positive radial solutions and non radial bifurcation for semilinear elliptic equations in annular domains abstract we discuss the existence and multiplicity of positive radial solutions and the non radial bifurcation of u f u 0 in and u 0 on where is an annular domain of r n n 2 we prove that if f u 0 for u 0 and lim u f u u then there exists 0 such that there are at least two positive radial solutions for each 0 at least one for and none for if f 0 0 lim u 0 f u u 1 and uf u 1 e f u for u 0 e 0 then there exists a variational solution for 0 1 where 1 is the least eigenvalue of if f 0 0 lim u 0 f u u 0 and lim u f u u then there exists at least one positive radial solution for any 0 we obtain some precise multiplicity results for narrow annulus and show that the non radial bifurcation occurs if the growth of f u is rapid enough as u

non existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities abstract in this paper we study the limit cycles of the planar polynomial differential systems x a x y p n x y y x a y q n x y where p n and q n are homogeneous polynomials of degree n 2 and a r consider the functions p n cos sin cos q n cos sin sin q n cos sin cos p n cos sin sin 1 a 2 n 1 2 a first we prove that these differential systems have at most 1 limit cycle if there exists a linear combination of 1 and 2 with definite sign this result improves previous known results furthermore if 1 1 a 2 0 for some 1 2 0 we provide necessary and sufficient conditions for the non existence and the existence and uniqueness of the limit cycles of these differential systems when one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin

a semi smooth newton method for projection equations and linear complementarity problems with respect to the second order cone abstract in this paper a special semi smooth equation associated to the second order cone is studied it is shown that under mild assumptions the semi smooth newton method applied to this equation is well defined and the generated sequence is globally and q linearly convergent to a solution as an application the obtained results are used to study the linear second order cone complementarity problem with special emphasis on the particular case of positive definite matrices moreover some computational experiments designed to investigate the practical viability of the method are presented

a krylov subspace method for large scale second order cone linear complementarity problem in this paper we first show that the second order cone linear complementarity problem soclcp can be solved by finding a positive zero s in mathbb r of a particular rational function h s and we then propose a krylov subspace method to reduce h s to h ell s as in the model reduction the zero s of h s can be accurately approximated by that of h ell s 0 which itself can be cast as a small eigenvalue problem the new method is made possible by a complete characterization of the curve of h s and it has several advantages over the bisection newton sf bn iteration recently proposed by l h zhang and w h yang math comp 83 2013 pp 1701 1720 and shown to be very efficient for small to medium size problems the method is tested and compared against the sf bn iteration and two other state of the art packages sf sdpt3 and sf sedumi our numerical results show that the method is very efficient for both small to medium dense problems and large scale ones

uniqueness of positive radial solutions for semilinear elliptic equations on annular domains where 5 5a b x rn 0iai x ib is an annular domain in rn n 3 it is well known that the uniqueness problems of solutions of 1 1 and 1 2 are both fundamental and often di8cult during the last decade there has been tremendous progress in studying these problems when 5 is a ball in rn or entire rn n 3 see e g 1 2 5 10 11 however very little was known concerning the annular domain until the work of ni and nussbaum 12 until very recently co man 3 showed that when n 3 f u u up 1ip 3 1 1 and 1 2 have unique positive radial solution for any annular domain see also 13 then it is of interest to extend co man s result to a more general situation for example we may ask the following question

a regularized smoothing newton method for solving soccps based on a new smoothing c function abstract in this paper an soc complementarity function is constructed based on this function a regularized smoothing newton method is proposed for solving monotone second order cone complementarity problems denoted by soccps the proposed algorithm is proved to be globally and quadratically convergent under mild conditions unlike some existing smoothing newton type methods we take the regularization parameter the same as the smoothing parameter and treat the parameter as independent variables in our algorithm some numerical results are reported and indicate that the proposed method is quite effective

