multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice i homogeneous problems we present a computational approach for the wkb approximation of the wave function of an electron moving in a periodic one dimensional crystal lattice we derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the bloch spectrum of the schrodinger operator relying on the framework of the multibranch entropy solutions introduced by brenier and corrias we compute efficiently multiphase solutions using well adapted and simple numerical schemes in this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method

a bloch decomposition based split step pseudospectral method for quantum dynamics with periodic potentials we present a new numerical method for accurate computations of solutions to linear one dimensional schro dinger equations with periodic potentials this is a prominent model in solid state physics where we also allow for perturbations by nonperiodic potentials describing external electric fields our approach is based on the classical bloch decomposition method which allows us to diagonalize the periodic part of the hamiltonian operator hence the dominant effects from dispersion and periodic lattice potential are computed together while the nonperiodic potential acts only as a perturbation because the split step communicator error between the periodic and nonperiodic parts is relatively small the step size can be chosen substantially larger than for the traditional splitting of the dispersion and potential operators indeed it is shown by the given examples that our method is unconditionally stable and more efficient than the traditional split step pseudospectral schemes to this end a particular focus is on the semiclassical regime where the new algorithm naturally incorporates the adiabatic splitting of slow and fast degrees of freedom

new geometry inspired relaxations and algorithms for the metric steiner tree problem determining the integrality gap of the bidirected cut relaxation for the metric steiner tree problem and exploiting it algorithmically is a long standing open problem we use geometry to define an lp whose dual is equivalent to this relaxation this opens up the possibility of using the primal dual schema in a geometric setting for designing an algorithm for this problem using this approach we obtain a 4 3 factor algorithm and integrality gap bound for the case of quasi bipartite graphs the previous best integrality gap upper bound being 3 2 rajagopalan and vazirani in on the bidirected cut relaxation for the metric steiner tree problem 1999 we also obtain a factor sqrt 2 strongly polynomial algorithm for this class of graphs a key difficulty experienced by researchers in working with the bidirected cut relaxation was that any reasonable dual growth procedure produces extremely unwieldy dual solutions a new algorithmic idea helps finesse this difficulty that of reducing the cost of certain edges and constructing the dual in this altered instance and this idea can be extracted into a new technique for running the primal dual schema in the setting of approximation algorithms

on the sample size of random convex programs with structured dependence on the uncertainty the scenario approach provides an intuitive method to address chance constrained problems arising in control design for uncertain systems it addresses these problems by replacing the chance constraint with a finite number of sampled constraints scenarios the sample size critically depends on helly s dimension a quantity always upper bounded by the number of decision variables however this standard bound can lead to computationally expensive programs whose solutions are conservative in terms of cost and violation probability we derive improved bounds of helly s dimension for problems where the chance constraint has certain structural properties the improved bounds lower the number of scenarios required for these problems leading both to improved objective value and reduced computational complexity our results are generally applicable to randomized model predictive control of chance constrained linear systems with additive uncertainty and affine disturbance feedback

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

a distributionally robust perspective on uncertainty quantification and chance constrained programming the objective of uncertainty quantification is to certify that a given physical engineering or economic system satisfies multiple safety conditions with high probability a more ambitious goal is to actively influence the system so as to guarantee and maintain its safety a scenario which can be modeled through a chance constrained program in this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry unimodality or independence patterns we delineate the watershed between tractability and intractability in ambiguity averse uncertainty quantification and chance constrained programming using tools from distributionally robust optimization we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice ii impurities confinement and bloch oscillations we present a computational approach for the wkb approximation of the wavefunction of an electron moving in a periodic one dimensional crystal lattice by means of a nonstrictly hyperbolic system whose flux function stems from the bloch spectrum of the schrodinger operator this second part focuses on the handling of the source terms which originate from adding a slowly varying exterior potential physically relevant examples are the occurrence of bloch oscillations in case it is linear a quadratic one modelling a confining field and the harmonic coulomb term resulting from the inclusion of a donor impurity inside an otherwise perfectly homogeneous lattice

data driven chance constrained stochastic program in this paper we study data driven chance constrained stochastic programs or more specifically stochastic programs with distributionally robust chance constraints dccs in a data driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters we consider a family of density based confidence sets based on a general phi divergence measure and formulate dcc from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set we derive an equivalent reformulation for dcc and show that it is equivalent to a classical chance constraint with a perturbed risk level we also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general phi divergence measures in several special cases our results can be strengthened such that we can derive closed form expressions for the perturbed risk levels in addition we show that the conservatism of dcc vanishes as the size of historical data goes to infinity furthermore we analyze the relationship between the conservatism of dcc and the size of historical data which can help indicate the value of data finally we conduct extensive computational experiments to test the performance of the proposed dcc model and compare various phi divergence measures based on a capacitated lot sizing problem with a quality of service requirement

semi classical asymptotics in solid state physics this article studies the schrodinger equation for an electron in a lattice of ions with an external magnetic field in a suitable physical scaling the ionic potential becomes rapidly oscillating and one can build asymptotic solutions for the limit of zero magnetic field by multiple scale methods from homogenization for the time dependent schrodinger equation this construction yields wave packets which follow the trajectories of the semiclassical model for the time independent equation one gets asymptotic eigenfunctions or quasimodes for the energy levels predicted by onsager s relation

