on stabilizing effect of elasticity in the rayleigh taylor problem of stratified viscoelastic fluids abstract we investigate the stabilizing effect of elasticity in the rayleigh taylor rt problem of stratified immiscible viscoelastic fluids separated by a free interface and in the presence of a uniform gravitational field in a horizontally periodic domain where the velocities of the fluids are non slip on both upper and lower fixed flat boundaries while the internal surface tension is omitted we establish a discriminant c r for the stability of the stratified viscoelastic rt problem more precisely if c r 1 then the stratified viscoelastic rt equilibrium state is exponentially stable this means that a sufficiently large elasticity coefficient has stabilizing effect so that it can inhibit viscoelastic rt instability on the other hand if c r 1 then we show that the rt equilibrium state is linearly unstable in the hadamard sense moreover for the case of a nonhomogeneous incompressible viscoelastic fluid the condition c r 1 will lead to the nonlinear instability of the rt equilibrium state and this shows that the rt instability still occurs in viscoelastic fluids when the elasticity coefficient is small

an efficient trajectory method for probabilistic production inventory distribution problems we consider a supply chain operating in an uncertain environment the customers demand is characterized by a discrete probability distribution a probabilistic programming approach is adopted for constructing an inventory production distribution plan over a multiperiod planning horizon the plan does not allow the backlogging of the unsatisfied demand and minimizes the costs of the supply chain while enabling it to reach a prescribed nonstockout service level it is a strategic plan that hedges against undesirable outcomes and that can be adjusted to account for possible favorable realizations of uncertain quantities a modular integrated and computationally tractable method is proposed for the solution of the associated stochastic mixed integer optimization problems containing joint probabilistic constraints with dependent right hand side variables the concept of p efficiency is used to construct a finite number of demand trajectories which in turn are employed to solve problems with joint probabilistic constraints we complement this idea by designing a preordered set based preprocessing algorithm that selects a subset of promising p efficient demand trajectories finally to solve the resulting disjunctive mixed integer programming problem we implement a special column generation algorithm that limits the risk of congestion in the resources of the supply chain the methodology is validated on an industrial problem faced by a large chemical supply chain and turns out to be very efficient it finds a solution with a minimal integrality gap and provides substantial cost savings

lagrangian decomposition and mixed integer quadratic programming reformulations for probabilistically constrained quadratic programs abstract probabilistically constrained quadratic programming pcqp problems arise naturally from many real world applications and have posed a great challenge in front of the optimization society for years due to the nonconvex and discrete nature of its feasible set we consider in this paper a special case of pcqp where the random vector has a finite discrete distribution we first derive second order cone programming socp relaxation and semidefinite programming sdp relaxation for the problem via a new lagrangian decomposition scheme we then give a mixed integer quadratic programming miqp reformulation of the pcqp and show that the continuous relaxation of the miqp is exactly the socp relaxation this new miqp reformulation is more efficient than the standard miqp reformulation in the sense that its continuous relaxation is tighter than or at least as tight as that of the standard miqp we report preliminary computational results to demonstrate the tightness of the new convex relaxations and the effectiveness of the new miqp reformulation

kochen specker sets and the rank 1 quantum chromatic number the quantum chromatic number of a graph g is sandwiched between its chromatic number and its clique number which are well known np hard quantities we restrict our attention to the rank 1 quantum chromatic number q 1 g which upper bounds the quantum chromatic number but is defined under stronger constraints we study its relation with the chromatic number g and the minimum dimension of orthogonal representations g it is known that g q 1 g g we answer three open questions about these relations we give a necessary and sufficient condition to have g q 1 g we exhibit a class of graphs such that g q 1 g and we give a necessary and sufficient condition to have q 1 g g our main tools are kochen specker sets collections of vectors with a traditionally important role in the study of contextuality of physical theories and more recently in the quantification of quantum zero error capacities finally as a corollary of our results and a result by avis et al on the quantum chromatic number we give a family of kochen specker sets of growing dimension

global strong solution to the three dimensional stochastic incompressible magnetohydrodynamic equations the three dimensional incompressible magnetohydrodynamic equations with stochastic external forces are considered first the existence and uniqueness of local strong solution to the stochastic magnetohydrodynamic equations are proved when the external forces satisfy some conditions the proof is based on the contraction mapping principle stopping time and stochastic estimates the strong solution is a weak solution for the fluid variables with a given complete probability space and a given brownian motion then the global existence of strong solutions in probability is established if the initial data are sufficiently small and the noise is multiplicative and non degenerate