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities the smoothing newton method for solving a system of nonsmooth equations f x 0 which may arise from the nonlinear complementarity problem the variational inequality problem or other problems can be regarded as a variant of the smoothing method at the kth step the nonsmooth function f is approximated by a smooth function f and the derivative of f at x k is used as the newton iterative matrix the merits of smoothing methods and smoothing newton methods are global convergence and convenience in handling in this paper we show that the smoothing newton method is also superlinearly convergent if f is semismooth at the solution and f satisfies a jacobian consistency property we show that most common smooth functions such as the gabriel more function have this property as an application we show that for box constrained variational inequalities if the involved function is p uniform the iteration sequence generated by the smoothing newton method will converge to the unique solution of the problem globally and superlinearly quadratically

modified newton methods for solving a semismooth reformulation of monotone complementarity problems in this paper we propose a newton type method for solving a semismooth reformulation of monotone complementarity problems in this method a direction finding subproblem which is a system of linear equations is uniquely solvable at each iteration moreover the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem we show that the algorithm is globally convergent under some mild assumptions next by slightly modifying the direction finding problem we propose another newton type method which may be considered a restricted version of the first algorithm we show that this algorithm has a superlinear or possibly quadratic rate of convergence under suitable assumptions finally some numerical results are presented

on the number of limit cycles for perturbed pendulum equations abstract we consider perturbed pendulum like equations on the cylinder of the form x sin x e s 0 m q n s x x s where q n s are trigonometric polynomials of degree n and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case e 0 in terms of m and n our first result gives upper bounds on the number of zeros of its associated first order melnikov function in both the oscillatory and the rotary regions these upper bounds are obtained expressing the corresponding abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind some further results give sharp bounds on the number of zeros of these integrals by identifying subfamilies which are shown to be chebyshev systems

on implementing a primal dual interior point method for conic quadratic optimization based on the work of the nesterov and todd on self scaled cones an implementation of a primal dual interior point method for solving large scale sparse conic quadratic optimization problems is presented the main features of the implementation are it is based on a homogeneous and self dual model it handles rotated quadratic cones directly it employs a mehrotra type predictor corrector extension and sparse linear algebra to improve the computational efficiency finally the implementation exploits fixed variables which naturally occurs in many conic quadratic optimization problems this is a novel feature for our implementation computational results are also presented to document that the implementation can solve very large problems robustly and efficiently

merit functions for semi definite complementarity problems merit functions such as the gap function the regularized gap function the implicit lagrangian and the norm squared of the fischer burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors we study the extension of these merit functions to complementarity problems defined over the cone of block diagonal symmetric positive semi definite real matrices the extension suggests new solution methods for the latter problems

a combined smoothing and regularization method for monotone second order cone complementarity problems the second order cone complementarity problem soccp is a wide class of problems containing the nonlinear complementarity problem ncp and the second order cone programming problem socp recently fukushima luo and tseng siam j optim 12 2001 pp 436 460 extended some merit functions and their smoothing functions for ncp to soccp moreover they derived computable formulas for the jacobians of the smoothing functions and gave conditions for the jacobians to be invertible in this paper we propose a globally and quadratically convergent algorithm which is based on smoothing and regularization methods for solving monotone soccp in particular we study strong semismoothness and jacobian consistency which play an important role in establishing quadratic convergence of the algorithm furthermore we examine the effectiveness of the algorithm by means of numerical experiments

convexity and differentiability properties of spectral functions and spectral mappings on euclidean jordan algebras abstract we study in this paper several properties of the eigenvalues function of a euclidean jordan algebra extending several known results in the framework of symmetric matrices in particular we give a concise form for the directional differential of a single eigenvalue we especially focus on spectral functions f on euclidean jordan algebras which are the composition of a symmetric real valued function f with the eigenvalues function we explore several properties of f that are transferred to f in particular convexity strong convexity and differentiability spectral mappings are also considered a special case of which is the gradient mapping of a spectral function answering a problem proposed by h sendov we give a formula for the jacobian of these functions

interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones we study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones while the analysis in l faybusovich positivity 1 1997 pp 331 357 depends on the optimization theory of convex log barrier functions our approach is based on the paper of monteiro and pang math oper res 23 1998 pp 39 60 where a vast set of conclusions concerning continuous trajectories is shown for monotone complementarity problems over the cone of symmetric positive semidefinite matrices as an application of the results we propose a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and discuss its theoretical aspects consequently we show the existence of a path having the following properties a the path is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions b any accumulation point of the path is a solution of the homogeneous model c if the original problem is solvable then every accumulation point of the path gives us a finite solution d if the original problem is strongly infeasible then under the assumption of lipschitz continuity any accumulation point of the path gives us a finite certificate proving infeasibility