a constraint sampling approach for multi stage robust optimization we propose a tractable approximation scheme for convex not necessarily linear multi stage robust optimization problems we approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computational cost through constraint randomization we obtain a priori probabilistic guarantees on the feasibility properties of the optimal decision rule by applying existing constraint sampling techniques to the semi infinite problem arising from the decision rule approximation we demonstrate that for a suitable choice of basis functions the approximation converges as the size of the basis and the number of sampled constraints tend to infinity the approach yields an algorithm parameterized in the basis size the probability of constraint violation and the confidence that this probability will not be exceeded these three parameters serve to tune the trade off between optimality and feasibility of the decision rules and the computational cost of the algorithm we assess the convergence and scalability properties of our approach in the context of two inventory management problems

on safe tractable approximations of chance constraints abstract a natural way to handle optimization problem with data affected by stochastic uncertainty is to pass to a chance constrained version of the problem where candidate solutions should satisfy the randomly perturbed constraints with probability at least 1 while being attractive from modeling viewpoint chance constrained problems as they are are in general computationally intractable in this survey paper we overview several simulation based and simulation free computationally tractable approximations of chance constrained convex programs primarily those of chance constrained linear conic quadratic and semidefinite programming

reliable approximations of probability constrained stochastic linear quadratic control here we consider a state constrained stochastic linear quadratic control problem this problem has linear dynamics and a quadratic cost and states are required to satisfy a probabilistic constraint in this paper the joint probabilistic constraint in the model is converted to a conservative deterministic constraint using a multi dimensional chebyshev bound a maximum volume inscribed ellipsoid problem is solved to obtain this probability bound using the probability bound we develop a recursive state feedback control algorithm for a special class of state constrained stochastic linear quadratic regulator lqr the performance of this approach is explored in a numerical example

exact results for the nonlinear diffusion equations delta u delta t delta delta x u 4 3 delta u delta x and delta u delta t delta delta x u 2 3 delta u delta x the author exploits local and non local symmetries of the two equations of the title and of some related equations some new exact solutions are derived and some specific boundary value problems of physical significance are considered

on safe tractable approximations of chance constrained linear matrix inequalities in the paper we consider the chance constrained version of an affinely perturbed linear matrix inequality lmi constraint assuming the primitive perturbations to be independent with light tail distributions e g bounded or gaussian constraints of this type playing a central role in chance constrained linear conic quadratic semidefinite programming are typically computationally intractable the goal of this paper is to develop a tractable approximation to these chance constraints our approximation is based on measure concentration results and is given by an explicit system of lmis thus the approximation is computationally tractable moreover it is also safe meaning that a feasible solution of the approximation is feasible for the chance constraint

ambiguous chance constrained problems and robust optimization in this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain we focus primarily on the special case where the uncertainty set of the distributions is of the form where p denotes the prohorov metric the ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure the main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability this result is established using the strassen dudley representation theorem that states that when the distributions of two random variables are close in the prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability we also show that the robust sampled problem can be solved efficiently both in theory and in practice

the exact feasibility of randomized solutions of uncertain convex programs many optimization problems are naturally delivered in an uncertain framework and one would like to exercise prudence against the uncertainty elements present in the problem in previous contributions it has been shown that solutions to uncertain convex programs that bear a high probability to satisfy uncertain constraints can be obtained at low computational cost through constraint randomization in this paper we establish new feasibility results for randomized algorithms specifically the exact feasibility for the class of the so called fully supported problems is obtained it turns out that all fully supported problems share the same feasibility properties revealing a deep kinship among problems of this class it is further proven that the feasibility of the randomized solutions for all other convex programs can be bounded based on the feasibility for the prototype class of fully supported problems the feasibility result of this paper outperforms previous bounds and is not improvable because it is exact for fully supported problems

spectral shift function and resonances for slowly varying perturbations of periodic schrodinger operators we study the spectral shift function s h and the resonances of the operator p h v x w hx here v is a periodic potential w a decreasing perturbation and h a small positive constant we give a representation of the derivative of s h related to the resonances of p h and we obtain a weyl type asymptotics of s h we establish an upper bound o h n 1 for the number of the resonances of p h lying in a disk of radius h

distributionally robust joint chance constraints with second order moment information we develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints assuming that only the first and second order moments as well as the support of the uncertain parameters are given it is known that robust chance constraints can be conservatively approximated by worst case conditional value at risk cvar constraints we first prove that this approximation is exact for robust individual chance constraints with concave or not necessarily concave quadratic constraint functions and we demonstrate that the worst case cvar can be computed efficiently for these classes of constraint functions next we study the worst case cvar approximation for joint chance constraints this approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations the tightness depends on a set of scaling parameters which can be tuned via a sequential convex optimization algorithm we show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the worst case cvar can be evaluated efficiently if the scaling parameters are kept constant we evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations

