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

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

tight second stage formulations in two stage stochastic mixed integer programs we study two stage stochastic mixed integer programs tss mips with integer variables in the second stage we show that under suitable conditions the second stage mips can be convexified by adding parametric cuts a priori as special cases we extend the results of miller and wolsey math program 98 2003 pp 73 88 to tss mips furthermore we consider second stage programs that are generalizations of the well known mixing and continuous mixing set or certain piecewise linear convex objective integer programs these results allow us to relax the integrality restrictions on the second stage integer variables without effecting the integrality of the optimal solution of the tss mip we also use four variants of the two stage stochastic capacitated lot sizing problems as test problems for computational experiments and present tight second stage formulations for these problems our computational results show that adding parametric inequalities that a priori convexify the second stage formulation si

infinitely many solutions for a p x laplacian equation in rn abstract this paper deals with a p x laplacian equation in r n by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish the existence of infinitely many distinct homoclinic radially symmetric solutions whose w 1 p x r n norms tend to zero to infinity respectively under weaker hypotheses about nonlinearity at zero at infinity respectively

oscillation criteria for half linear elliptic inequalities with p x laplacians via riccati method abstract riccati inequality for half linear elliptic inequalities with p x laplacians is established and oscillation criteria are derived by using the riccati inequality our method is to reduce oscillation problems for half linear elliptic inequalities with p x laplacians to one dimensional riccati inequality with variable exponents generalizations to more general elliptic inequalities with p x laplacians are also discussed

existence of radial solutions for p x laplacian equations in rn abstract in this paper we obtain sufficient conditions of the existence of radial solutions for p x laplacian equations in r n

weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions abstract in this paper the weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions is derived through asymptotic expansions and justified by employing the energy estimates compared with the work of g q chen c d levermore and t p liu 1994 comm pure appl math 47 787 830 the main difficulty we confront in our analyzes lies in the fact that the problem of the existence of an convex entropy which was essential in the proof of their estimate 5 29 for the system under our consideration is still an open problem since their theorem 3 2 was proved only for the 2 2 case

valid inequalities based on simple mixed integer sets in this paper we use facets of simple mixed integer sets with three variables to derive a parametric family of valid inequalities for general mixed integer sets we call these inequalities two step mir inequalities as they can be derived by applying the simple mixed integer rounding mir principle of wolsey 1998 twice the two step mir inequalities define facets of the master cyclic group polyhedron of gomory 1969 in addition they dominate the strong fractional cuts of letchford and lodi 2002

oscillation criteria of kamenev type for pde with p laplacian some oscillation criteria are given for the partial differential equation with p laplacian and the nonlinearity of the emden fowler typediv email protected email protected p email protected u c x u p 2u 0 p 1 the results are extension of integral averaging technique of kamenev and improve the result of usami

relaxed most negative cycle and most positive cut canceling algorithms for minimum cost flow this paper presents two new scaling algorithms for the minimum cost network flow problem one a primal cycle canceling algorithm the other a dual cut canceling algorithm both algorithms scale a relaxed optimality parameter and create a second inner relaxation the primal algorithm uses the inner relaxation to cancel a most negative node disjoint family of cycles w r t the scaled parameter the dual algorithm uses it to cancel most positive cuts w r t the scaled parameter we show that in a network with n nodes and m arcs both algorithms need to cancel only omn objects per scaling phase furthermore we show how to efficiently implement both algorithms to yield weakly polynomial running times that are as fast as any other cycle or cut canceling algorithms our algorithms have potential practical advantages compared to some other canceling algorithms as well along the way we give a comprehensive survey of cycle and cut canceling algorithms for min cost flow we also clarify the formal duality between cycles and cuts

