the level sets e and compute the free energy of the associated logarithmically correlated random energy model in full generality moreover our results complete the previous works on multifractal analysis by including the levels which do not belong to the range of the gradient of the free energy this covers in particular a situation that was until now badly understood namely the case where a first order phase transition occurs as a consequence of our study we can also describe the whole singularity spectrum of mandelbrot measures as well as the associated free energy function or lq spectrum when a first order phase transition occurs

non euclidean restricted memory level method for large scale convex optimization we propose a new subgradient type method for minimizing extremely large scale nonsmooth convex functions over simple domains the characteristic features of the method are a the possibility to adjust the scheme to the geometry of the feasible set thus allowing to get nearly dimension independent and nearly optimal in the large scale case rate of convergence results for minimization of a convex lipschitz continuous function over a euclidean ball a standard simplex and a spectahedron the set of positive semidefinite symmetric matrices of given size with unit trace b flexible handling of accumulated information allowing for tradeoff between the level of utilizing this information and iteration s complexity we present extensions of the scheme for the cases of minimizing non lipschitzian convex objectives finding saddle points of convex concave functions and solving variational inequalities with monotone operators finally we report on encouraging numerical results of experiments with test problems of dimensions up to 66 000

analysis of stochastic dual dynamic programming method abstract in this paper we discuss statistical properties and convergence of the stochastic dual dynamic programming sddp method applied to multistage linear stochastic programming problems we assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original true distribution is generated and consequently the sddp algorithm is applied to the constructed sample average approximation saa problem then we proceed to analysis of the sddp solutions of the saa problem and their relations to solutions of the true problem finally we discuss an extension of the sddp method to a risk averse formulation of multistage stochastic programs we argue that the computational complexity of the corresponding sddp algorithm is almost the same as in the risk neutral case

accuracy certificates for computational problems with convex structure the goal of this paper is to introduce the notion of certificates which verify the accuracy of solutions of computational problems with convex structure such problems include minimizing convex functions variational inequalities with monotone operators computing saddle points of convex concave functions and solving convex nash equilibrium problems we demonstrate how the implementation of the ellipsoid method and other cutting plane algorithms can be augmented with the computation of such certificates without significant increase of the computational effort further we show that computable certificates exist for any algorithm that is capable of producing solutions of guaranteed accuracy

partially observable nonlinear risk sensitive control problems dynamic programming and verification theorems in this paper we consider continuous time partially observable optimal control problems with exponential of integral cost criteria we derive a rigorous verification theorem when the state and control enter nonlinear in the dynamics in addition we show that the quadratic sensor problem is estimation solvable with respect to a certain cost criterion the framework relies on dynamic programming and the hamilton jacobi theory

risk sensitive control and dynamic games for partially observed discrete time nonlinear systems solves a finite horizon partially observed risk sensitive stochastic optimal control problem for discrete time nonlinear systems and obtains small noise and small risk limits the small noise limit is interpreted as a deterministic partially observed dynamic game and new insights into the optimal solution of such game problems are obtained both the risk sensitive stochastic control problem and the deterministic dynamic game problem are solved using information states dynamic programming and associated separated policies a certainty equivalence principle is also discussed the authors results have implications for the nonlinear robust stabilization problem the small risk limit is a standard partially observed risk neutral stochastic optimal control problem

complexity estimates of some cutting plane methods based on the analytic barrier in this paper we establish the efficiency estimates for two cutting plane methods based on the analytic barrier we prove that the rate of convergence of the second method is optimal uniformly in the number of variables we present a modification of the second method in this modified version each test point satisfies an approximate centering condition we also use the standard strategy for updating approximate hessians of the logarithmic barrier function we prove that the rate of convergence of the modified scheme remains optimal and demonstrate that the number of newton steps in the auxiliary minimization processes is bounded by an absolute constant we also show that the approximate hessian strategy significantly improves the total arithmetical complexity of the method

exact packing dimension in random recursive constructions we explore the exact packing dimension of certain random recursive constructions in case of polynomial decay at 0 of the distribution function of random variable x associated with the construction we prove that it does not exist and in case of exponential decay it is t log logt where is the fractal dimension of the limit set and 1 is the rate of exponential decay

a finite dimensional risk sensitive control problem a partially observed stochastic control problem with exponential running cost is considered the dynamics are linear and the running cost is quadratic although the control may enter nonlinearly explicit solutions are found to a modified zakai equation and a backward adjoint equation this enables the problem to be expressed in terms of observable finite dimensional dynamics and a separation principle to be applied

conditional gradient algorithms for norm regularized smooth convex optimization motivated by some applications in signal processing and machine learning we consider two convex optimization problems where given a cone k k a norm vert cdot vert and a smooth convex function f f we want either 1 to minimize the norm over the intersection of the cone and a level set of f f or 2 to minimize over the cone the sum of f f and a multiple of the norm we focus on the case where a the dimension of the problem is too large to allow for interior point algorithms b vert cdot vert is too complicated to allow for computationally cheap bregman projections required in the first order proximal gradient algorithms on the other hand we assume that it is relatively easy to minimize linear forms over the intersection of k k and the unit vert cdot vert ball motivating examples are given by the nuclear norm with k k being the entire space of matrices or the positive semidefinite cone in the space of symmetric matrices and the total variation norm on the space of 2d images we discuss versions of the conditional gradient algorithm capable to handle our problems of interest provide the related theoretical efficiency estimates and outline some applications

