a branch and cut decomposition algorithm for solving chance constrained mathematical programs with finite support we present a new approach for exactly solving chance constrained mathematical programs having discrete distributions with finite support and random polyhedral constraints such problems have been notoriously difficult to solve due to nonconvexity of the feasible region and most available methods are only able to find provably good solutions in certain very special cases our approach uses both decomposition to enable processing subproblems corresponding to one possible outcome at a time and integer programming techniques to combine the results of these subproblems to yield strong valid inequalities computational results on a chance constrained formulation of a resource planning problem inspired by a call center staffing application indicate the approach works significantly better than both an existing mixed integer programming formulation and a simple decomposition approach that does not use strong valid inequalities we also demonstrate how the approach can be used to efficiently solve for a sequence of risk levels as would be done when solving for the efficient frontier of risk and cost

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

distinguishing classical correlations from quantum correlations the aim of this paper is to distinguish classical correlations from quantum ones first a new characterization of a classical correlated cc state is established corresponding results for the left and right classical correlations are also obtained second a sufficient and necessary condition for a convex combination of two cc states to be cc is given it is proved that the set cc ha hb of all cc states on ha hb is a perfect nowhere dense and compact subset of the metric space d ha hb of all states density operators on ha hb based on our new characterization of cc states a quantity q is associated with a state it is shown that a state is cc if and only if q 0 in particular we prove that the werner state w in a two qubit system with a single real parameter is cc if and only if 0 25

duality and statistical tests of optimality for two stage stochastic programs we present alternative methods for verifying the quality of a proposed solution to a two stage stochastic program with recourse our methods revolve around implications of a dual problem in which dual multipliers on the nonanticipativity constraints play a critical role using randomly sampled observations of the stochastic elements we introduce notions of statistical dual feasibility and sampled error bounds additionally we use the nonanticipativity multipliers to develop connections to reduced gradient methods finally we propose a statistical test based on directional derivatives we illustrate the applicability of these tests via some examples

partial correlations in multipartite quantum systems abstract quantum correlations are more general than entanglement a problem on how to characterize and quantify quantum correlations has attracted substantial attention quantum correlations in bipartite systems have been researched extensively in this paper partial correlations in a multipartite quantum system a 1 a 2 a n are discussed in detail in order to reveal correlations of a given state with respect to some subsystems for a nonempty subset of the index set 1 2 n classical correlations cc quantum correlations qc partially classical correlations pcc and genuinely quantum correlations gqc are introduced in a multipartite system subsequently characterizations of cc are obtained secondly a measure function g of partial correlations of a state is defined and called the quantum discord it is proved that g is nonnegative independent of the choice of basis and invariant under local unitary transformations it is also proved that a state is cc if and only if g 0 and then it is qc if and only if g 0 moreover some relationships among partial correlations with respect to different are discussed in light of the measure function finally an illustrative example with 6 qubit ghz state is given

a finite branch and bound algorithm for two stage stochastic integer programs this paper addresses a general class of two stage stochastic programs with integer recourse and discrete distributions we exploit the structure of the value function of the second stage integer problem to develop a novel global optimization algorithm the proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature

on the convergence of algorithms with implications for stochastic and nondifferentiable optimization studies of the convergence of algorithms often revolve around the existence of a function with respect to which monotonic descent is required in this paper we show that under relatively lenient conditions stage dependent descent not necessarily monotonic is sufficient to guarantee convergence this development also provides the impetus to examine optimization algorithms we show that one of the important avenues in the study of convergence namely the theory of epi convergence imposes stronger conditions than are necessary to establish the convergence of an optimization algorithm working from a relaxation of epi convergence we introduce the notion of compatibility and prove several results that permit relaxations of conditions imposed by previous approaches to algorithmic convergence finally to illustrate the usefulness of the concepts we combine stage dependent descent with results derivable from compatibility to provide a basis for the convergence of a general algorithmic statement that might be used for stochastic and nondifferentiable optimization