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

a picone identity for half linear elliptic equations and its applications to oscillation theory abstract a picone identity is established for a class of half linear elliptic operators and sturmian comparison and oscillation results are derived on the basis of the picone identity wirtinger inequalities and riccati inequalities are also obtained as corollaries of the picone identity and a new class of half linear elliptic equations is introduced

interval criteria for oscillation of second order linear ordinary differential equations abstract new oscillation criteria are established for the equation py qy 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are more natural according to the sturm separation theorem and sharper than some previous results and can be applied to extreme cases such as t 0 q t dt

lot sizing with constant batches formulation and valid inequalities we consider the classical lot sizing problem with constant production capacities lcc and a variant in which the capacity in each period is an integer multiple of some basic batch size lcb we first show that the classical dynamic program for lcc simplifies for lcb leading to an on2 min n c algorithm where n is the number of periods and c the batch size using this new algorithm we show how to formulate both problems as linear programs with on3 variables and constraints a class of so called k l s i inequalities are described for lcb which capture both the dynamic nature of the problem as well as the capacity aspects for lcb we prove that these inequalities are the only facet defining inequalities of a certain form for lcc we show that these inequalities include all the known classes of valid inequalities finally we discuss several open questions and possible extensions

negative cycle detection algorithms we study the problem of finding a negative length cycle in a network an algorithm for the negative cycle problem combines a shortest path algorithm and a cycle detection strategy we survey cycle detection strategies study various combinations of shortest path algorithms and cycle detection strategies and find the best combinations one of out discoveries is that a cycle detection strategy of tarjan greatly improves computational performance of a classical shortest path algorithm making it competitive with the fastest known algorithms on a wide range of problems as a part of out study we develop problem families for testing negative cycle algorithms

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

n step mingling inequalities new facets for the mixed integer knapsack set the n step mixed integer rounding mir inequalities of kianfar and fathi math program 120 2 313 346 2009 are valid inequalities for the mixed integer knapsack set that are derived by using periodic n step mir functions and define facets for group problems the mingling and 2 step mingling inequalities of atamturk and gunluk math program 123 2 315 338 2010 are also derived based on mir and they incorporate upper bounds on the integer variables it has been shown that these inequalities are facet defining for the mixed integer knapsack set under certain conditions and generalize several well known valid inequalities in this paper we introduce new classes of valid inequalities for the mixed integer knapsack set with bounded integer variables which we call n step mingling inequalities for positive integer n these inequalities incorporate upper bounds on integer variables into n step mir and therefore unify the concepts of n step mir and mingling in a single class of inequalities furthermore we show that for each n the n step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions for n 2 we extend the results of atamturk and gunluk on facet defining properties of 2 step mingling inequalities for n 3 we present new facets for the mixed integer knapsack set as a special case we also derive conditions under which the n step mir inequalities define facets for the mixed integer knapsack set in addition we show that n step mingling can be used to generate new valid inequalities and facets based on covers and packs defined for mixed integer knapsack sets

generating facets for finite master cyclic group polyhedra using n step mixed integer rounding functions the n step mixed integer rounding mir functions generate n step mir inequalities for mip problems and are facets for the infinite group problems we show that the n step mir functions also directly generate facets for the finite master cyclic group polyhedra especially in many cases where the breakpoints of the n step mir function are not necessarily at the elements of the group hence the linear interpolation of the facet coefficients obtained has more than two slopes

a multiplicity result for p x laplacian problem in rn abstract in this paper we discuss the solutions of the following p x laplacian problem div u p x 2 u u p x 2 u f x u x r n where 1 p p x p n based on the theory of the variable exponent sobolev spaces w 1 p x r n we get that there exist at least two non trivial weak solutions

generalized mixed integer rounding inequalities facets for infinite group polyhedra we present a generalization of the mixed integer rounding mir approach for generating valid inequalities for mixed integer programming mip problems for any positive integer n we develop n facets for a certain n 1 dimensional single constraint polyhedron in a sequential manner we then show that for any n the last of these facets which we call the n step mir facet can be used to generate a family of valid inequalities for the feasible set of a general mixed ip constraint which we refer to as the n step mir inequalities the gomory mixed integer cut and the 2 step mir inequality of dash and gunluk math program 105 1 29 53 2006 are the first two families corresponding to n 1 2 respectively the n step mir inequalities are easily produced using periodic functions which we refer to as the n step mir functions none of these functions dominates the other on its whole period finally we prove that the n step mir inequalities generate two slope facets for the infinite group polyhedra and hence are potentially strong

