stable solutions with zeros to the ginzburg landau equation with neumann boundary condition abstract this paper is devoted to the ginzburg landau equation 1 2 0 u1 iu2in a bounded domain r nwith the homogeneous neumann boundary condition the previous works 12 14 showed that for large there exist stable non constant solutions with no zeros in domains which are topologically non trivial in a certain sense in this aritcle it is proved that for a domain containing a non trivial domaindas a subset there exist stable solutions with zeros provided that the volume of dis sufficiently small

a norm relaxed method of feasible directions for finely discretized problems from semi infinite programming abstract in this paper a class of finely discretized semi infinite programming sip problems is discussed combining the idea of the norm relaxed method of feasible directions mfd and the technique of updating discretization index set we present a new algorithm for solving the discretized semi infinite dsi problems from sip at each iteration the iteration point is feasible for the discretized problem and an improved search direction is computed by solving only one direction finding subproblem i e a quadratic program and some appropriate constraints are chosen to reduce the computational cost a high order correction direction can be obtained by solving another quadratic programming subproblem with only equality constraints under weak conditions such as mangasarian fromovitz constraint qualification mfcq the proposed algorithm possesses weak global convergence moreover the superlinear convergence is obtained under linearly independent constraint qualification licq and other assumptions in the end some elementary numerical experiments are reported

a new norm relaxed method of strongly sub feasible direction for inequality constrained optimization combining the norm relaxed method of feasible direction mfd with the idea of strongly sub feasible direction method we present a new convergent algorithm with arbitrary initial point for inequality constrained optimization at each iteration the new algorithm solves one direction finding subproblem dfs which always possesses a solution some good properties of the new algorithm are that it can unify automatically the operations of initialization phase i and optimization phase ii and the number of the functions satisfying the inequality constrains is nondecreasing particularly a feasible direction of descent can be obtained by solving dfs whenever the iteration point gets into the feasible set under some mild assumptions without the linear independence the global and strong convergence of the algorithm can be obtained

random convex hulls in a product of balls the convex hull of a set of points sampled independently and uniformly from the cartesian product of balls of various dimensions is investigated bounds on the asymptotic behavior of the expected combinatorial complexity volume and mean width are derived when the distribution is held fixed and the sample size approaches infinity the expected combinational complexity and volume are found to depend up to constant factors only on the greatest dimension of any factor ball and the number of balls of that dimension on the other hand the expected mean width depends only on the number of balls and the dimensions of the product

random polytopes and the efron stein jackknife inequality let k be a smooth convex body the convex hull of independent random points in k is a random polytope estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained the essential step is the use of the efron stein jackknife inequality for the variance of symmetric statistics consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope a conjecture of barany concerning random and best approximation of convex bodies is confirmed analogous results for random polytopes with vertices on the boundary of the convex body are given

central limit theorems for gaussian polytopes choose n random independent points in r d according to the standard normal distribution their convex hull k n is the gaussian random polytope we prove that the volume and the number of faces of k n satisfy the central limit theorem settling a well known conjecture in the field

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

random points and lattice points in convex bodies assume k subset of r d is a convex body and x is a large finite subset of k how many convex polytopes are there whose vertices belong to x is there a typical shape of such polytopes how well does the maximal such polytope which is actually the convex hull of x approximate k we are interested in these questions mainly in two cases the first is when x is a random sample of n uniform independent points from k in this case motivation comes from sylvester s famous four point problem and from the theory of random polytopes the second case is when x k boolean and z d where z d is the lattice of integer points in r d and the questions come from integer programming and geometry of numbers surprisingly or not so surprisingly the answers in the two cases are rather similar

dynamics for a class of non autonomous degenerate p laplacian equations abstract in this paper we investigate a class of non autonomous degenerate p laplacian equations t u div a x u p 2 u u f u g x t in where a x is allowed to vanish on a nonempty closed subset with lebesgue measure zero g x t l l o c p r d 1 p a and an unbounded domain in r n we first establish the well posedness of these equations by constructing a compact embedding then we show the existence of the minimal pullback d attractor and prove that it indeed attracts the d class in l 2 norm for any 0 our results extend some known ones in previously published papers