a simulation based approach to two stage stochastic programming with recourse in this paper we consider stochastic programming problems where the objective function is given as an expected value function we discuss monte carlo simulation based approaches to a numerical solution of such problems in particular we discuss in detail and present numerical results for two stage stochastic programming with recourse where the random data have a continuous multivariate normal distribution we think that the novelty of the numerical approach developed in this paper is twofold first various variance reduction techniques are applied in order to enhance the rate of convergence successful application of those techniques is what makes the whole approach numerically feasible second a statistical inference is developed and applied to estimation of the error validation of optimality of a calculated solution and statistically based stopping criteria for an iterative alogrithm 1998 the mathematical programming society inc published by elsevier science b v

a gaussian upper bound for gaussian multi stage stochastic linear programs this paper deals with two stage and multi stage stochastic programs in which the right hand sides of the constraints are gaussian random variables such problems are of interest since the use of gaussian estimators of random variables is widespread we introduce algorithms to find upper bounds on the optimal value of two stage and multi stage stochastic minimization programs with gaussian right hand sides the upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size the algorithm for the two stage problem involves the solution of a deterministic linear program and a simple semidefinite program the algorithm for the multi stage problem invovles the solution of a quadratically constrained convex programming problem

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

aggregation and discretization in multistage stochastic programming multistage stochastic programs have applications in many areas and support policy makers in finding rational decisions that hedge against unforeseen negative events in order to ensure computational tractability continuous state stochastic programs are usually discretized and frequently the curse of dimensionality dictates that decision stages must be aggregated in this article we construct two discrete stage aggregated stochastic programs which provide upper and lower bounds on the optimal value of the original problem the approximate problems involve finitely many decisions and constraints thus principally allowing for numerical solution

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

multidimensional color image storage retrieval and compression based on quantum amplitudes and phases abstract in this study we propose a new representation method for multidimensional color images called an n qubit normal arbitrary superposition state nass where n qubits represent the colors and coordinates of 2 n pixels e g a three dimensional color image of 1024 1024 1024 using only 30 qubits based on nass we present an n 1 qubit normal arbitrary superposition state with relative phases nassrp and an n 2 qubit normal arbitrary superposition state with three components nasstc for lossless and lossy quantum compression respectively we also design three general quantum circuits to generate nass nassrp and nasstc states where we retrieve an image from a quantum system using different projection measurement operators finally we define the quantum compression ratio and analyze lossless and lossy quantum compression algorithms of multidimensional quantum images for the first time we implemented the compression of multidimensional color images on a quantum computer thus we address the theoretical and practical aspects of image processing on a quantum computer

traveling waves for a model of gravity driven film flows in cylindrical domains abstract traveling wave solutions are studied for a recently derived model of a falling viscous film on the interior of a vertical rigid tube by identifying a hopf bifurcation and using numerical continuation software families of non trivial traveling wave solutions may be traced out in parameter space these families all contain a single solution at a turnaround point with larger film thickness than all others in the family in an earlier paper it was conjectured that this turnaround point may represent a critical thickness separating two distinct flow regimes observed in physical experiments as well as two distinct types of behavior in transient solutions to the model here these hypotheses are verified over a range of parameter values using a combination of numerical and analytical techniques the linear stability of these solutions is also discussed both large and small amplitude solutions are shown to be unstable though the instability mechanisms are different for each wave type specifically for small amplitude waves the region of relatively flat film away from the localized wave crest is subject to the same instability that makes the trivial flat film solution unstable for large amplitude waves this mechanism is present but dwarfed by a much stronger tendency to relax to a regime close to that followed by small amplitude waves

convex approximations for totally unimodular integer recourse models a uniform error bound we consider a class of convex approximations for totally unimodular tu integer recourse models and derive a uniform error bound by exploiting properties of the total variation of the probability density functions involved for simple integer recourse models this error bound is tight and improves the existing one by a factor 2 whereas for tu integer recourse models this is the first nontrivial error bound available the bound ensures that the performance of the approximations is good as long as the total variations of the densities of all random variables in the model are small enough

finite master programs in regularized stochastic decomposition stochastic decomposition is a stochastic analog of benders decomposition in which randomly generated observations of random variables are used to construct statistical estimates of supports of the objective function in contrast to deterministic benders decomposition for two stage stochastic programs the stochastic version requires infinitely many inequalities to ensure convergence we show that asymptotic optimality can be achieved with a finite master program provided that a quadratic regularizing term is included our computational results suggest that the elimination of the cutting planes impacts neither the number of iterations required nor the statistical properties of the terminal solution

