p matrices are just sufficient abstract the classes of sufficient matrices and of p matrices have recently arisen in connection with the linear complementarity problem it is known that p matrices are sufficient we show that conversely every sufficient matrix is a p matrix

high order infeasible interior point methods for solving sufficient linear complementarity problems in this paper we develop systematically infeasible interior point methods of arbitrarily high order for solving horizontal linear complementarity problems that are sufficient in the sense ofcottle pang and venkateswaran 1989 the results apply to degenerate problems and problem shaving no strictly complementary solution variants of these methods are described that eventually avoid recentering steps and for which all components of the approximate solutions converge superlinearly at a high order and other variants which even terminate with a solution of the complementarity problem after finitely many steps

sufficient matrices and the linear complementarity problem abstract we pose and answer two questions about solutions of the linear complementarity problem lcp the first question is concerned with the conditions on a matrix m which guarantee that for every vector q the solutions of the lcp q m are identical to the karush kuhn tucker points of the natural quadratic program associated with q m in answering this question we introduce the class of row sufficient matrices the transpose of such a matrix is what we call column sufficient the latter matrices turn out to furnish the answer to our second question which asks for the conditions on m under which the solution set of q m is convex for every q in addition to these two main results we discuss the connections of these twonew matrix classes with other well known matrix classes in linear complementarity theory

on the uniqueness of solutions to linear complementarity problems this paper characterizes the classu of all realn n matricesm for which the linear complementarity problem q m has a unique solution for all realn vectorsq interior to the conek m of vectors for which q m has any solution at all it is shown that restricting the uniqueness property to the interior ofk m is necessary because whenu the problem q m has infinitely many solutions ifq belongs to the boundary of intk m it is shown thatm must have nonnegative principal minors whenu andk m is convex finally it is shown that whenm has nonnegative principal minors only one of which is 0 andk m r n thenu andk m is a closed half space

periodic orbits and non integrability of henon heiles systems we apply the averaging theory of second order to study the periodic orbits for a generalized henon heiles system with two parameters which contains the classical henon heiles system two main results are shown the first result provides sufficient conditions on the two parameters of these generalized systems which guarantee that at any positive energy level the hamiltonian system has periodic orbits these periodic orbits form in the whole phase space a continuous family of periodic orbits parameterized by the energy the second result shows that for the non integrable henon heiles systems in the sense of liouville arnol d which have the periodic orbits analytically found with averaging theory cannot exist any second first integral of class in particular for any second first integral of class we prove that the classical henon heiles system and many generalizations of it are not integrable in the sense of liouville arnol d moreover the tools we use for studying the periodic orbits and the non liouville arnol d integrability can be applied to hamiltonian systems with an arbitrary number of degrees of freedom

a class of linear complementarity problems solvable in polynomial time we describe a condition number for the linear complementarity problem lcp which characterizes the degree of difficulty for its solution when a potential reduction algorithm is used consequently we develop a class of lcps solvable in polynomial time the result suggests that the convexity or positive semidefiniteness of the lcp may not be the basic issue that separates lcps solvable and not solvable in polynomial time

analysis of infeasible interior point paths arising with semidefinite linear complementarity problems we consider semidefinite monotone linear complementarity problems sdlcp in the space equation here n of real symmetric n n matrices equipped with the cone equation here n of all symmetric positive semidefinite matrices one may define weighted using any m equation here n as weight infeasible interior point paths by replacing the standard condition xy ri r 0 that defines the usual central path by xy yx 2 rm under some mild assumptions the most stringent is the existence of some strictly complementary solution of sdlcp these paths have a limit as r 0 and they depend analytically on all path parameters such as r and m even at the limit point r 0

criteria for sufficient matrices abstract column sufficient row sufficient and sufficient matrices have recently arisen in connection with the linear complementarity problem we review and supplement the basic theory of these matrix classes propose new criteria for identifying them and compare these criteria with the existing ones our main mathematical tool is principal pivoting

