tight formulations for some simple mixed integer programs and convex objective integer programs we study the polyhedral structure of simple mixed integer sets that generalize the two variable set s z r 1 z 1 s z b these sets form basic building blocks that can be used to derive tight formulations for more complicated mixed integer programs for four such sets we give a complete description by valid inequalities and or an integral extended formulation and we also indicate what constraints can be added without destroying integrality we apply these results to provide tight formulations for certain piecewise linear convex objective integer programs and in a companion paper we exploit them to provide polyhedral descriptions and computationally effective mixed integer programming formulations for discrete lot sizing problems

transition probabilities for degenerate diffusions arising in population genetics we provide a detailed description of the structure of the transition probabilities and of the hitting distributions on boundary components of a manifold with corners for a degenerate strong markov process arising in population genetics the markov processes that we study are a generalization of the classical wright fisher process the main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward kolmogorov equation defined on a compact manifold with corners which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities

mixing mir inequalities with two divisible coefficients this article is a polyhedral study of a generalization of the mixing set where two different divisible coefficients are allowed for the integral variables our results generalize earlier work on mixed integer rounding mixing and extensions they also directly apply to applications such as production planning problems involving lower bounds or start ups on production when these are modeled as mixed integer linear programs we define a new class of valid inequalities and give two proofs that they suffice to describe the convex hull of this mixed integer set we give a characterization of each of the maximal faces of the convex hull as well as a closed form description of its extreme points and rays and show how to separate over this set in mathcal o n log n finally we give several extended formulations of polynomial size and study conditions under which adding certain simple constraints on the integer variables preserves our main result

compact formulations as a union of polyhedra we explore one method for finding the convex hull of certain mixed integer sets the approach is to break up the original set into a small number of subsets find a compact polyhedral description of the convex hull of each subset and then take the convex hull of the union of these polyhedra the resulting extended formulation is then compact its projection is the convex hull of the original set and optimization over the mixed integer set is reduced to solving a linear program over the extended formulation the approach is demonstrated on three different sets a continuous mixing set with an upper bound and a mixing set with two divisible capacities both arising in lot sizing and a single node flow model with divisible capacities that arises as a subproblem in network design

schauder estimates for a degenerate second order elliptic operator on a cube abstract in the present article we are concerned with a class of degenerate second order differential operators l a b defined on the cube 0 1 d with d 1 under suitable assumptions on the coefficients a and b among them the assumption of their holder regularity we show that the operator l a b defined on c 2 0 1 d is closable and its closure is m dissipative in particular its closure l a b is the generator of a c 0 semigroup of contractions on c 0 1 d and c 2 0 1 d is a core for it the proof of such result is obtained by studying the solvability in holder spaces of functions of the elliptic problem u x l a b u x f x x 0 1 d for a sufficiently large class of functions f

on splittable and unsplittable flow capacitated network design arc set polyhedra we study the polyhedra of splittable and unsplittable single arc set relaxations of multicommodity flow capacitated network design problems we investigate the optimization problems over these sets and the separation and lifting problems of valid inequalities for them in particular we give a linear time separation algorithm for the residual capacity inequalities 19 and show that the separation problem of c strong in equalities 7 is np hard but can be solved over the subspace of fractional variables only we introduce two classes of inequalities for the unsplittable flow problems we present a summary of computational experiments with a branch and cut algorithm for multicommodity flow capacitated network design problems to test the effectiveness of the results presented here empirically

mixing mixed integer inequalities mixed integer rounding mir inequalities play a central role in the development of strong cutting planes for mixed integer programs in this paper we investigate how known mir inequalities can be combined in order to generate new strong valid inequalities given a mixed integer region s and a collection of valid base mixed integer inequalities we develop a procedure for generating new valid inequalities for s the starting point of our procedure is to consider the mir inequalities related with the base inequalities for any subset of these mir inequalities we generate two new inequalities by combining or mixing them we show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed integer region associated with the base inequalities we discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed integer programming problems in particular we present examples for production planning capacitated facility location capacitated network design and multiple knapsack problems we also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs finally we study some extensions of this mixing procedure