differentiable exact penalty functions for nonlinear second order cone programs we propose a method for solving nonlinear second order cone programs socps based on a continuously differentiable exact penalty function the construction of the penalty function is given by incorporating a multipliers estimate in the augmented lagrangian for socps under the nondegeneracy assumption and the strong second order sufficient condition we show that a generalized newton method has global and superlinear convergence we also present some preliminary numerical experiments

a chebyshev criterion with applications abstract we show that a family of certain definite integrals forms a chebyshev system if two families of associated functions appearing in their integrands are chebyshev systems as well we apply this criterion to several examples which appear in the context of perturbations of periodic non autonomous odes to determine bounds on the number of isolated periodic solutions as well as to persistence problems of periodic solutions for perturbed hamiltonian systems

semismooth karush kuhn tucker equations and convergence analysis of newton and quasi newton methods for solving these equations there are several forms of systems of nonsmooth equations which are equivalent to the karush kuhn tucker kkt system of a nonlinearly constrained optimization problem nlp if the nlp is twice continuously differentiable and the hessian functions of its objective and constraint functions are locally lipschitzian then these kkt equations are strongly semismooth if furthermore the linear independence condition and the strong second order sufficiency condition are satisfied at a kkt point then the generalized jacobians of these kkt equations are nonsingular at that point and the sequence generated by the generalized newton method converges to this point q quadratically however direct application of quasi newton methods cannot guarantee q superlinear convergence we present a mixed quasi newton method which converges q superlinearly with common symmetrical updating rules under the above conditions for the generalized newton method superlinear convergence of the primal variables and global convergence are also discussed

global bifurcations of solutions of emden fowler type equation δu x λf u x on an annulus in rn n 3 abstract in this article we study global and symmetry breaking bifurcations of solutions of emden fowler type elliptic differential equations as the main tool we use a topological invariant which is suitable for study of global bifurcations and continuations of critical points of equivariant functionals namely we will apply degree for s 1 equivariant orthogonal maps defined in rybicki 1994 nonlinear anal tma 23 83 102

lower bounds for the number of limit cycles of trigonometric abel equations abstract we consider the abel equation x a t x 3 b t x 2 where a t and b t are trigonometric polynomials of degree n and m respectively and we give lower bounds for its number of isolated periodic orbits for some values of n and m these lower bounds are obtained by two different methods the study of the perturbations of some abel equations having a continuum of periodic orbits and the hopf type bifurcation of periodic orbits from the solution x 0

nonlinear elliptic problems with critical exponent in shrinking annuli soit dans r n n 2 l anneau a x p 0 dans a u 0 sur a et u u n 2 n 2 0 u 0 dans a u n 0 quand x u 0 quand x r

analysis of nonsmooth vector valued functions associated with infinite dimensional second order cones abstract given a hilbert space h the infinite dimensional lorentz second order cone k is introduced for any x h a spectral decomposition is introduced and for any function f r r we define a corresponding vector valued function f h x on hilbert space h by applying f to the spectral values of the spectral decomposition of x h with respect to k we show that this vector valued function inherits from f the properties of continuity lipschitz continuity differentiability smoothness as well as s semismoothness these results can be helpful for designing and analyzing solution methods for solving infinite dimensional second order cone programs and complementarity problems

a merit function method for infinite dimensional soccps abstract we introduce the jordan product associated with the second order cone k into the real hilbert space h and then define a one parametric class of complementarity functions t on h h with the parameter t 0 2 we show that the squared norm of t with t 0 2 is a continuously f rechet differentiable merit function by this the second order cone complementarity problem soccp in h can be converted into an unconstrained smooth minimization problem involving this class of merit functions and furthermore under the monotonicity assumption every stationary point of this minimization problem is shown to be a solution of the soccp