sample path optimization of convex stochastic performance functions in this paper we propose a method for optimizing convex performance functions in stochastic systems these functions can include expected performance in static systems and steady state performance in discrete event dynamic systems they may be nonsmooth the method is closely related to retrospective simulation optimization it appears to overcome some limitations of stochastic approximation which is often applied to such problems we explain the method and give computational results for two classes of problems tandem production lines with up to 50 machines and stochastic pert program evaluation and review technique problems with up to 70 nodes and 110 arcs

instabilities of one dimensional trivial phase solutions of the two dimensional cubic nonlinear schrodinger equation abstract the two dimensional cubic nonlinear schrodinger equation nls is used as a model of a wide variety of physical phenomena in this paper we study the stability of a class of its one dimensional periodic traveling wave solutions first we establish that all such solutions are unstable with respect to two dimensional perturbations with long wavelengths in the transverse dimension second we establish that all such solutions are unstable with respect to two dimensional perturbations with arbitrarily short wavelengths if the coefficients of the linear dispersion terms in the nls have opposite signs both arguments rely on formal perturbation methods third we use the fourier floquet hill numerical method to examine the spectral stability problem we present detailed spectra for twelve different solutions and demonstrate strong agreement between the numerical and asymptotic results

a numerical study of the large period limit of a zakharov shabat eigenvalue problem with periodic potentials deconinck and kutz 2006 j comput phys 219 296 321 developed an efficient algorithm for solving the zakharov shabat eigenvalue problem with periodic potentials numerically it is natural to use the same algorithm for solving the problem for non periodic potential decaying potentials defined over the whole real line using large periods in this paper we propose the use of a specific value of the floquet exponent our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of gardner 1997 j reine angew math 491 149 81 to be valid we also illustrate the rather complicated path to convergence of some nonlinear schr dinger potentials

numerical inverse scattering for the focusing and defocusing nonlinear schrodinger equations we solve the focusing and defocusing nonlinear schrodinger nls equations numerically by implementing the inverse scattering transform the computation of the scattering data and of the nls solution are both spectrally con vergent initial conditions in a suitable space are treated using the approach of biondini and bui 3 we numerically solve homogeneous robin boundary value problems on the half line finally using recent theoretical developments in the numerical approximation of riemann hilbert problems we prove that under mild assumptions our method of approximating solutions to the nls equations is uniformly accurate in their domain of definition

convergence of hill s method for nonselfadjoint operators by the introduction of a generalized evans function defined by an appropriate 2 modified fredholm determinant we give a simple proof of convergence in location and multiplicity of hill s method for numerical approximation of spectra of periodic coefficient ordinary differential operators our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite automatically satisfied in the scalar case notably this includes a large class of non self adjoint operators which previously had not been treated in a simple way the case of general coefficients depends on an interesting operator theoretic question regarding properties of toeplitz matrices

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

asymptotic results of stochastic decomposition for two stage stochastic quadratic programming this paper presents the stochastic decomposition sd algorithms for two classes of stochastic programming problems 1 two stage stochastic quadratic linear programming sqlp in which a quadrati

adaptive multicut aggregation for two stage stochastic linear programs with recourse outer linearization methods for two stage stochastic linear programs with recourse such as the l shaped algorithm generally apply a single optimality cut on the nonlinear objective at each major iteration while the multicut version of the algorithm allows for several cuts to be placed at once in general the l shaped algorithm tends to have more major iterations than the multicut algorithm however the trade offs in terms of computational time are problem dependent this paper investigates the computational trade offs of adjusting the level of optimality cut aggregation from single cut to pure multicut specifically an adaptive multicut algorithm that dynamically adjusts the aggregation level of the optimality cuts in the master program is presented and tested on standard large scale instances from the literature computational results reveal that a cut aggregation level that is between the single cut and the multicut can result in substantial computational savings over the single cut method

quantum discord a measure of the quantumness of correlations two classically identical expressions for the mutual information generally differ when the systems involved are quantum this difference defines the quantum discord it can be used as a measure of the quantumness of correlations separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord thus showing that absence of entanglement does not imply classicality we relate this to the quantum superposition principle and consider the vanishing of discord as a criterion for the preferred effectively classical states of a system i e the pointer states