using cuts for mixed integer knapsack sets to generate cuts for mixed integer polyhedral conic sets based on a bijective mapping between two mixed integer sets we introduce a new perspective on developing cuts for the mixed integer polyhedral conic mipc set by establishing a one to one correspondence between the cuts for this set and those for a related mixed integer knapsack mik set the face facet defining properties of the corresponding cuts are identical for their respective sets we further show that the cut generation approach for the mipc set resulting from this new perspective always produces cuts that dominate those generated based on any of the two individual mik constraints corresponding to the mipc constraint our computational results show this dominance can be quite significant as a special case of this new perspective the conic mir inequality of atamturk and narayanan for the mipc set and its properties can be directly derived from the mir inequality for the mik set and its properties we also generalize these cuts to the n step conic mir inequalities which are directly derived form the n step mir inequalities for the mik set

existence of solution for a degenerate p x laplacian equation in rn abstract using variational methods we establish the existence of nontrivial solutions for the following class of problems div a x u p x 2 u u p x 2 u k x u q x 2 u in r n where p q a k r n r are continuous functions satisfying some conditions and the set x r n a x 0 is not empty

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

n n step cycle inequalities facets for continuous multi mixing set and strong cuts for multi module capacitated lot sizing problem in this paper we introduce a generalization of the continuous mixing set which we refer to as the continuous multi mixing set this set is closely related to the feasible set of the multi module capacitated lot sizing mml problem with out backlogging we present a family of valid inequalities for the continuous multi mixing set and identify conditions under which they are facet defining the cycle inequalities n n step mir inequalities and mixed n n step mir inequalities are special cases of these inequalities we also present an exact separation algorithm for our inequalities we then use these inequalities to generate valid inequalities for mml with out backlogging our computational results show that our cuts are very effective in solving the mml instances resulting in substantial reduction in the integrality gap number of nodes and total solution time

picone identity for quasilinear elliptic equations with p x laplacians and sturmian comparison theory picone identities for quasilinear elliptic equations with p x laplacians are established and sturmian comparison theory is developed on the basis of picone identities as specializations various elliptic equations such as linear elliptic equations half linear elliptic equations with p laplacians half linear elliptic equations with p x laplacians pseudo p laplacian with variable power are discussed generalizations are also investigated

decomposition algorithms for two stage distributionally robust mixed binary programs in this paper we introduce and study a two stage distributionally robust mixed binary problem tsdr mbp in which the random parameters follow the worst case distribution belonging to an uncertain

mingling mixed integer rounding with bounds mixed integer rounding mir is a simple yet powerful procedure for generating valid inequalities for mixed integer programs when used as cutting planes mir inequalities are very effective for mixed integer programming problems with unbounded integer variables for problems with bounded integer variables however cutting planes based on lifting techniques appear to be more effective this is not surprising as lifting techniques make explicit use of the bounds on variables whereas the mir procedure does not in this paper we describe a simple procedure which we call mingling for incorporating variable bound information into mir by explicitly using the variable bounds the mingling procedure leads to strong inequalities for mixed integer sets with bounded variables we show that facets of mixed integer knapsack sets derived earlier by superadditive lifting techniques can be obtained by the mingling procedure in particular the mingling inequalities developed in this paper subsume the continuous cover and reverse continuous cover inequalities of marchand and wolsey math program 85 15 33 1999 as well as the continuous integer knapsack cover and pack inequalities of atamturk math program 98 145 175 2003 ann oper res 139 21 38 2005 in addition mingling inequalities give a generalization of the two step mir inequalities of dash and gunluk math program 105 29 53 2006 under some conditions

on kamenev type oscillation theorems for second order differential equations with damping abstract oscillation criteria for the nonlinear second order ordinary differential equation with damping x p t x f x 0 are given the results are extensions of integral averaging technique of kamenev and include earlier results of yeh yan and chen