a generalized newton method for absolute value equations associated with second order cones in this paper we introduce the absolute value equations associated with second order cones socave in short which is a generalization of the absolute value equations discussed recently in the literature it is proved that the socave is equivalent to a class of second order cone linear complementarity problems soclcp in short in particular we propose a generalized newton method for solving the socave and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions we also report some preliminary numerical results of the proposed method for solving the socave and the soclcp which show the efficiency of the proposed method

an inexact smoothing method for soccps based on a one parametric class of smoothing function in this paper we introduce a one parametric class of smoothing functions which contains the fischer burmeister smoothing function as special case based on this class of smoothing functions an inexact smoothing method for solving second order cone complementarity problems soccps is proposed in each iteration the corresponding linear system is solved only approximately under mild assumptions it is proved that the proposed method has global convergence and local superlinear convergence properties preliminary numerical results indicate that the method is effective for large scale problems

on non radially symmetric bifurcation in the annulus we discuss the radially symmetric solutions and the non radially symmetric bifurcation of the semilinear elliptic equation u 2 eu 0 in and u 0 on where x r2 a2 0 a the upper branch of radial solutions has a non radially symmetric bifurcation symmetry breaking at each k a k 1 as a 0 the radial solutions will tend to the radial solutions on the disk and 0 a 1 the critical number on the disk

numerical comparison between prediction correction methods for general variational inequalities in this paper we propose two prediction correction methods for solving general variational inequalities the first method makes use of a descent direction to produce the new iterate while the second method generates the corrector by the improved extragradient method under certain conditions the global convergence of both methods is proved it is proved theoretically that the lower bound of the progress obtained by the second method is greater than that by the first one numerical results are given to verify and compare the numerical efficiency of these two predictor corrector methods

existence of positive nonradial solutions for nonlinear elliptic equations in annular domains we study the existence of positive nonradial solutions of equation a f u 0 in sia u 0 on d la where s x 6 w a 2 and is positive and superlinear at both 0 and oo we use a bifurcation method to show that there is a nonradial bifurcation with mode k at ak 0 1 for any positive integer k if is subcritical and for large k if is supercritical when is subcritical then a nehari type variational method can be used to prove that there exists a e 0 1 such that for any a e a 1 the equation has a nonradial solution on 0 a

new family of abelian integrals satisfying chebyshev property abstract this paper is motivated by the chebyshev property of a family of abelian integrals along non algebraic level ovals a new criterion is established to study the chebyshev property of these abelian integrals as its applications we give three examples to show that the ordered sets defined by abelian integrals are ect systems moreover the number of the abelian integrals can be an arbitrary positive integer

on the nonlinear eigenvalue problem δu λe u 0 on etudie la structure de l ensemble 7b c des solutions du probleme aux valeurs propres non lineaire u exp u 0 sous des conditions de dirichlet dans un domaine borne simplement connexe

infinitely many non radial solutions to a critical equation on annulus abstract in this paper we build infinitely many non radial sign changing solutions to the critical problem p u u 4 n 2 u in u 0 on on the annulus x r n a x b n 3 in particular for any integer k large enough we build a non radial solution which look like the unique positive solution u 0 to p crowned by k negative bubbles arranged on a regular polygon with radius r 0 such that r 0 n 2 2 u 0 r 0 max a r b r n 2 2 u 0 r

a regularized explicit exchange method for semi infinite programs with an infinite number of conic constraints the semi infinite program sip is normally represented with infinitely many inequality constraints and has been studied extensively so far however there have been very few studies on the sip involving conic constraints even though it has important applications such as chebyshev like approximation filter design and so on in this paper we focus on the sip with infinitely many conic constraints called an sicp for short we show that under the robinson constraint qualification a local optimum of the sicp satisfies the kkt conditions that can be represented only with a finite subset of the conic constraints we also introduce two exchange type algorithms for solving the convex sicp we first provide an explicit exchange method and show that it has global convergence under the strict convexity assumption on the objective function we then propose an algorithm combining a regularization method with the explicit exchange method and establish global convergence of the hybrid algorithm without the strict c