new bounding and decomposition approaches for milp investment problems multi area transmission and generation planning under policy constraints abstract we propose a novel two phase bounding and decomposition approach to compute optimal and near optimal solutions to large scale mixed integer investment planning problems that have to consider a large number of operating subproblems each of which is a convex optimization our motivating application is the planning of power transmission and generation in which policy constraints are designed to incentivize high amounts of intermittent generation in electric power systems the bounding phase exploits jensen s inequality to define a lower bound which we extend to stochastic programs that use expected value constraints to enforce policy objectives the decomposition phase in which the bounds are tightened improves upon the standard benders algorithm by accelerating the convergence of the bounds the lower bound is tightened by using a jensen s inequality based approach to introduce an auxiliary lower bound into the benders master problem upper bounds for both phases are computed using a sub sampling approach executed on a parallel computer system numerical results show that only the bounding phase is necessary if loose optimality gaps are acceptable however the decomposition phase is required to attain optimality gaps use of both phases performs better in terms of convergence speed than attempting to solve the problem using just the bounding phase or regular benders decomposition separately

on the spectra of periodic waves for infinite dimensional hamiltonian systems abstract we consider the problem of determining the spectrum for the linearization of an infinite dimensional hamiltonian system about a spatially periodic traveling wave by using a bloch wave decomposition we recast the problem as determining the point spectra for a family of operators j l where j is skew symmetric with bounded inverse and l is symmetric with compact inverse our main result relates the number of unstable eigenvalues of the operator j l to the number of negative eigenvalues of the symmetric operator l the compactness of the resolvent operators allows us to greatly simplify the proofs as compared to those where similar results are obtained for linearizations about localized waves the theoretical results are general and apply to a larger class of problems than those considered herein the theory is applied to a study of the spectra associated with periodic and quasi periodic solutions to the nonlinear schrodinger equation as well as periodic solutions to the generalized korteweg de vries equation with power nonlinearity

whitham modulation theory for the kadomtsev petviashvili equation the genus 1 kadomtsev petviashvili kp whitham system is derived for both variants of the kp equation namely the kpi and kpii equations the basic properties of the kp whitham system including symmetries exact reductions and its possible complete integrability together with the appropriate generalization of the one dimensional riemann problem for the korteweg de vries equation are discussed finally the kp whitham system is used to study the linear stability properties of the genus 1 solutions of the kpi and kpii equations it is shown that all genus 1 solutions of kpi are linearly unstable while all genus 1 solutions of kpii are linearly stable within the context of whitham theory

metastability of solitary roll wave solutions of the st venant equations with viscosity we study by a combination of numerical and analytical evans function techniques the stability of solitary wave solutions of the st venant equations for viscous shallowwater ow down an incline and related models our main result is to exhibit examples of metastable solitary waves for the st venant equations with stable point spectrum indicating coherence of the wave prole but unstable essential spectrum indicating oscillatory convective instabilities shed in its wake we propose a mechanism based on dynamic spectrum of the wave prole by which a wave train of solitary pulses can stabilize each other by de amplication of convective instabilities as they pass through successive waves we present numerical time evolution studies supporting these conclusions which bear also on the possibility of stable periodic solutions close to the homoclinic for the closely related viscous jin xin model by contrast for which the essential spectrum is stable we show using the stability index of gardner zumbrun that solitary wave pulses are always exponentially unstable possessing point spectra with positive real part

quantum cryptographic algorithm for color images using quantum fourier transform and double random phase encoding abstract a novel encryption decryption method for color images is proposed in which quantum fourier transform qft and double random phase encoding drpe techniques have been used the encryption process can be realized by performing two secret random phase encoding operations respectively in the input and the qft planes because all quantum operations are invertible the decryption process is the inverse of the encryption process supported by detailed numerical simulation and theoretical analysis the method has clarified its robustness security and computational complexity over its classical counterpart and it opens the door for introducing color image encryption into quantum scenarios