the mixing mir set with divisible capacities we study the set s x y in re times z n x b j y j geq b j j 1 ldots n where b j b j in re 0 j 1 n and b 1 b n the set s generalizes the mixed integer rounding mir set of nemhauser and wolsey and the mixing mir set of gunluk and pochet in addition it arises as a substructure in general mixed integer programming mip such as in lot sizing despite its importance a number of basic questions about s remain unanswered including the tractability of optimization over s and how to efficiently find a most violated cutting plane valid for p conv s we address these questions by analyzing the extreme points and extreme rays of p we give all extreme points and extreme rays of p in the worst case the number of extreme points grows exponentially with n however we show that in some interesting cases it is bounded by a polynomial of n in such cases it is possible to derive strong cutting planes for p efficiently finally we use our results on the extreme points of p to give a polynomial time algorithm for solving optimization over s

c0 estimates and smoothness of solutions to the parabolic equation defined by kimura operators abstract kimura diffusions serve as a stochastic model for the evolution of gene frequencies in population genetics their infinitesimal generator is an elliptic differential operator whose second order coefficients matrix degenerates on the boundary of the domain in this article we consider the inhomogeneous initial value problem defined by generators of kimura diffusions and we establish c 0 estimates which allows us to prove that solutions to the inhomogeneous initial value problem are smooth up to the boundary of the domain where the operator degenerates even when the initial data is only assumed to be continuous

the continuous mixing polyhedron we analyze the polyhedral structure of the setspcmix s r z r xr nxz n s r j z j f j j 1 n andp cmix pcmixn s 0 the setp cmix is a natural generalization of the mixing set studied by pochet and wolsey 15 16 and ga nla k and pochet 8 and recently has been introduced by miller and wolsey 12 we introduce a new class of valid inequalities that has proven to be sufficient for describing conv pcmix we give an extended formulation of sizeo n xo n2 variables and constraints and indicate how to separate over conv pcmix ino n3 time finally we show how the mixed integer rounding mir inequalities of nemhauser and wolsey 14 and the mixing inequalities of ga nla k and pochet 8 constitute special cases of the cycle inequalities

polyhedra for lot sizing with wagner whitin costs we examine the single item lot sizing problem with wagner whitin costs over ann period horizon i e pt ht pt 1 fort 1 n 1 wherept ht are the unit production and storage costs in periodt respectively so it always pays to produce as late as possible

network formulations of mixed integer programs we consider mixed integer sets described by system of linear inequalities in which the constraint matrix a is totally unimodular the right hand side is arbitrary vector and a subset of the variables is required to be integer we show that the problem of checking nonemptiness of a set of this type is np complete even in the case in which the linear system describes mixed integer network flows with half integral requirement on the nodes this is in contrast to the case in which a is totally unimodular and contains at most two nonzeros per row in this case we provide an extended formulation for the convex hull of solutions whose constraint matrix is a dual network matrix with an integral right hand side vector the size of this formulation depends on the number of distinct fractional parts taken by the continuous variables in the extreme points of the convex hull of the given set when this number is polynomial in the dimension of a the extended formulation is of polynomial size if in addition the corresponding list of fractional parts can be computed efficiently then our result provides a polynomial algorithm for the optimization problem over these sets we show that there are instances for which this list is of exponential size and we also give conditions under which it is short and can be efficiently computed finally we show that these results provide a unified framework leading to polynomial size extended formulations for several generalizations of mixing sets and lot sizing sets studied in the last few years

mixing sets linked by bidirected paths recently there has been considerable research on simple mixed integer sets called mixing sets and closely related sets arising in uncapacitated and constant capacity lot sizing this in turn has led to study of more general sets called network dual sets for which it is possible to derive extended formulations whose projection gives the convex hull of the network dual set unfortunately this formulation cannot be used in general to optimize in polynomial time furthermore the inequalities defining the convex hull of a network dual set in the original space of variables are known only for some special cases here we study two new cases in which the continuous variables of the network dual set are linked by a bidirected path in the first case which is motivated by lot sizing problems with lost sales we provide a description of the convex hull as the intersection of the convex hulls of 2 n mixing sets where n is the number of continuous variables of the set however optimization is polynomial as only n 1 of the sets are required for any given objective function in the second case generalizing single arc flow sets we describe again the convex hull as the intersection of an exponential number of mixing sets and also give a combinatorial polynomial time separation algorithm