a one parametric class of merit functions for the symmetric cone complementarity problem abstract in this paper we extend the one parametric class of merit functions proposed by kanzow and kleinmichel c kanzow h kleinmichel a new class of semismooth newton type methods for nonlinear complementarity problems comput optim appl 11 1998 227 251 for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem sccp we show that the class of merit functions is continuously differentiable everywhere and has a globally lipschitz continuous gradient mapping from this we particularly obtain the smoothness of the fischer burmeister merit function associated with symmetric cones and the lipschitz continuity of its gradient in addition we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the ncp function classes studied by c kanzow y yamashita m fukushima new ncp functions and their properties j optim theory appl 97 1997 115 135 to the sccp by exploiting the cartesian p properties for a nonlinear transformation we show that the class of regularized merit functions provides a global error bound for the solution of the sccp and moreover has bounded level sets under a rather weak condition which can be satisfied by the monotone sccp with a strictly feasible point or the sccp with the joint cartesian r 02 property all of these results generalize some recent important works in j s chen p tseng an unconstrained smooth minimization reformulation of the second order cone complementarity problem math program 104 2005 293 327 c k sim j sun d ralph a note on the lipschitz continuity of the gradient of the squared norm of the matrix valued fischer burmeister function math program 107 2006 547 553 p tseng merit function for semidefinite complementarity problems math program 83 1998 159 185 under a unified framework

a new chebyshev family with applications to abel equations we prove that a family of functions defined through some definite integrals forms an extended complete chebyshev system the key point of our proof consists of reducing the study of certain wronskians to the gram determinants of a suitable set of new functions our result is then applied to give upper bounds for the number of isolated periodic solutions of some perturbed abel equations

polynomial convergence of primal dual algorithms for the second order cone program based on the mz family of directions in this paper we study primal dual path following algorithms for the second order cone programming socp based on a family of directions that is a natural extension of the monteiro zhang mz family for semidefinite programming we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of socp that is they have an o loge 1 iteration complexity to reduce the duality gap by a factor of e where n is the number of second order cones since the mz type family studied in this paper includes an analogue of the alizadeh haeberly and overton pure newton direction we establish for the first time the polynomial convergence of primal dual algorithms for socp based on this search direction

soc monotone and soc convex functions vs matrix monotone and matrix convex functions abstract the soc monotone function respectively soc convex function is a scalar valued function that induces a map to preserve the monotone order respectively the convex order when imposed on the spectral factorization of vectors associated with second order cones socs in general hilbert spaces in this paper we provide the sufficient and necessary characterizations for the two classes of functions and particularly establish that the set of continuous soc monotone respectively soc convex functions coincides with that of continuous matrix monotone respectively matrix convex functions of order 2

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

periodic orbits from second order perturbation via rational trigonometric integrals abstract the second order poincare pontryagin melnikov perturbation theory is used in this paper to study the number of bifurcated periodic orbits from certain centers this approach also allows us to give the shape and the period up to the first order we address these problems for some classes of abel differential equations and quadratic isochronous vector fields in the plane we prove that two is the maximum number of hyperbolic periodic orbits bifurcating from the isochronous quadratic centers with a birational linearization under quadratic perturbations of second order in particular the configurations 2 0 and 1 1 are realizable when two centers are perturbed simultaneously the required computations show that all the considered families share the same iterated rational trigonometric integrals

symmetry breaking for solutions of semilinear elliptic equations with general boundary conditions we study the bifurcation of radially symmetric solutions of f u 0 onn balls into asymmetric ones we show that ifu satisfies homogeneous neumann boundary conditions the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension for general boundary conditions we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions and on the structure of the kernels of the linearized operators

on the existence of positive radial solutions for nonlinear elliptic equations in annular domains abstract we study the existence of positive radially symmetric solutions of u g x f u 0 in annulus with dirichlet dirichlet neumann boundary conditions we show that the equation has a positive radial solution on any annulus if f and f are positive and f is superlinear at 0 and