a probability metrics approach for reducing the bias of optimality gap estimators in two stage stochastic linear programming monte carlo sampling based estimators of optimality gaps for stochastic programs are known to be biased when bias is a prominent factor estimates of optimality gaps tend to be large on average even for high quality solutions this diminishes our ability to recognize high quality solutions in this paper we present a method for reducing the bias of the optimality gap estimators for two stage stochastic linear programs with recourse via a probability metrics approach motivated by stability results in stochastic programming we apply this method to the averaged two replication procedure a2rp by partitioning the observations in an effort to reduce bias which can be done in polynomial time in sample size we call the resulting procedure the averaged two replication procedure with bias reduction a2rp b we provide conditions under which a2rp b produces strongly consistent point estimators and an asymptotically valid confidence interval we illustrate the effectiveness of our approach analytically on a newsvendor problem and test the small sample behavior of a2rp b on a number of two stage stochastic linear programs from the literature our computational results indicate that the procedure effectively reduces bias we also observe variance reduction in certain circumstances

parametric resonance in immersed elastic boundaries in this paper we investigate the stability of a fluid structure interaction problem in which a flexible elastic membrane immersed in a fluid is excited via periodic variations in the elastic stiffness parameter this model can be viewed as a prototype for active biological tissues such as the basilar membrane in the inner ear or heart muscle fibers immersed in blood problems such as this in which the system is subjected to internal forcing through a parameter can give rise to parametric resonance we formulate the equations of motion in two dimensions using the immersed boundary formulation assuming small amplitude motions we can apply floquet theory to the linearized equations and derive an eigenvalue problem whose solution defines the marginal stability boundaries in parameter space the eigenvalue equation is solved numerically to determine values of fiber stiffness and fluid viscosity for which the problem is linearly unstable we present direct numerical simulations of the fluid structure interaction problem using the immersed boundary method that verify the existence of the parametric resonances suggested by our analysis

numerical inverse scattering for the korteweg de vries and modified korteweg de vries equations abstract recent advances in the numerical solution of riemann hilbert problems allow for the implementation of a cauchy initial value problem solver for the korteweg de vries equation kdv and the defocusing modified korteweg de vries equation mkdv without any boundary approximation borrowing ideas from the method of nonlinear steepest descent this method is demonstrated to be asymptotically accurate the method is straightforward for the case of defocusing mkdv due to the lack of poles in the riemann hilbert problem and the boundedness properties of the reflection coefficient solving kdv requires the introduction of poles in the riemann hilbert problem and more complicated deformations the introduction of a new deformation for kdv allows for the stable asymptotic computation of the solution in the entire spacial and temporal plane kdv and mkdv are dispersive equations and this method can fully capture the dispersion with spectral accuracy thus this method can be used as a benchmarking tool for determining the effectiveness of future numerical methods designed to capture dispersion this method can easily be adapted to other integrable equations with riemann hilbert formulations such as the nonlinear schrodinger equation

numerical continuation methods for studying periodic travelling wave wavetrain solutions of partial differential equations abstract periodic travelling waves wavetrains are an important solution type for many partial differential equations in this paper i review the use of numerical continuation for studying these solutions i discuss the calculation of the form and stability of a given periodic travelling wave and the calculation of boundaries in a two dimensional parameter plane for wave existence and stability i also describe the automated implementation of these numerical continuation procedures via the software package wavetrain http www ma hw ac uk wavetrain i conclude by discussing ongoing work on numerical continuation methods for determining the absolute stability of periodic travelling waves

computing spectra of linear operators using the floquet fourier hill method in order to establish the stability of an equilibrium solution u of an infinite dimensional dynamical system u x u one is interested in the spectrum of the linear operator lu obtained by linearizing the dynamical system around u we use a spectrally accurate method for the computation of the spectrum of the maximal extension of the operator lu the method is particularly well suited to the case of periodic u although no periodic boundary conditions on the perturbations are imposed by incorporating the fundamentals of floquet theory an almost uniform approximation to the entire spectrum of the maximal extension is obtained as opposed to an approximation of a few selected elements the numerical component of the method is limited to i choosing the size of the matrices to be used and ii an eigenvalue solver such as the qr algorithm compared to often used finite difference approaches the method is an order of magnitude faster for comparable accuracy we illustrate that the method is efficiently extended to problems defined on the whole line

elliptic solutions of the defocusing nls equation are stable the stability of the stationary periodic solutions of the integrable one dimensional cubic defocusing nonlinear schrodinger nls equation is reasonably well understood especially for solutions of small amplitude in this paper we exploit the integrability of the nls equation to establish the spectral stability of all such stationary solutions this time by explicitly computing the spectrum and the corresponding eigenfunctions associated with their linear stability problem an additional argument using an appropriate krein signature allows us to conclude the nonlinear orbital stability of all stationary solutions of the defocusing nls equation with respect to so called subharmonic perturbations perturbations that have period equal to an integer multiple of the period of the amplitude of the solution all results presented here are independent of the size of the amplitude of the solutions and apply equally to solutions with trivial and nontrivial phase profiles