new characterizations of row sufficient matrices in dealing with a linear complementarity problem much depends on knowing that the matrix through which the particular lcp is defined belongs to a suitable matrix class two such classes are su the so called sufficient matrices and l which were introduced in r w cottle j s pang v venkateswaran sufficient matrices and the linear complementarity problem linear algebra appl 114 115 1989 231 249 b c eaves the linear complementarity problem manage sci 17 1971 612 634 respectively in an earlier article i adler r w cottle s verma sufficient matrices belong to l math prog 106 2006 391 401 the authors proved that su is a subclass of l by definition the class su is the intersection of two distinct classes rsu the row sufficient matrices and csu the column sufficient matrices in the present work we strengthen the aforementioned inclusion by showing that all row sufficient matrices belong to l using what we call structural properties of certain matrix classes we add to the existing characterizations of rsu in r w cottle s m guu two characterizations of sufficient matrices linear algebra appl 170 1992 65 74 s m guu r w cottle on a subclass of p0 linear algebra appl 223 224 1995 325 335 h valiaho criteria for sufficient matrices linear algebra appl 233 1996 109 129 this line of inquiry was inspired by asking what must be true of a row sufficient l matrix we establish three new characterizations of rsu in terms of the matrix classes l e0 and q0 and the structural properties of sign change invariance completeness and fullness the new characterizations of rsu provide new characterizations of su by adjoining a fourth structural property we call reflectiveness

characterizing q linear transformations for semidefinite linear complementarity problems abstract in this paper we introduce a new class called f of linear transformations defined from the space of real n n symmetric matrices into itself within this new class we show the equivalence between q and q b transformations we also provide conditions under which a linear transformation belongs to f moreover this class when specialized to square matrices of size n turns out to be the largest class of matrices for which such equivalence holds true in the context of standard linear complementary problems

sufficient matrices belong to l in this paper we establish a significant matrix class inclusion that seems to have been overlooked in the literature of the linear complementarity problem we show that p the class of sufficient matrices is a subclass of l in the course of demonstrating this inclusion we introduce other new matrix classes that forge interesting new connections between known matrix classes

a constructive characterization of q 0 matrices with nonnegative principal minors in 2 we characterized the class of matrices with nonnegative principla minors for which the linear complementarity problem always has a solution that class is contained in the one we study here our main result gives a finitely testable set of necessary and sufficient conditions under which a matrix with nonnegative principal minors has the property that if a corresponding linear complementarity problem is feasible then it is solvable in short we constructively characterize the matrix class known asqo po

on a new homotopy continuation trajectory for nonlinear complementarity problems most known continuation methods for p 0 complementarity problems require some restrictive assumptions such as the strictly feasible condition and the properness condition to guarantee the existence and the boundedness of certain homotopy continuation trajectory to relax such restrictions we propose a new homotopy formulation for the complementarity problem based on which a new homotopy continuation trajectory is generated for p 0 complementarity problems the most promising feature of this trajectory is the assurance of the existence and the boundedness of the trajectory under a condition that is strictly weaker than the standard ones used widely in the literature of continuation methods particularly the often assumed strictly feasible condition is not required here when applied to p complementarity problems the boundedness of the proposed trajectory turns out to be equivalent to the solvability of the problem and the entire trajectory converges to the unique least element solution provided that it exists moreover for monotone complementarity problems the whole trajectory always converges to a least 2 norm solution provided that the solution set of the problem is nonempty the results presented in this paper can serve as a theoretical basis for constructing a new path following algorithm for solving complementarity problems even for the situations where the solution set is unbounded

complementarity properties of singular m matrices abstract for a matrix a whose off diagonal entries are nonpositive its nonnegative invertibility namely that a is an invertible m matrix is equivalent to a being a p matrix which is necessary and sufficient for the unique solvability of the linear complementarity problem defined by a this in turn is equivalent to the statement that a is strictly semimonotone in this paper an analogue of this result is proved for singular symmetric z matrices this is achieved by replacing the inverse of a by the group generalized inverse and by introducing the matrix classes of strictly range semimonotonicity and range column sufficiency a recently proposed idea of p matrices plays a pivotal role some interconnections between these matrix classes are also obtained

error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem consider the affine variational inequality problem it is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point this bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem this latter result improves upon a recent result of luo and tseng that further assumes the problem to be monotone

almost copositive matrices abstract a real n n matrix is termed almost copositive if it is not copositive but all its principal submatrices of order n 1 are a real copositive n n matrix is called almost copositive plus almost strictly copositive if it is not copositive plus strictly copositive but all its principal submatrices of order n 1 are such matrices are of crucial importance in deriving criteria for copositivity the properties of these matrix classes are investigated the emphasis being on the less known class of almost copositive plus matrices the basic mathematical tools used are principal pivoting and quadratic programming