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

a version of the bundle idea for minimizing a nonsmooth function conceptual idea convergence analysis numerical results during recent years various proposals for the minimization of a nonsmooth functional have been made amongst these the bundle concept turned out to be an especially fruitful idea based on this concept a number of authors have developed codes that can successfully deal with nonsmooth problems the aim of the paper is to show that by adding some features of the trust region philosophy to the bundle concept the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions the reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real life problems

a minimax resource allocation problem with variable resources abstract the problem of optimally allocating the resources to competing activities where the amounts of the resources are not only previosly given but are also to be determined subject to certain linear constraints is considered the objective is to find such activity levels such that their weighted deviation from a prespecified target is as small as possible a primal decomposition approach for the problem and derive an explicit formula for the piecewise affine linear convex objective function of the upper level problem are suggested the lower level problem is a parametric optimization problem with a bottleneck objective function if the constraints on the amounts of the resources are all of knapsack type or if there is only one such constraint then the upper level problem is equivalent to the lower level problem with fixed right hand sides in the general case the problem can efficiently be solved by means of the level method

h control for non linear stochastic systems the output feedback case the h output feedback control problem for non linear stochastic systems is considered a solution for a large class of non linear stochastic systems is introduced including non linear diffusion systems as a subclass this solution is based on a bounded real lemma for non linear stochastic systems that was previously established via a stochastic dissipativity concept the theory yields sufficient conditions for the closed loop system to possess a prescribed l 2 gain bound in terms of two hamilton jacobi inequalities one that is associated with the state feedback part of the problem is n dimensional where n is the underlying system s state dimension and the other inequality that stems from the estimation part is 2n dimensional both stationary and non stationary systems are considered stability of the closed loop system is established both in the mean square and the in probability senses as the solution to the hamilton jacobi inequalities may in general lead to a non realisable state estimator

the role of information state and adjoint in relating nonlinear output feedback risk sensitive control and dynamic games this paper employs logarithmic transformations to establish relations between continuous time nonlinear partially observable risk sensitive control problems and analogous output feedback dynamic games the first logarithmic transformation is introduced to relate the stochastic information state to a deterministic information state the second logarithmic transformation is applied to the risk sensitive cost function using the laplace varadhan lemma in the small noise limit this cost function is shown to be logarithmically equivalent to the cost function of an analogous dynamic game

optimal control of partially observable stochastic systems with an exponential of integral performance index the stochastic control problem with linear stochastic differential equations driven by brownian motion processes and as cost functional the exponential of a quadratic form is considered the soluti

on the optimal control of stochastic systems with an exponential of integral performance index abstract in this paper a theory of optimal control is developed for stochastic systems whose performance is measured by the exponential of an integral form such a formulation of the cost function is shown to be not only general and useful but also analytically tractable starting with very general classes of stochastic systems optimality conditions are obtained which exploit the multiplicative decomposability of the exponential of integral form specializing to partially observed systems of stochastic differential equations with brownian motion disturbances optimality conditions are obtained which parallel those for systems with integral costs also treated are the special cases of linear systems with exponential of quadratic costs for which explicit optimal controls are obtainable in addition several general results of independent interest are obtained which concern optimality of stochastic systems

a max plus based algorithm for a hamilton jacobi bellman equation of nonlinear filtering the hamilton jacobi bellman hjb equation associated with the robust hinfty filter as well as the mortensen filter is considered these filters employ a model where the disturbances have finite power the hjb equation for the filter information state is a first order equation with a term that is quadratic in the gradient yet the solution operator is linear in the max plus algebra this property is exploited by the development of a numerical algorithm where the effect of the solution operator on a set of basis functions is computed off line the precomputed solutions are stored as vectors of coefficients of the basis functions these coefficients are then used directly in the real time computations

guaranteed cost lqg control for uncertain systems with a normalized coprime factor uncertainty structure the paper extends the robust minimax lqg control design methodology to stochastic uncertain systems with a general uncertainty structure which includes normalized coprime factor uncertainty passive uncertainty and sector norm bounded uncertainty as special cases the derivation of the result uses a special parameter dependent girsanov measure transformation the efficacy of the proposed methodology is illustrated using the problem of frequency locking of an optical cavity which occurs in the area of experimental quantum optics

piecewise linear approximations in nonconvex nonsmooth optimization we present a bundle type method for minimizing nonconvex nondifferentiable functions of several variables the algorithm is based on the construction of both a lower and an upper polyhedral approximation of the objective function in particular at each iteration a search direction is computed by solving a quadratic program aiming at maximizing the difference between the lower and the upper model a proximal approach is used to guarantee convergence to a stationary point under the hypothesis of weak semismoothness

