an optimal approximate dynamic programming algorithm for the lagged asset acquisition problem we consider a multistage asset acquisition problem where assets are purchased now at a price that varies randomly over time to be used to satisfy a random demand at a particular point in time in the future we provide a rare proof of convergence for an approximate dynamic programming algorithm using pure exploitation where the states we visit depend on the decisions produced by solving the approximate problem the resulting algorithm does not require knowing the probability distribution of prices or demands nor does it require any assumptions about its functional form the algorithm and its proof rely on the fact that the true value function is a family of piecewise linear concave functions

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

dirichlet problem for semilinear hyperbolic equation multidimensional case abstract we discuss the solvability of the dirichlet problem for the semilinear equation of the vibrating string x t t t y x t y f t y x t y 0 in higher dimensions with sides length being irrational numbers and superlinear nonlinearity to this effect we derive a new dual variational method

dirichlet problem for semilinear hyperbolic equation abstract we discuss the solvability of the dirichlet problem for the semilinear equation of the vibrating string x tt t y x yy t y f t y x t y 0 in rectangle with sides length being irrational numbers and superlinear nonlinearity to this effect we derive new dual variational method

stochastic decomposition an algorithm for two stage linear programs with recourse we present a cutting plane algorithm for two stage stochastic linear programs with recourse motivated by benders decomposition our method uses randomly generated observations of random variables to construct statistical estimates of supports of the objective function in general the resulting piecewise linear approximations do not agree with the objective function in finite time however certain subsequences of the estimated supports are shown to accumulate at supports of the objective function with probability one from this we establish the convergence of the algorithm under relatively mild assumptions

an algorithm for multistage dynamic networks with random arc capacities with an application to dynamic fleet management we consider the class of multistage dynamic networks with random arc capacities a framework that is well suited to model dynamic fleet management problems we propose a successive convex approximation approach that produces an approximation to the expected recourse function which captures the future effects of current decisions under uncertainty this method decomposes the network in each stage into tree subproblems whose expected recourse functions are easy to obtain we also compare this method with two alternative methods on a set of dynamic fleet management problems the numerical results show that this method is superior to the two alternative methods

a linearization method for nonsmooth stochastic programming problems for a stochastic programming problem with a nonsmooth objective a recursive stochastic subgradient method is proposed in which successive directions are obtained from quadratic programming subproblems the subproblems result from linearization of original constrains and from approximation of the objective by a quadratic function involving stochastic subgradient estimates constructed in the course of computation a special lyapunov function technique is used to show that the method is convergent with probability 1 to stationary points and recovers the subgradient and lagrange multipliers that appear in necessary conditions of optimality

an adaptive distribution free algorithm for the newsvendor problem with censored demands with applications to inventory and distribution we consider the problem of optimizing inventories for problems where the demand distribution is unknown and where it does not necessarily follow a standard form such as the normal we address problems where the process of deciding the inventory and then realizing the demand occurs repeatedly the only information we use is the amount of inventory left over rather than attempting to estimate the demand distribution we directly estimate the value function using a technique called the concave adaptive value estimation cave algorithm cave constructs a sequence of concave piecewise linear approximations using sample gradients of the recourse function at different points in the domain since it is a sampling based method cave does not require knowledge of the underlying sample distribution the result is a nonlinear approximation that is more responsive than traditional linear stochastic quasi gradient methods and more flexible than analytical techniques that require distribution information in addition we demonstrate near optimal behavior of the cave approximation in experiments involving two different types of stochastic programs the newsvendor stochastic inventory problem and two stage distribution problems

on the rate of convergence of optimal solutions of monte carlo approximations of stochastic programs in this paper we discuss monte carlo simulation based approximations of a stochastic programming problem we show that if the corresponding random functions are convex piecewise linear and the distribution is discrete then an optimal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size moreover by using the theory of large deviations we show that the probability of such an event approaches one exponentially fast with increase of the sample size in particular this happens in the case of linear two or multi stage stochastic programming with recourse if the corresponding distributions are discrete the obtained results suggest that in such cases monte carlo simulation based methods could be very efficient we present some numerical examples to illustrate the ideas involved

using stochastic approximation methods to compute optimal base stock levels in inventory control problems in this paper we consider numerous inventory control problems for which the base stock policies are known to be optimal and we propose stochastic approximation methods to compute the optimal base stock levels the existing stochastic approximation methods in the literature guarantee that their iterates converge but not necessarily to the optimal base stock levels in contrast we prove that the iterates of our methods converge to the optimal base stock levels moreover our methods continue to enjoy the well known advantages of the existing stochastic approximation methods in particular they only require the ability to obtain samples of the demand random variables rather than to compute expectations explicitly and they are applicable even when the demand information is censored by the amount of available inventory

a piecewise linear upper bound on the network recourse function we consider the optimal value of a pure minimum cost network flow problem as a function of supply demand and arc capacities we present a new piecewise linear upper bound on this function which is called the network recourse function the bound is compared to the standard madansky bound and is shown computationally to be a little weaker but much faster to find the amount of work is linear in the number of stochastic variables not exponential as is the case for the madansky bound therefore the reduction in work increases as the number of stochastic variables increases computational results are presented