pseudomonotone and copositive star matrices abstract we study general and complementarity properties of matrices which are either pseudomonotone or copositive star on a closed convex cone in particular we show that if a matrix is pseudomonotone on the nonnegative orthant of r n then it belongs to p 0 q 0 we also show that if a matrix t is copositive star on a closed convex cone k then the linear complementarity problem lcp t k q is solvable for all q in r n iff zero is the only solution of the homogeneous problem lcp t k 0 we characterize pseudomonotonicity of a matrix t by the nonnegativity of the quadratic form tx x on a certain set

the basic theorem of complementarity revisited the basic theorm of linear complementarity was stated in a 1971 paper 6 by b c eaves who credited c e lemke for giving a constructive proof based on his almost complementary pivot algorithm this theorem asserts that associated with an arbitrary linear complementarity problem a certain augmented problem always possesses a solution many well known existence results pertaining to the linear complementarity problem are consequences of this fundamental theorem in this paper we explore some further implications of the basic theorem of complementarity and derive new existence results for the linear complementarity problem based on these results conditions for the existence of a solution to a linear complementarity problem with a fully semimonotone matrix are examined the class of the linear complementarity problems with ag matrix is also investigated

integer solution for linear complementarity problem we consider the problem of finding an integer solution to a linear complementarity problem we introduce the class i of matrices for which the corresponding linear complementarity problem has an integer complementary solution for every vector q for which it has a solution strong principal unimodularity forms a sufficient condition for inclusion in the class i it is also shown to be necessary for some well known classes of matrices including the class of positive semidefinite matrices this is used in deriving necessary and sufficient conditions for a convex quadratic program to have an integer optimal solution for all b and c for which it has an optimal solution characterizations are also derived for some other well known classes of matrices in linear complementarity to belong to the class i in the end a peeling algorithm for finding integer solution for linear complementarity problems is given and related results are derived

non interleaving bisimulation equivalences on basic parallel processes we show polynomial time algorithms for deciding hereditary history preserving bisimilarity in o n 3logn and history preserving bisimilarity in o n 6 on the class basic parallel processes the latter algorithm also decides a number of other non interleaving behavioural equivalences e g distributed bisimilarity which are known to coincide with history preserving bisimilarity on this class the common general scheme of both algorithms is based on a fixpoint characterization of the equivalences for tree like labelled event structures the technique for realizing the greatest fixpoint computation in the case of hereditary history preserving bisimilarity is based on the revealed tight relationship between equivalent tree like labelled event structures in the case of history preserving bisimilarity a technique of deciding classical bisimilarity on acyclic petri nets is used

decidability of performance equivalence for basic parallel processes we study an extension of the class of basic parallel processes bpp in which actions are durational and urgent and parallel components have independent local clocks the main result is decidability of strong bisimilarity known also as performance equivalence in this class this extends the earlier decidability result for plain bpp by christensen et al our decision procedure is based on decidability of the validity problem for presburger arithmetic we prove also polynomial complexity in positive duration fragment thus properly extending a previous result by berard et al both ill timed and well timed semantics are treated

on singular n0 matrices and the class q abstract the class of n and n 0 matrices arises in the theory of global univalence of functions multivariate analysis and in complementarity theory in this paper we study singular n 0 matrices with q property

on hidden z matrices and the linear complementarity problem abstract in this paper we explore various matrix theoretic aspects of the hidden z class and demonstrate how the concept of principal pivot transform can be effectively used to extend many existing results in fact we revisit various results obtained for hidden z class by mangasarian cottle and pang in context of solving linear complementarity problems as linear programs we identify hidden z matrices of special category and discuss the number of solutions of the associated linear complementarity problems we also present game theoretic interpretation of various results related to hidden z class and obtain proofs following the game theoretic approach of raghavan for a subclass of z matrices

on the transpose of a pseudomonotone matrix and the lcp abstract in the study of linear complementary problem it is known that pseudomonotone matrices belong to the class p 0 q 0 in this note we show that under certain conditions such as invertibility or normality the transpose of a pseudomonotone matrix belongs to the class q 0