the feynman kac formula and harnack inequality for degenerate diffusions we study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in population genetics the so called generalized kimura diffusion operators epstein and mazzeo siam j math anal 42 2010 568 608 degenerate diffusion operators arising in population biology 2013 princeton university press applied mathematics research express 2016 our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower order coefficients and the proof of the scale invariant harnack inequality for nonnegative solutions to the kimura parabolic equation the stochastic representation of solutions that we establish is a considerable generalization of the classical results on feynman kac formulas concerning the assumptions on the degeneracy of the diffusion matrix the boundedness of the drift coefficients and the a priori regularity of the weak solutions

seismic data interpolation using a fast generalized fourier transform we have found a fast and efficient method for the interpolation of nonstationary seismic data the method uses the fast generalized fourier transform fgft to identify the space wavenumber evolution of nonstationary spatial signals at each temporal frequency the nonredundant nature of fgft renders a big computational advantage to this interpolation method a least squares fitting scheme is used next to retrieve the optimal fgft coefficients representative of the ideal interpolated data for randomly sampled data on a regular grid we seek a sparse representation of fgft coefficients to retrieve the missing samples in addition to interpolate the regularly sampled seismic data at a given frequency we use a mask function derived from the fgft coefficients of the low frequencies synthetic and real data examples can be used to examine the performance of the method

doubling properties of second order parabolic equations we prove the doubling property of l caloric measure corresponding to the second order parabolic equation in the whole space and in lipschitz domains for parabolic equations in the divergence form a weaker form of the doubling property follows easily from a recent result the backward harnack inequality and known estimates of green s function our method works for both the divergence and nondivergence cases moreover the backward harnack inequality and estimates of green s function are not needed in the course of proof

the convex hull of two core capacitated network design problems the network loading problem nlp is a specialized capacitated network design problem in which prescribed point to point demand between various pairs of nodes of a network must be met by installing loading a capacitated facility we can load any number of units of the facility on each of the arcs at a specified arc dependent cost the problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost

existence uniqueness and the strong markov property of solutions to kimura diffusions with singular drift motivated by applications to proving regularity of solutions to degenerate parabolic equations arising in population genetics we study existence uniqueness and the strong markov property of weak solutions to a class of degenerate stochastic differential equations the stochastic differential equations considered in our article admit solutions supported in the set 0 r and they are degenerate in the sense that the diffusion matrix is not strictly elliptic as the smallest eigenvalue converges to zero at a rate proportional to the distance to the boundary of the domain and the drift coefficients are allowed to have power type singularities in a neighborhood of the boundary of the domain under suitable regularity assumptions on the coefficients we establish existence of weak solutions that satisfy the strong markov property and uniqueness in law in the class of markov processes

degenerate stochastic differential equations and super markov chains we consider diffusions corresponding to the generator for continuous with i nonnegative we show uniqueness for the corresponding martingale problem under certain non degeneracy conditions on b i i and present a counter example when these conditions are not satisfied as a special case we establish uniqueness in law for some classes of super markov chains with state dependent branching rates and spatial motions

the uncapacitated lot sizing problem with sales and safety stocks we examine a variant of the uncapacitated lot sizing model of wagner whitin involving sales instead of fixed demands and lower bounds on stocks two extended formulations are presented as well as a dynamic programming algorithm and a complete description of the convex hull of solutions when the lower bounds on stocks are non decreasing over time it is possible to describe an extended formulation for the problem and a combinatorial separation algorithm for the convex hull of solutions finally when the lower bounds on stocks are constant a simpler polyhedral description is obtained for the case of wagner whitin costs

the fundamental solution to 1d degenerate diffusion equation with one sided boundary abstract in this work we adopt a combination of probabilistic approach and analytic method to study the fundamental solution to a certain type of one dimensional degenerate diffusion equation to be specific we consider a diffusion equation on 0 whose diffusion coefficient vanishes at the boundary 0 equipped with the cauchy initial data and the dirichlet boundary condition one such diffusion equation that has been extensively studied is the one whose diffusion coefficient vanishes linearly at 0 our main goal is to extend the study to cases when the diffusion coefficient has a general order of degeneracy with a primary focus on the fundamental solution to such a degenerate diffusion equation in particular we study the regularity properties of the fundamental solution near 0 and investigate how the order of degeneracy of the diffusion operator and the dirichlet boundary condition jointly affect these properties we also provide estimates for the fundamental solution and its derivatives near 0