shape a stochastic hybrid approximation procedure for two stage stochastic programs we consider the problem of approximating the expected recourse function for two stage stochastic programs our problem is motivated by applications that have special structure such as an underlying network that allows reasonable approximations to the expected recourse function to be developed in this paper we show how these approximations can be improved by combining them with sample gradient information from the true recourse function for the case of strictly convex nonlinear approximations we prove convergence for this hybrid approximation the method is attractive for practical reasons because it retains the structure of the approximation

quasi periodic solutions of hamiltonian perturbations of 2d linear schrodinger equations the general problem discussed here is the persistency of quasi periodic solutions of linear or integrable equations after hamiltonian perturbation this subject is closely related to the well known kam theory of invariant tori in smooth dynamical systems the main source of new difficulties is the fact that one considers here finite dimensional tori in an infinite dimensional phase space for instance already in the study of time periodic solutions the fact that the phase space is infinite dimensional leads to small divisor problems now the standard kam technology may be reworked in a form applicable to treat persistency problems in space dimension 1d in the case of dirichlet boundary conditions say where in particular the normal frequencies of the problem are well separated results along this line were obtained by s kuksin ki and c e wayne w among other works we mentioned ki as the main expository reference for the kam approach considering pde s in id with periodic boundary conditions and especially in space dimension 1 a significant new problem arises due to the presence of clusters of normal frequencies

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

decomposition coordination algorithms in stochastic optimization this paper considers an extension to the situation of stochastic programming of the auxiliary problem principle formerly introduced in a deterministic setting to serve as a general framework for decomposition coordination optimization algorithms the idea is based upon that of the stochastic gradient that is independent noise realizations are considered successively along the iterations as a consequence deterministic subproblems are solved at each iteration whereas iterations fulfill the two tasks of coordination and stochastic approximation at the same time coupling cost function expectation of some performance index and deterministic coupling constraints are considered price dual decomposition encompassing extensions of the uzawa and arrow hurwicz algorithms to this stochastic case are studied as well as resource allocation primal decomposition

generalized group characters and complex oriented cohomology theories let bg be the classifying space of a finite group g given a multiplicative cohomology theory e the assignment g 7i e bg is a functor from groups to rings endowed with induction transfer maps in this paper we investigate these functors for complex oriented cohomology theories e using the theory of complex representations of finite groups as a model for what one would like to know an analogue of artin s theorem is proved for all complex oriented e the abelian subgroups of g serve as a detecting family for e bg modulo torsion dividing the order of g when e is a complete local ring with residue field of characteristic p and associated formal group of height n we construct a character ring of class functions that computes 1 e bg the domain of the characters is gn p the set of n tuples of elements in g each of which has order a power of p a formula for induction is also found the ideas we use are related to the lubin tate theory of formal groups the construction applies to many cohomology theories of current interest completed versions of elliptic cohomology e n theory etc the nth morava k theory euler characteristic for bg is computed to be the number of g orbits in gn p for various groups g including all symmetric groups we prove that k n bg concentrated in even degrees our results about e bg extend to theorems about e eg gx where x is a finite g cw complex

random lsc functions an ergodic theorem an ergodic theorem for random lsc lower semicontinuous functions is obtained by relying on a scalarization of such functions in the process kolmogorov s extension theorem for random lsc functions is established applications to statistical estimation problems composite materials and stochastic optimization problems are briefly noted

multiplicity of periodic solutions of nonlinear wave equations abstract we prove multiplicity of small amplitude periodic solutions with fixed frequency of completely resonant wave equations with general nonlinearities as 1 the number n of 2 periodic solutions u1 un un tends to the minimal period of the nth solution un is 2 n the proofs are based on the variational lyapunov schmidt reduction comm math phys to appear and minimax arguments 1

diophantine approximation bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball the aim of this paper is to consider the radially symmetric periodic dirichlet problem on 0 t x b n a for the equation u tt deltau f t x u where delta is the classical laplacian operator and b n a denotes the open ball of center 0 and radius a in r n when alpha a t is a sufficiently large irrational with bounded partial quotients we combine some number theory techniques with the asymptotic properties of the bessel functions to show that 0 is not an accumulation point of the spectrum of the linear part this result is used to obtain existence conditions for the nonlinear problem

an adaptive algorithm for finding the optimal base stock policy in lost sales inventory systems with censored demand we consider a periodic review single location single product inventory system with lost sales and positive replenishment lead times it is well known that the optimal policy does not possess a simple structure motivated by recent results showing that base stock policies perform well in these systems we study the problem of finding the best base stock policy in such a system in contrast to the classical inventory literature we assume that the manager does not know the demand distribution a priori but must make the replenishment decision in each period based only on the past sales censored demand data we develop a nonparametric adaptive algorithm that generates a sequence of order up to levels whose running average of the inventory holding and lost sales penalty cost converges to the cost of the optimal base stock policy and we establish the cubic root convergence rate of the algorithm our analysis is based on recent advances in stochastic online convex optimization and on the uniform ergodicity of markov chains associated with bases stock policies

a separable piecewise linear upper bound for stochastic linear programs stochastic linear programs require the evaluation of an integral in which the integrand is itself the value of a linear program this integration is often approximated by discrete distributions that bound the integral from above or below a difficulty with previous upper bounds is that they generally require a number of function evaluations that grows exponentially in the number of variables we give a new upper bound that requires operations that only grow polynomially in the number of random variables we show that this bound is sharp if the function is linear and give computational results to illustrate its performance