criss cross methods a fresh view on pivot algorithms criss cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution the first finite criss cross method was invented by chang terlaky and wang independently unlike the simplex method that follows a monotonic edge path on the feasible region the trace of a criss cross method is neither monotonic with respect to the objective function nor feasibility preserving the main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss cross pivot methods a recent result on the existence of a short admissible pivot path to an optimal basis is given indicating shortest pivot paths from any basis might be indeed short for criss cross type algorithms the origins and the history of criss cross methods are also touched upon

determining the handicap of a sufficient matrix abstract any linear complementarity problem with a sufficient matrix can be solved by means of the unified interior point method the complexity bound of the method is the better the smaller the so called handicap of the matrix is we propose a method for determining the handicap of a sufficient matrix and show that a sufficient matrix and its transpose have the same handicap

an alternative theorem for generalized variational inequalities and solvability of nonlinear quasi p m complementarity problems in this paper an alternative theorem and hence a sufficient solution condition is established for generalized variational inequality problems the concept of exceptional family for generalized variational inequality is introduced this concept is general enough to include as special cases the notions of exceptional family of elements and the d orientation sequence for continuous functions particularly we apply the alternative theorem for investigating the solvability of the nonlinear complementarity problems with so called quasi p m maps which are broad enough to encompass the quasi monotone maps and p maps as the special cases an existence theorem for this class of complementarity problems is established which significantly generalizes several previous existence results in the literature

on pseudomatroid property of matrices abstract a necessary and sufficient condition for a matrix to have pseudomatroid property is presented pseudosymmetric matrices which possess pseudomatroid property are introduced as a generalization of quasisymmetric matrices considered by a bouchet a duchamp linear algebra and appl 146 1991 67 78 principal unimodularity and perfect balancedness of these matrices are investigated a result on nonbinary pseudomatroids is presented in the end

the principal pivoting method revisited the principal pivoting method ppm for the linear complementarity problem lcp is shown to be applicable to the class of lcps involving the newly identified class of sufficient matrices

periodic orbits and nonintegrability of generalized classical yang mills hamiltonian systems the averaging theory of first order is applied to study a generalized yang mills system with two parameters two main results are proved first we provide sufficient conditions on the two parameters of the generalized system to guarantee the existence of continuous families of isolated periodic orbits parameterized by the energy and these families are given up to first order in a small parameter second we prove that for the nonintegrable classical yang mills hamiltonian systems in the sense of liouville arnold which have the isolated periodic orbits found with averaging theory cannot exist in any second first integral of class c1 this is important because most of the results about integrability deals with analytic or meromorphic integrals of motion

more on positive subdefinite matrices and the linear complementarity problem abstract in this article we consider positive subdefinite matrices psbd recently studied by j p crouzeix et al siam j matrix anal appl 22 2000 66 and show that linear complementarity problems with psbd matrices of rank 2 are processable by lemke s algorithm and that a psbd matrix of rank 2 belongs to the class of sufficient matrices introduced by r w cottle et al linear algebra appl 114 115 1989 231 we also show that if a matrix a is a sum of a merely positive subdefinite copositive plus matrix and a copositive matrix and a feasibility condition is satisfied then lemke s algorithm solves lcp q a this supplements the results of jones and evers

complementarity problems over locally compact cones this paper proves an existence result for a complementarily problem on a locally convex space where the mapping is copositive positive homogeneous and of monotone type on a locally compact cone a perturbation theorem is proved that extends a result of mangasarian and doverspike proved for n times n matrices on the nonnegative orthant

integral solutions of linear complementarity problems we characterize the class of integral square matrices m having the property that for every integral vector q the linear complementarity problem with data m q has only integral basic solutions these matrices called principally unimodular matrices are those for which every principal nonsingular submatrix is unimodular as a consequence we show that if m is rank symmetric and principally unimodular and q is integral then the problem has an integral solution if it has a solution principal unimodularity can be regarded as an extension of total unimodularity and our results can be regarded as extensions of well known results on integral solutions to linear programs we summarize what is known about principally unimodular symmetric and skew symmetric matrices

some existence results for multivalued complementarity problems this paper presents some new results for the existence of a solution to a multivalued complementary problem these results are derived under a upper limiting homogeneity ulh property on the multifunction defining the complementarity problem and on a certain closedness interiority condition the closedness condition seems to be new in the context of existence proof for the multivalued complementarity problem we show that this closedness condition is satisfied by the class of polyhedral multifunctions