effective mass theorems for nonlinear schrodinger equations we consider time dependent nonlinear schrodinger equations subject to smooth lattice periodic potentials plus additional confining potentials slowly varying on the lattice scale after an appropriate scaling we study the homogenization limit for vanishing lattice spacing assuming well prepared initial data the resulting effective dynamics is governed by a homogenized nonlinear schrodinger equation with an effective mass tensor depending on the initially chosen bloch eigenvalue the given results rigorously justify the use of the effective mass formalism for the description of bose einstein condensates on optical lattices

a provisioning problem with stochastic payments abstract we consider the problem of determining the minimal requirement one must establish in order to meet a series of future random payments it is shown in a very general setting that this problem can be recast as a chance constrained model and how the technique of sample average approximation can be employed to find solutions we also use comonotonic theory to analyze analytical approximations in a restricted gaussian setting our numerical illustrations demonstrate that the sample average approximation is a viable and efficient way to solve the stated problem generally and outperforms the analytical approximations in passing we present a result that is related to stein s famous lemma stein 1981 and is of interest in itself

an integer programming approach for linear programs with probabilistic constraints linear programs with joint probabilistic constraints pclp are difficult to solve because the feasible region is not convex we consider a special case of pclp in which only the right hand side is random and this random vector has a finite distribution we give a mixed integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint we obtain two strengthened formulations as a byproduct of this analysis we obtain new results for the previously studied mixing set subject to an additional knapsack inequality we present computational results which indicate that by using our strengthened formulations instances that are considerably larger than have been considered before can be solved to optimality

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

risk and complexity in scenario optimization scenario optimization is a broad methodology to perform optimization based on empirical knowledge one collects previous cases called scenarios for the set up in which optimization is being performed and makes a decision that is optimal for the cases that have been collected for convex optimization a solid theory has been developed that provides guarantees of performance and constraint satisfaction of the scenario solution in this paper we open a new direction of investigation the risk that a performance is not achieved or that constraints are violated is studied jointly with the complexity as precisely defined in the paper of the solution it is shown that the joint probability distribution of risk and complexity is concentrated in such a way that the complexity carries fundamental information to tightly judge the risk this result is obtained without requiring extra knowledge on the underlying optimization problem than that carried by the scenarios in particular no extra knowledge on the distribution by which scenarios are generated is assumed so that the result is broadly applicable this deep seated result unveils a fundamental and general structure of data driven optimization and suggests practical approaches for risk assessment

on the connection between compression learning and scenario based single stage and cascading optimization problems we investigate the connections between compression learning and scenario based optimization we first show how to strengthen or relax the consistency assumption at the basis of compression learning and provide novel learnability conditions for the underlying algorithms we then consider different constrained optimization problems affected by uncertainty represented by means of scenarios we show that the compression learning perspective provides a unifying framework for scenario based optimization since the issue of providing guarantees on the probability of constraint violation reduces to a learning problem for an appropriately chosen algorithm that satisfies some consistency assumption to illustrate this we revisit the scenario approach within the developed context moreover using the compression learning machinery we provide novel results on the probability of constraint violation for the class of cascading optimization problems

a mathematical approach to the effective hamiltonian in perturbed periodic problems we describe a rigorous mathematical reduction of the spectral study for a class of periodic problems with perturbations which gives a justification of the method of effective hamiltonians in solid state physics we study the partial differential operators of the formp p hy y d y a hy onr n whenh 0 is small enough wherep x y is elliptic periodic iny with respect to some lattice and admits smooth bounded coefficients in x y a x is a magnetic potential with bounded derivatives we show that the spectral study ofp near any fixed energy level can be reduced to the study of a finite system ofh pseudodifferential operatorse x hd x h acting on some hilbert space depending on we then apply it to the study of the schrodinger operator when the electric potential is periodic and to some quasiperiodic potentials with vanishing magnetic field

effective dynamics for bloch electrons peierls substitution and beyond we consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential x and vector potential a x with x d and 1 we prove that associated to an isolated family of bloch bands there exists an almost invariant subspace of l 2 d and an effective hamiltonian governing the evolution inside this subspace to all orders in to leading order the effective hamiltonian is given through the peierls substitution we explicitly compute the first order correction from a semiclassical analysis of this effective quantum hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics

interaction of modulated pulses in the nonlinear schrodinger equation with periodic potential abstract we consider a cubic nonlinear schrodinger equation with periodic potential in a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several bloch bands is studied the notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic interaction of these pulses

a local trace formula for resonances of perturbed periodic schrodinger operators abstract we give a local trace formula for the pair p 1 h p 0 w hy p 0 where p 0 is a periodic schrodinger operator w is a decreasing perturbation and h is a small positive parameter we apply this result to establish the existence of h n resonances near some energy of p 0

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice iii from ab initio models to wkb for schrodinger poisson this work is concerned with the semiclassical approximation of the schrodinger poisson equation modeling ballistic transport in a 1d periodic potential by means of wkb techniques it is derived by considering the mean field limit of a n body quantum problem then k multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for pauli s exclusion principle numerical experiments display the behaviour of self consistent wave packets and screening effects