algorithmic superactivation of asymptotic quantum capacity of zero capacity quantum channels the superactivation of zero capacity quantum channels makes it possible to use two zero capacity quantum channels with a positive joint capacity for their output currently we have no theoretical background to describe all possible combinations of superactive zero capacity channels hence there may be many other possible combinations in practice to discover such superactive zero capacity channel pairs we must analyze an extremely large set of possible quantum states channel models and channel probabilities there is still no extremely efficient algorithmic tool for this purpose this paper shows an efficient algorithmical method of finding such combinations our method can be a very valuable tool for improving the results of fault tolerant quantum computation and possible communication techniques over very noisy quantum channels

augmented lagrangian method within l shaped method for stochastic linear programs stochastic programming is an optimization technique used in the presence of uncertainty and it typically leads to very large problem sizes in this paper a modified version of the l shaped method was used to solve two stage stochastic linear programs with recourse based on the projection method and the augmented lagrangian method using this modified version of the l shaped method allows us to reduce the number of iterations and the time of solving a two stage stochastic linear program with fixed recourse in comparison with traditional methods

confidence level solutions for stochastic programming we propose an alternative approach to stochastic programming based on monte carlo sampling and stochastic gradient optimization the procedure is by essence probabilistic and the computed solution is a random variable we propose a solution concept in which the probability that the random algorithm produces a solution with an expected objective value departing from the optimal one by more than e is small enough we derive complexity bounds on the number of iterations of this process we show that by repeating the basic process on independent samples one can significantly reduce the number of iterations

an arc exchange decomposition method for multistage dynamic networks with random arc capacities abstract multistage dynamic networks with random arc capacities mdnrac have been successfully used for modeling various resource allocation problems in the transportation area however solving these problems is generally computationally intensive and there is still a need to develop more efficient solution approaches in this paper we propose a new heuristic approach that solves the mdnrac problem by decomposing the network at each stage into a series of subproblems with tree structures each subproblem can be solved efficiently the main advantage is that this approach provides an efficient computational device to handle the large scale problem instances with fairly good solution quality we show that the objective value obtained from this decomposition approach is an upper bound for that of the mdnrac problem numerical results demonstrate that our proposed approach works very well

complexity of stochastic dual dynamic programming stochastic dual dynamic programming is a cutting plane type algorithm for multi stage stochastic optimization originated about 30 years ago in spite of its popularity in practice there does not exist any analysis on the convergence rates of this method in this paper we first establish the number of iterations i e iteration complexity required by a basic dual dynamic programming method for solving single scenario multi stage optimization problems by introducing novel mathematical tools including the saturation of search points we then refine these basic tools and establish the iteration complexity for an explorative dual dynamic programing method proposed herein and the classic stochastic dual dynamic programming method for solving more general multi stage stochastic optimization problems under the standard stage wise independence assumption our results indicate that the complexity of these methods mildly increases with the number of stages t in fact linearly dependent on t for discounted problems therefore they are efficient for strategic decision making which involves a large number of stages but with a relatively small number of decision variables in each stage without explicitly discretizing the state and action spaces these methods might also be pertinent to the related reinforcement learning and stochastic control areas

solving stochastic programs with integer recourse by enumeration a framework using gro bner basis reductions in this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right hand side vector using grobner basis methods from computational algebra to solve the numerous second stage integer programs using structural properties of the expected integer recourse function we prove that under mild conditions an optimal solution is contained in a finite set furthermore we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed

approximations of strongly continuous families of unbounded self adjoint operators the problem of approximating the discrete spectra of families of self adjoint operators that are merely strongly continuous is addressed it is well known that the spectrum need not vary continuously as a set under strong perturbations however it is shown that under an additional compactness assumption the spectrum does vary continuously and a family of symmetric finite dimensional approximations is constructed an important feature of these approximations is that they are valid for the entire family uniformly an application of this result to the study of plasma instabilities is illustrated