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

monotonicity of the first eigenvalue and the global bifurcation diagram for the branch of interior peak solutions abstract let b r n n 1 be a unit ball and d r m be a bounded domain we study the global branch consisting of interior single peak solutions of the elliptic neumann problem u u u p 0 in b u 0 on b and the monotonicity of the first eigenvalue along the branch for large where 1 p p s n 2 n 2 n 3 n 1 2 when the domain is replaced with b d using this monotonicity we show that the branch of solutions concentrating on 0 d has secondary bifurcation points for n 1 and p 2 an integer we determine the global bifurcation diagram by showing the monotonicity of the time map the period function the monotonicity of the first eigenvalue along the whole branch is also proved for n 1 and p 3

least energy solutions of the emden fowler equation in hollow thin symmetric domains abstract in this paper the emden fowler equation is studied in a hollow thin domain which is invariant under the action of a closed subgroup of the orthogonal group then it is proved that if the domain is thin enough a least energy solution is not group invariant

a globally convergent method based on fischer burmeister operators for solving second order cone constrained variational inequality problems the karush kuhn tucker system of a second order cone constrained variational inequality problem is transformed into a semismooth system of equations with the help of fischer burmeister operators over second order cones the clarke generalized differential of the semismooth mapping is presented a modified newton method with armijo line search is proved to have global convergence with local superlinear rate of convergence under certain assumptions on the variational inequality problem an illustrative example is given to show how the globally convergent method works

a uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities abstract this paper is devoted to study the planar polynomial system x a x y p n x y y x a y q n x y where a r and p n q n are homogeneous polynomials of degree n 2 denote cos q n cos sin sin p n cos sin we prove that the system has at most 1 limit cycle surrounding the origin provided n 1 a 0 furthermore this upper bound is sharp this is maybe the first uniqueness criterion which only depends on a linear condition of for the limit cycles of this kind of systems we show by examples that in many cases the criterion is applicable while the classical ones are invalid the tool that we mainly use is a new estimate for the number of limit cycles of abel equation with coefficients of indefinite signs employing this tool we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin

a smoothing method for second order cone complementarity problem in this paper the second order cone complementarity problem is studied based on a perturbed symmetrically smoothing function which has coerciveness under proper conditions we present a smoothing newton method for this problem the boundedness of the level set can be obtained from the coerciveness which plays an important role in the convergence analysis furthermore the proposed algorithm for the reformulation has no restrictions on the starting point and solves only one system of equations preliminary numerical results indicate that the algorithm is effective

strong semismoothness of the fischer burmeister sdc and soc complementarity functions we show that the fischer burmeister complementarity functions associated to the semidefinite cone sdc and the second order cone soc respectively are strongly semismooth everywhere interestingly enough the proof relys on a relationship between the singular value decomposition of a nonsymmetric matrix and the spectral decomposition of a symmetric matrix

primal dual algorithms and infinite dimensional jordan algebras of finite rank we consider primal dual algorithms for certain types of infinite dimensional optimization problems our approach is based on the generalization of the technique of finite dimensional euclidean jordan algebras to the case of infinite dimensional jb algebras of finite rank this generalization enables us to develop polynomial time primal dual algorithms for infinite dimensional second order cone programs we consider as an example a long step primal dual algorithm based on the nesterov todd direction it is shown that this algorithm can be generalized along with complexity estimates to the infinite dimensional situation under consideration an application is given to an important problem of control theory multi criteria analytic design of the linear regulator the calculation of the nesterov todd direction requires in this case solving one matrix differential riccati equation plus solving a finite dimensional system of linear algebraic equations on each iteration the number of equations and unknown variables of this algebraic system is m 1 where m is a number of quadratic performance criteria

expected residual minimization formulation for a class of stochastic linear second order cone complementarity problems abstract this paper considers a class of stochastic linear second order cone complementarity problems slsoccp noticing that the slsoccp does not have a solution suitable to all realizations in general we present a deterministic formulation called the expected residual minimization erm formulation for it the coercive property of the erm problem and the robustness of its solutions are discussed due to the existence of expectation in the erm problem we employ the monte carlo approximation techniques to approximate the erm problem and show that under mild conditions this approximation approach possesses exponential convergence rate then we extend the above results to a general mixed slsoccp furthermore we apply the theoretical results to a stochastic optimal power flow model in radial network and report some numerical dispatching experiments for real world southern california edison 47 bus network