robust h sub spl infin filtering for nonlinear stochastic systems this paper describes the robust h sub spl infin filtering analysis and synthesis of nonlinear stochastic systems with state and exogenous disturbance dependent noise we assume that the state and measurement are corrupted by stochastic uncertain exogenous disturbance and that the system dynamic is modeled by ito spl circ type stochastic differential equations for general nonlinear stochastic systems the h sub spl infin filter can be obtained by solving second order nonlinear hamilton jacobi inequalities when the worst case disturbance is considered in the design procedure a mixed h sub 2 h sub spl infin filtering problem is also solved by minimizing the total estimation error energy it is found that for a class of special nonlinear stochastic systems the h sub spl infin filtering design can be given via solving several linear matrix inequalities instead of hamilton jacobi inequalities a few examples show that the proposed methods are effective

using regularization and second order information in outer approximation for convex minlp in this paper we present two new methods for solving convex mixed integer nonlinear programming problems based on the outer approximation method the first method is inspired by the level method and uses a regularization technique to reduce the step size when choosing new integer combinations the second method combines ideas from both the level method and the sequential quadratic programming technique and uses a second order approximation of the lagrangean when choosing the new integer combinations the main idea behind the methods is to choose the integer combination more carefully at each iteration in order to obtain the optimal solution in fewer iterations compared to the original outer approximation method we prove rigorously that both methods will find and verify the optimal solution in a finite number of iterations furthermore we present a numerical comparison of the methods based on 109 test problems to illustrate their advantages

the approximation algorithm for solving a sort of non smooth programming an approximation algorithm for solving a sort of non smooth programming is proposed in the programming the objective function is the holder function and the feasible region is contained in a rectangle viz hyper rectangle to establish the algorithm the properties of the holder function and an approximation of the function by using bernstein polynomial are studied according to the properties of the approximation polynomial and the algorithm for solving geometric programming the strategy for branching and bounding and the branch and bound algorithm are constructed to solve the programming the convergence of the algorithm can be guaranteed by the exhaustive of the bisection of the rectangle the feasibility of the algorithm is validated by solving an example

proximity control in bundle methods for convex proximal bundle methods for minimizing a convex functionf generate a sequence x k by takingxk 1 to be the minimizer of hat f k x u k x x k 2 2 where hat f k is a sufficiently accurate polyhedral approximation tof andu k 0 the usual choice ofu k 1 may yield very slow convergence a technique is given for choosing u k adaptively that eliminates sensitivity to objective scaling some encouraging numerical experience is reported

a sequential linear programming algorithm for solving monotone variational inequalities applied to strongly monotone variational inequalities newton s algorithm achieves local quadratic convergence in this paper it is shown how the basic newton method can be modified to yield an algorithm whose global convergence can be guaranteed by monitoring the monotone decrease of the gap function associated with the variational inequality each iteration consists in the solution of a linear program in the space of primal dual variables and of a linesearch convergence does not depend on strong monotonicity however under strong monotonicity and geometric stability assumptions the set of active constraints at the solution is implicitly identified and quadratic convergence is achieved

asynchronous level bundle methods in this paper we consider nonsmooth convex optimization problems with additive structure featuring independent oracles black boxes working in parallel existing methods for solving these distributed problems in a general form are synchronous in the sense that they wait for the responses of all the oracles before performing a new iteration in this paper we propose level bundle methods handling asynchronous oracles these methods require original upper bounds using upper models or scarce coordinations to deal with asynchronicity we prove their convergence using variational analysis techniques and illustrate their practical performance on a lagrangian decomposition problem

a bundle method for solving equilibrium problems we present a bundle method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle first we consider a general algorithm that we prove to be convergent then we explain how to make this algorithm implementable the strategy is to approximate the nonsmooth convex functions by piecewise linear convex functions in such a way that the subproblems are easy to solve and the convergence is preserved in particular we introduce a stopping criterion which is satisfied after finitely many iterations and which gives rise to stationary points finally we apply our implementable algorithm for solving the particular case of singlevalued and multivalued variational inequalities and we find again the results obtained recently by salmon et al 18

a risk sensitive maximum principle the case of imperfect state observation the risk sensitive maximum principle for optimal stochastic control derived by the author in an earlier work system control letters vol 15 1990 is restated this is an immediate generalization of the classic pontryagin principle to which it reduces in the deterministic case and is expressed immediately in terms of observables it is derived on the assumption that the criterion function is the exponential of an additive cost function and is exact under linear quadratic gaussian assumptions but is otherwise valid as a large deviation approximation the principle is extended to the case of imperfect state observation after preliminary establishment of a certainty equivalence principle the derivation yields as byproduct a large deviation version of the updating equation for nonlinear filtering the development is heuristic it is believed that the mathematical arguments given are the essential ones and provide a self contained treatment at this level

on the asymptotics of the eigenvalue counting function for random recursive sierpinski gaskets we consider natural laplace operators on random recursive affine nested fractals based on the sierpinski gasket and prove an analogue of weyl s classical result on their eigenvalue asymptotics the eigenvalue counting function n is shown to be of order ds 2 as where we can explicitly compute the spectral dimension d s moreover the limit n ds 2 will typically exist and can be expressed as a deterministic constant multiplied by a random variable this random variable is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume of the fractal