nonlinear convective instability in the compressible magnetic convection problem without heat conductivity abstract we investigate the convective instability problem for a full compressible viscous magnetohydrodynamic mhd fluid with zero resistivity and zero heat conductivity in the presence of a uniform gravitational force in a bounded domain r 3 first we establish an instability criterion for the magnetic convection problem and construct unstable solutions of linearized magnetic convection problem that grow in time in the sobolev space h 3 then based on the linearly unstable solutions we further modify the initial data of the linearly unstable solutions to be ones of the original magnetic convection problem finally using the local well posedness of classical solutions to the original magnetic convection problem and a modified bootstrap instability method we can construct unstable solutions for the original magnetic convection problem in the sense of hadamard under the instability criterion

pattern based modeling and solution of probabilistically constrained optimization problems we propose a new modeling and solution method for probabilistically constrained optimization problems the methodology is based on the integration of the stochastic programming and combinatorial pattern recognition fields it permits the fast solution of stochastic optimization problems in which the random variables are represented by an extremely large number of scenarios the method involves the binarization of the probability distribution and the generation of a consistent partially defined boolean function pdbf representing the combination f p of the binarized probability distribution f and the enforced probability level p we show that the pdbf representing f p can be compactly extended as a disjunctive normal form dnf the dnf is a collection of combinatorial p patterns each defining sufficient conditions for a probabilistic constraint to hold we propose two linear programming formulations for the generation of p patterns that can be subsequently used to derive a linear programming inner approximation of the original stochastic problem a formulation allowing for the concurrent generation of a p pattern and the solution of the deterministic equivalent of the stochastic problem is also proposed the number of binary variables included in the deterministic equivalent formulation is not an increasing function of the number of scenarios used to represent uncertainty results show that large scale stochastic problems in which up to 50 000 scenarios are used to describe the stochastic variables can be consistently solved to optimality within a few seconds

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

cell and bound algorithm for chance constrained programs with discrete distributions abstract chance constrained programing ccp is often encountered in real world applications when there is uncertainty in the data and parameters we consider in this paper a special case of ccp with finite discrete distributions we propose a novel approach for solving ccp the methodology is based on the connection between ccp and arrangement of hyperplanes by involving cell enumeration methods for an arrangement of hyperplanes in discrete geometry we develop a cell and bound algorithm to identify an exact solution to ccp which is much more efficient than branch and bound algorithms especially in the worst case furthermore based on the cell and bound algorithm a new polynomial solvable subclass of ccp is discovered we also find that the probabilistic version of the classical transportation problem is polynomially solvable when the number of customers is fixed we report preliminary computational results to demonstrate the effectiveness of our algorithm

on instability and stability of three dimensional gravity driven viscous flows in a bounded domain abstract we investigate the instability and stability of some steady states of a three dimensional nonhomogeneous incompressible viscous flow driven by gravity in a bounded domain of class c 2 when the steady density is heavier with increasing height i e the rayleigh taylor steady state we show that the steady state is linear unstable i e the linear solution grows in time in h 2 by constructing a standard energy functional and exploiting the modified variational method then by introducing a new energy functional and using a careful bootstrap argument we further show that the steady state is nonlinear unstable in the sense of hadamard when the steady density is lighter with increasing height we show with the help of a restricted condition imposed on steady density that the steady state is linearly globally stable and nonlinearly asymptotically stable in the sense of hadamard

on the rayleigh taylor instability for incompressible inviscid magnetohydrodynamic flows we study the rayleigh taylor instabolity for two incompressible immiscible inviscid magnetohydrodynamic mhd uids with zero resistivity evolving with a free interface in the presence of a uniform gravitational fleld we flrst construct the rayleigh taylor steady state solution with a denser uid lying above the light one then we turn to an analysis of the equations obtained from linearizing around such a steady state by solving a system of ordinary difierential equations we construct the normal mode solutions to the linearized problem that grow exponentially in time a fourier synthesis of these normal mode solutions allows us to construct solutions that grow arbitrarily quickly in the sobolev space h k thus leading to an ill posedness result for the linearized problem in the sense of hadamard using these pathological solutions we can then demonstrate the ill posedness of the original non linear problem in some sense

compactness of weak solutions to the three dimensional compressible magnetohydrodynamic equations abstract the compactness of weak solutions to the magnetohydrodynamic equations for the viscous compressible heat conducting fluids is considered in both the three dimensional space r 3 and the three dimensional periodic domains the viscosities the heat conductivity as well as the magnetic coefficient are allowed to depend on the density and may vanish on the vacuum this paper provides a different idea from x hu d wang global solutions to the three dimensional full compressible magnetohydrodynamic flows comm math phys 2008 in press to show the compactness of solutions of viscous compressible heat conducting magnetohydrodynamic flows derives a new entropy identity and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations

instability solutions for the rayleigh taylor problem of non homogeneous viscoelastic fluids in bounded domains abstract it is well known that the rayleigh taylor rt problem of an inhomogeneous viscoelastic fluid defined on a bounded domain is unstable if the elasticity coefficient is less than some threshold c in this paper we rigorously prove the existence of a unique unstable strong solution in the sense of l 1 norm for the rt problem in lagrangian coordinates based on a bootstrap instability method when c applying an inverse transformation of lagrangian coordinates to the obtained unstable solution we can further get a unique unstable solution for the rt problem of inhomogeneous viscoelastic fluids in eulerian coordinates