two characterizations of sufficient matrices abstract two characterizations are given for the class of sufficient matrices defined by cottle pang and venkateswaran the first is a direct translation of the definition into linear programming terms the second can be thought of as a generalization of a theorem of t d parsons on p matrices

process rewrite systems many formal models for infinite state concurrent systems are equivalent to special classes of rewrite systems we classify these models by their expressiveness and define a hierarchy of classes of rewrite systems we show that this hierarchy is strict with respect to bisimulation equivalence the most general and most expressive class of systems in this hierarchy is called process rewrite systems prs they subsume petri nets pa processes and pushdown processes and are strictly more expressive than any of these intuitively prs can be seen as an extension of petri nets by subroutines that can return a value to their caller we show that the reachability problem is decidable for prs it is even decidable if there is a reachable state that satisfies certain properties that can be encoded in a simple logic thus prs are more expressive than petri nets but not turing powerful

the linear complementarity problem under asymptotic analysis in this work we study the classical linear complementarity problem lcp by describing the asymptotic behavior of the approximate solutions to its variational inequality formulation thus some properties satisfied by the directions which are limits of the normalized unbounded approximate solutions will be established based on this analysis various equivalent conditions guaranteeing the existence of solutions to lcp are given in particular the sufficient condition of gowda and pang expressed in terms of the solutions to augmented linear complementarity problems is written in a way that is more easily verifiable our approach allows us to deal with garca a matrices semimonotone copositive q pseudomonotone matrices among others in a unified framework furthermore we introduce a larger class of matrices for which many of the results including a sensitivity one due to gowda and pang are still valid in addition some conditions ensuring the boundedness of the solution set are also provided and some estimates for the asymptotic cone of the solution set for different classes of matrices are given as well hence the present approach sheds new light and offers an alternative to view classical results

on a global projection type error bound for the linear complementarity problem abstract we consider a projection type error bound for the linear complementarity problem involving a matrix m and vector q first we show that the mangasarian ren sufficient condition on m for this error bound to hold globally for all q such that the problem is solvable is also necessary second we derive necessary and sufficient conditions on m and q for this error bound to hold globally thus extending a result of luo and tseng which in addition assumes m is psd plus third we derive necessary and sufficient conditions on m for this error bound to hold globally independent of q these conditions are closely related to m being lipschitzian we also discuss application of our result to the affine variational inequality problem

a fully polynomial time approximation algorithm for computing a stationary point of the general linear complementarity problem we apply a potential reduction algorithm to solve the general linear complementarity problem glcp minimize xty subject to ax by cz q and x y z 0 we show that the algorithm is a fully polynomial time approximation scheme fptas for computing an e approximate stationary point of the glcp note that there are some glcps in which every stationary point is a solution xty 0 these include the lcps with row sufficient matrices we also show that the algorithm is a polynomial time algorithm for a special class of glcps

conditions ensuring the applicability of cutting plane methods for solving variational inequalities let vip f c denote the variational inequality problem associated with the mapping f and the closed convex set c in this paper we introduce weak conditions on the mapping f that allow the development of a convergent cutting plane framework for solving vip f c in the process we introduce in a natural way new and useful notions of generalized monotonicity for which first order characterizations are presented

total dual integrality and integral solutions of the linear complementarity problem abstract this paper deals with the problem of finding an integer solution to a linear complementarity problem lcp chandrasekaran et al 1 introduced the class i of integral matrices for which the corresponding lcp has an integer solution for every integral vector q for which it has a solution and proved that for some well known matrix classes principal unimodularity forms a necessary and sufficient condition for inclusion in the class i in this paper we identify some more well known matrix classes for which principal unimodularity forms a necessary and sufficient condition for inclusion in the class i the concept of total dual integrality is utilized to obtain a necessary and sufficient condition for existence of an integer solution to lcp with a hidden k matrix we interconnect the concept of hilbert basis with principal unimodularity of a matrix and the corresponding complementary cones a necessary and sufficient condition is given for the existence of an integer solution of a linear fractional programming problem by using its lcp formulation

on a subclass of p0 abstract in 1966 fiedler and ptak wrote the first systematic investigation of the matrix class p 0 consisting of all real square matrices with nonnegative principal minors in this note we focus on a particular subclass of p 0 that has arisen within the field of linear complementarity theory namely the class su of sufficient matrices our principal result is that su contains another class p we conjecture that the latter two classes are in fact the same