semismooth matrix valued functions matrix valued functions play an important role in the development of algorithms for semidefinite programming problems this paper studies generalized differential properties of such functions related to nonsmooth smoothing newton methods the first part of this paper discusses basic properties such as the generalized derivative rademacher s theorem b derivative directional derivative and semismoothness the second part shows that the matrix absolute value function the matrix semidefinite projection function and the matrix projective residual function are strongly semismooth

a new non interior continuation method for solving the second order cone complementarity problem abstract in this paper based on a symmetrically perturbed smoothing fischer burmeister function a non interior continuation method is proposed for solving the second order cone complementarity problem soccp the proposed algorithm solves only one linear system of equations and performs only one line search at each iteration under monotonicity it is shown that our algorithm is globally and locally superlinearly convergent without requiring strict complementarity assumption at the soccp solution furthermore the proposed algorithm has local quadratic convergence under mild conditions some numerical results are reported which indicate the effectiveness of the proposed algorithm

existence of radial solutions for nonlinear elliptic equations with gradient terms in annular domains abstract this paper is concerned with the problem of elliptic equations in annular domains involving derivative terms using nonlinear analysis methods some results regarding existence of solutions are established more precisely at least one radial solution is obtained basing on schauder s fixed point theorem and contraction mapping theorem respectively some iterative technique is also applied which allows us to overcome difficulties that the nonlinearity is related to derivative terms

a generalized newton method for absolute value equations associated with circular cones in this paper we study a class of absolute value equations associated with circular cone ccave for short which is a generalization of the absolute value equations discussed recently in the literature analogously to the fact that circular cone is the very generalization of the second order cone we show that the ccave is equivalent to a class of circular cone linear complementarity problems and hence generalize the well known equivalence between absolute value equations and linear complementarity problems useful properties of the generalized differential of the absolute value function over the circular cone are investigated which helps to propose a generalized newton method for solving the ccave the convergence of this method is established under mild conditions as well as the efficiency of which is illustrated by some preliminary numerical results

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

on q and r0 properties of a quadratic representation in linear complementarity problems over the second order cone abstract this paper studies the linear complementarity problem lcp m q over the second order lorentz or ice cream cone denoted by n where m is a n n real square matrix and q rn this problem is denoted as solcp m q the study of second order cone programming problems and hence an independent study of solcp is motivated by a number of applications though the second order cone is a special case of the cone of squares symmetric cone in a euclidean jordan algebra the geometry of its faces is much simpler and hence an independent study of lcp over n may yield interesting results in this paper we characterize the r0 property x n m x n and x m x 0 x 0 of a quadratic representation pa x 2a a x a2 x of n for a x n where is a jordan product and show that the r0 property of pa is equivalent to stating that solcp pa q has a solution for all q n

exceptional family and solvability of the second order cone complementarity problems abstract in this paper we introduce a concept of exceptional family for second order cone complementarity problems denoted by soccp which is the particular case of the concept of exceptional family of elements introduced by isac and carbone 1999 25 for the nonlinear complementarity problems and the exceptional family for second order cone complementarity problems has some pretty unique properties furthermore we propose a new sufficient existence condition of a solution to soccp and give a particular example to show that the new condition is not stronger than known isac carbone s condition

a geometric criterion for equation x i 0mai t xi having at most m isolated periodic solutions abstract this paper is devoted to the investigation of generalized abel equation x s x t i 0 m a i t x i where a i c 0 1 a solution x t is called a periodic solution if x 0 x 1 in order to estimate the number of isolated periodic solutions of the equation we propose a hypothesis h which is only concerned with s x t on m straight lines there exist m real numbers 1 m such that either 1 i s i t 0 for i 1 m or 1 i s i t 0 for i 1 m by means of lagrange interpolation formula we prove that the equation has at most m isolated periodic solutions counted with multiplicities if hypothesis h holds and the upper bound is sharp furthermore this conclusion is also valid under some weaker geometric hypotheses applying our main result for the trigonometrical generalized abel equation with coefficients of degree one we give a criterion to obtain the upper bound for the number of isolated periodic solutions this criterion is almost equivalent to hypothesis h and can be much more effectively checked