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

convex approximations of chance constrained programs we consider a chance constrained problem where one seeks to minimize a convex objective over solutions satisfying with a given close to one probability a system of randomly perturbed convex constraints this problem may happen to be computationally intractable our goal is to build its computationally tractable approximation i e an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem we construct a general class of such convex conservative approximations of the corresponding chance constrained problem moreover under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent of each other random variables we build a large deviation type approximation referred to as bernstein approximation of the chance constrained problem this approximation is convex and efficiently solvable we propose a simulation based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the bernstein and well known scenario approximation approaches finally we extend our construction to the case of ambiguous chance constrained problems where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly while the chance constraint should be satisfied for every distribution given by this set

a sample approximation approach for optimization with probabilistic constraints we study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample we show that such a sample approximation problem with a risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast this leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound we then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability once again we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem finally we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints

on mixing sets arising in chance constrained programming the mixing set with a knapsack constraint arises in deterministic equivalent of chance constrained programming problems with finite discrete distributions we first consider the case that the chance constrained program has equal probabilities for each scenario we study the resulting mixing set with a cardinality constraint and propose facet defining inequalities that subsume known explicit inequalities for this set we extend these inequalities to obtain valid inequalities for the mixing set with a knapsack constraint in addition we propose a compact extended reformulation with polynomial number of variables and constraints that characterizes a linear programming equivalent of a single chance constraint with equal scenario probabilities we introduce a blending procedure to find valid inequalities for intersection of multiple mixing sets we propose a polynomial size extended formulation for the intersection of multiple mixing sets with a knapsack constraint that is stronger than the original mixing formulation we also give a compact extended linear program for the intersection of multiple mixing sets and a cardinality constraint for a special case we illustrate the effectiveness of the proposed inequalities in our computational experiments with probabilistic lot sizing problems

instability of periodic bgk equilibria a collisionless plasma is described by the vlasov poisson equations the bgk equilibria were proposed in 1957 as the simplest spatially dependent equilibria since that time the question of their instability has been an important open problem we prove that certain periodic bgk equilibria are unstable 1995 john wiley sons inc

probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra we consider stochastic programming problems with probabilistic constraints involving random variables with discrete distributions they can be reformulated as large scale mixed integer programming problems with knapsack constraints using specific properties of stochastic programming problems and bounds on the probability of the union of events we develop new valid inequalities for these mixed integer programming problems we also develop methods for lifting these inequalities these procedures are used in a general iterative algorithm for solving probabilistically constrained problems the results are illustrated with a numerical example

holder and lipschitz stability of solution sets in programs with probabilistic constraints we study perturbations of a stochastic program with a probabilistic constraint and r concave original probability distribution first we improve our earlier results substantially and provide conditions implying holder continuity properties of the solution sets w r t the kolmogorov distance of probability distributions secondly we derive an upper lipschitz continuity property for solution sets under more restrictive conditions on the original program and on the perturbed probability measures the latter analysis applies to linear quadratic models and is based on work by bonnans and shapiro the stability results are illustrated by numerical tests showing the different asymptotic behaviour of parametric and nonparametric estimates in a program with a normal probabilistic constraint

threshold boolean form for joint probabilistic constraints with random technology matrix we develop a new modeling and solution method for stochastic programming problems that include a joint probabilistic constraint in which the multirow random technology matrix is discretely distributed we binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined boolean function pdbf representing the probabilistic constraint we then construct a tight threshold boolean minorant for the pdbf any separating structure of the tight threshold boolean minorant defines sufficient conditions for the satisfaction of the probabilistic constraint and takes the form of a system of linear constraints we use the separating structure to derive three new deterministic formulations for the studied stochastic problem and we derive a set of strengthening valid inequalities a crucial feature of the new integer formulations is that the number of integer variables does not depend on the number of scenarios used to represent uncertainty the computational study based on instances of the stochastic capital rationing problem shows that the mixed integer linear programming formulations are orders of magnitude faster to solve than the mixed integer nonlinear programming formulation the method integrating the valid inequalities in a branch and bound algorithm has the best performance