on the convergence of hill s method hill s method is a means to numerically approximate spectra of linear differential operators with periodic coefficients in this paper we address different issues related to the convergence of hill s method we show the method does not produce any spurious approximations and that for selfadjoint operators the method converges in a restricted sense furthermore assuming convergence of an eigenvalue we prove convergence of the associated eigenfunction approximation in the l 2 norm these results are not restricted to selfadjoint operators finally for certain selfadjoint operators we prove that the rate of convergence of hill s method to the least eigenvalue is faster than any polynomial power

no local broadcasting theorem for multipartite quantum correlations we prove that the correlations present in a multipartite quantum state have an operational quantum character even if the state is unentangled as long as it does not simply encode a multipartite classical probability distribution said quantumness is revealed by the new task of local broadcasting i e of locally sharing preestablished correlations which is feasible if and only if correlations are stricly classical our operational approach leads to natural definitions of measures for quantumness of correlations it also reproduces the standard no broadcasting theorem as a special case

on stability of generalized short crested water waves abstract in this paper we take up the question of dynamic stability of genuinely two dimensional generalized hexagonal traveling wave patterns on the surface of a three dimensional ideal fluid i e stability of generalized short crested wave gscw solutions of the water wave problem we restrict ourselves to a study of spectral stability which considers the linearization of the water wave operator about one of these traveling generalized hexagonal waves and draws conclusions about stability from the spectral data of the resulting linear operator within the class of perturbations we are allowed to study for a wide range of geometrical parameters we find stable traveling waveforms which eventually destabilize with features that depend strongly on the problem configuration in particular we find zones of instability for patterns shaped as symmetric diamonds while such zones are absent for asymmetric configurations furthermore we note that within a geometrical configuration as a generalized scw ratio is varied essentially the character of the linear solution these waves become more unstable as the waves become more asymmetric

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

on the stability of periodic travelling waves with large spatial period abstract in many circumstances a pulse to a partial differential equation pde on the real line is accompanied by periodic wave trains that have arbitrarily large period it is then interesting to investigate the pde stability of the periodic wave trains given that the pulse is stable using the evans function gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves in this article the precise location of these circles is determined it is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse as a consequence periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi hump pulses are created that is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations the proof uses lyapunov schmidt reduction and relies on the existence of exponential dichotomies the approach is also applicable to periodic waves with large spatial period of elliptic problems on r n or on unbounded cylinders r with bounded

assessing solution quality in stochastic programs determining whether a solution is of high quality optimal or near optimal is fundamental in optimization theory and algorithms in this paper we develop monte carlo sampling based procedures for assessing solution quality in stochastic programs quality is defined via the optimality gap and our procedures output is a confidence interval on this gap we review a multiple replications procedure that requires solution of say 30 optimization problems and then we present a result that justifies a computationally simplified single replication procedure that only requires solving one optimization problem even though the single replication procedure is computationally significantly less demanding the resulting confidence interval might have low coverage probability for small sample sizes for some problems we provide variants of this procedure that require two replications instead of one and that perform better empirically we present computational results for a newsvendor problem and for two stage stochastic linear programs from the literature we also discuss when the procedures perform well and when they fail and we propose using optimal solutions to strengthen the performance of our procedures

solving slp recourse problems with arbitrary multivariate distributions the dependent case bounding the expectation of a convex function of a multivariate random variable is of great importance for solving stochastic linear programming problems slp with recourse the classic bounds are those of jensen and edmundson madansky whereby jensen s inequality is available in the independent as well as in the dependent case in contrast with the edmundson madansky e m inequality madansky a 1959 bounds on the expectation of a convex function of multivariate random variable ann math statist 30 743 746 which is valid only in the independent case this paper presents a straightforward extension of the e m inequality for the dependent case yielding a discrete distribution that is extremal with respect to a partial ordering of a set of distribution functions further we prove that the bounds for the expectation behave monotonically by applying the obtained inequality to smaller and smaller subintervals as this requires the evaluation of conditional probabilities and expectations we replace the given distribution by a discrete one resulting from sampling and determine the sampling size that keeps the statistical error negligible finally we conclude with the application to slp recourse problems and state some computational results illustrating the effort for solving recourse problems