on the classes of fully copositive and fully semimonotone matrices abstract in this paper we consider the class c 0 f of fully copositive and the class e 0 f of fully semimonotone matrices we show that c 0 f matrices with positive diagonal entries are column sufficient we settle a conjecture made by murthy and parthasarathy to the effect that a c 0 f q 0 matrix is positive semidefinite by providing a counterexample we finally consider e 0 f matrices introduced by cottle and stone math program 27 1983 191 213 and partially address stone s conjecture to the effect that e 0 f q 0 p 0 by showing that e 0 f d c matrices are p 0 where d c is the doverspike class of matrices

pc matrices and the linear complementarity problem abstract we introduce a new matrix class pc which consists of those matrices m for which the solution set of the corresponding linear complementarity problem is connected for every q rn we consider lemke s pivotal method from the perspective of piecewise linear homotopies and normal maps and show that lemke s method processes all matrices in pc q0 we further investigate the relationship of the class pc to other known matrix classes and show that column sufficient matrices are a subclass of pc as are 2 2 p0 matrices

the linear complimentarity problem sufficient matrices and the criss cross method specially structured linear complementarity problems lcps and their solution by the criss cross method are examined the criss cross method is known to be finite for lcps with positive semidefinite bisymmetric matrices and with p matrices it is also a simple finite algorithm for oriented matroid programming problems recently cottle pang and venkateswaran identified the class of column row sufficient matrices they showed that sufficient matrices are a common generalization of p and psd matrices cottle also showed that the principal pivoting method with a clever modification can be applied to row sufficient lcps in this paper the finiteness of the criss cross method for sufficient lcps is proved further it is shown that a matrix is sufficient if and only if the criss cross method processes all the lcps defined by this matrix and all the lcps defined by the transpose of this matrix and any parameter vector

undecidability of domino games and hhp bisimilarity history preserving bisimilarity hp bisimilarity and hereditary history preserving bisimilarity hhp bisimilarity are behavioural equivalences taking into account causal relationships between events of concurrent systems their prominent feature is that they are preserved under action refinement an operation important for the top down design of concurrent systems it is shown that in contrast to hp bisimilarity checking hhp bisimilarity for finite labelled asynchronous transition systems is undecidable by a reduction from the halting problem of 2 counter machines to make the proof more transparent a novel intermediate problem of checking domino bisimilarity for origin constrained tiling systems is introduced and shown undecidable it is also shown that the unlabelled domino bisimilarity problem is undecidable which implies undecidability of hhp bisimilarity for unlabelled finite asynchronous systems moreover it is argued that the undecidability of hhp bisimilarity holds for finite elementary net systems and 1 safe petri nets

the linear complementarity problem and a subclass of fully semimonotone matrices abstract in this paper we investigate a subclass w of the n n real matrices a matrix m belongs to w if and only if certain pairs of its complementary cones intersect only in the zero vector we show that w u where u is the class of matrices m for which lcp q m the linear complementarity problem has a unique solution whenever q belongs in the interior of the union of its complementary cones membership in w can be established by solving 2n 1 systems of linear equations finally several sufficient conditions are given on a matrix in w so that it possesses nonnegative principal minors

principal pivot transforms of some classes of matrices in this paper we study and characterize various classes of matrices that are defined based on principal pivot transforms we show that matrices in these classes have nonnegative principal minors

notes on sufficient matrices abstract we give some constructive characterizations for sufficient matrices and present some matrix classes whose elements are identified as column sufficient matrices in particular we give the structure of column row sufficient matrices which leads to efficient testing methods for column row sufficiency

on the periodic orbits and the integrability of the regularized hill lunar problem the classical hill s problem is a simplified version of the restricted three body problem where the distance of the two massive bodies say primary for the largest one and secondary for the smallest one is made infinity through the use of hill s variables the levi civita regularization takes the hamiltonian of the hill lunar problem into the form of two uncoupled harmonic oscillators perturbed by the coriolis force and the sun action polynomials of degree 4 and 6 respectively in this paper we study periodic orbits of the planar hill problem using the averaging theory moreover we provide information about the c1 integrability or non integrability of the regularized hill lunar problem