the stability spectrum for elliptic solutions to the focusing nls equation abstract we present an analysis of the stability spectrum of all stationary elliptic type solutions to the focusing nonlinear schrodinger equation nls an analytical expression for the spectrum is given from this expression various quantitative and qualitative results about the spectrum are derived specifically the solution parameter space is shown to be split into four regions of distinct qualitative behavior of the spectrum additional results on the stability of solutions with respect to perturbations of an integer multiple of the period are given as well as a procedure for approximating the greatest real part of the spectrum

on the spectral stability of solitary wave solutions of the vector nonlinear schrodinger equation we consider a system of coupled cubic nonlinear schrodinger nls equations where the interaction coefficients jk are real the spectral stability of solitary wave solutions both bright and dark is examined both analytically and numerically our results build on preceding work by nguyen et al and others specifically we present closed form solitary wave solutions with trivial and nontrivial phase profiles their spectral stability is examined analytically by determining the locus of their essential spectrum their full stability spectrum is computed numerically using a large period limit of hill s method we find that all nontrivial phase solutions are unstable while some trivial phase solutions are spectrally stable to our knowledge this paper presents the first investigation of the stability of the solitary waves of the coupled cubic nls equation without the restriction that all components j are proportional to sech

a multicut algorithm for two stage stochastic linear programs outer linearization methods such as van slyke and wets s l shaped method for stochastic linear programs generally apply a single cut on the nonlinear objective at each major iteration the structure of stochastic programs allows for several cuts to be placed at once this paper describes a multicut algorithm to carry out this procedure it presents experimental and theoretical justification for reductions in major iterations

periodic finite genus solutions of the kdv equation are orbitally stable abstract the stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade the kdv equation is known to have large families of periodic solutions that are parameterized by hyperelliptic riemann surfaces they are generalizations of the famous multi soliton solutions we show that all such periodic solutions are orbitally stable with respect to subharmonic perturbations perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution

restricted recourse bounds for stochastic linear programming we consider the problem of bounding the expected value of a linear program lp containing random coefficients with applications to solving two stage stochastic programs an upper bound for minimizations is derived from a restriction of an equivalent penalty based formulation of the primal stochastic lp and a lower bound is obtained from a restriction of a reformulation of the dual our restricted recourse bounds are more general and more easily computed than most other bounds because random coefficients may appear anywhere in the lp neither independence nor boundedness of the coefficients is needed and the bound is computed by solving a single lp or nonlinear program analytical examples demonstrate that the new bounds can be stronger than complementary jensen bounds an upper bound is complementary to a lower bound and vice versa in computational work we apply the bounds to a two stage stochastic program for semiconductor manufacturing with uncertain demand and production rates

stopping rules for a class of sampling based stochastic programming algorithms monte carlo sampling based algorithms hold much promise for solving stochastic programs with many scenarios a critical component of such algorithms is a stopping criterion to ensure the quality of the solution in this paper we develop a stopping rule theory for a class of algorithms that estimate bounds on the optimal objective function value by sampling we provide rules for selecting sample sizes and terminating the algorithm under which asymptotic validity of confidence intervals for the quality of the proposed solution can be verified empirical coverage results are given for a simple example

robustness of quantum correlations against linear noise relative robustness of quantum correlations rroqc of a bipartite state is firstly introduced relative to a classically correlated state robustness of quantum correlations roqc of a bipartite state is then defined as the minimum of rroqc of the state relative to all classically correlated ones it is proved that as a function on quantum states roqc is nonnegative lower semi continuous and neither convex nor concave especially it is zero if and only if the state is classically correlated thus roqc not only quantifies the endurance of quantum correlations of a state against linear noise but also can be used to distinguish between quantum and classically correlated states furthermore the effects of local quantum channels on the robustness are explored and characterized

stability analysis of 1 1 dimensional cnoidal waves in media with cubic nonlinearity in the present paper we perform stability analysis of stationary 1 1 dimensional cnoidal waves of cn and dn types anomalous group velocity dispersion and sn type normal group velocity dispersion the mathematical model is based on the nonlinear schrodinger equation with this aim we developed a method that takes into consideration the properties of complex eigenvalues of cauchy matrix for perturbation vectors we show that cnoidal sn wave is stable in the whole domain of its existence whereas cn and dn waves are unstable the instability of cn and dn waves is suppressed in the limiting case of strong localization when waves evolve into a set of well separated fundamental bright solitons