intrinsic volumes and f vectors of random polytopes on appelle polytope aleatoire k n l enveloppe convexe de n points choisis de facon aleatoire dans un corps convexe k on s interesse a l approximation stochastique de k par k n plus particulierement on obtient des resultats d approximation pour les esperances de v s k v s k n ou v s est le s ieme volume intrinseque du nombre de faces s dimensionnelles de k n et de la distance de hausdorff entre k et k n

central limit theorems for random polytopes let k be a smooth convex set the convex hull of independent random points in k is a random polytope central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity one essential step is to determine the precise asymptotic order of the occurring variances

the combinatorial structure of random polytopes abstract choose n random points in r d let p n be their convex hull and denote by f i p n the number of i dimensional faces of p n a general method for computing the expectation of f i p n i 0 d 1 is presented this generalizes classical results of efron in the case i 0 and renyi and sulanke in the case i d 1 to arbitrary i for random points chosen in a smooth convex body a limit law for f i p n is proved as n for random points chosen in a polytope the expectation of f i p n is determined as n this implies an extremal property for random points chosen in a simplex

a strongly convergent norm relaxed method of strongly sub feasible direction for optimization with nonlinear equality and inequality constraints in this paper a class of optimization problems with nonlinear equality and inequality constraints is discussed firstly the original problem is transformed to an associated simpler auxiliary optimization problem with only inequality constraints and a penalty parameter and the later problem is showed to be equivalent to the original problem if the parameter is large enough but finite then combining the norm relaxed method of feasible direction mfd with the idea of method of strongly sub feasible direction mssfd we present an algorithm with arbitrary initial point for the original problem at each iteration of the auxiliary problem an improved search direction is obtained by solving one direction finding subproblem dfs i e a quadratic program which always possesses a solution in the process of iteration the feasibility of the iteration points is monotone increasing furthermore whenever an iteration point enters the feasible set the proposed algorithm reduces to a feasible and decent method for the auxiliary problem under some suitable assumptions the global and strong convergence of the proposed algorithm can be obtained finally some elementary numerical experiments are reported

a random approximation of set valued cadlag functions abstract in this paper we study the problem of estimating a cadlag function f whose values are compact convex sets for this purpose a random selection of points in the interval 0 1 is considered and for each selected point x a random sample in f x on the basis of this a sequence of approximants f n m is constructed where n and m are the respective sample sizes under general conditions rates of convergence are obtained for skorokhod s j 1 topology and in case of continuity of the estimated function also for the uniform one

stochastische approximation konvexer polygone let k be a convex polygon with perimeter l and area f denote the perimeter of the convex hull of n uniformly and independently distributed points in k by ln and by fn its area as n tends to infinity l e l n c 1 k sqrt n o 1 n 1 varepsilon for any fixed 0 and f e f n c 2 k log n n o 1 n where the constants c1 k and c2 k are given explicitly this generalizes a result of renyi and sulanke

on normal approximations of distributions in terms of dependency graphs l auteur interprete les bornes de l erreur introduites par stein dans l approximation normale de sommes de variables aleatoires dependantes en termes de graphes de dependance ceci mene a des ameliorations d un theoreme de limite centrale de petrovskaya et leontovich et de recentes applications de baldi et rinatt en particulier on obtient des bornes pour des taux de convergence on etudie ensuite l approximation normale du nombre de maximums locaux d une fonction aleatoire sur un graphe

a new superlinearly convergent norm relaxed method of strongly sub feasible direction for inequality constrained optimization method of feasible directions mfd is an important method for solving nonlinearly constrained optimization however various types of mfd all need an initial feasible point which can not be found easily in generally in addition the computational cost of some mfd with superlinearly convergent property is rather high on the other hand the strongly sub feasible direction method does not need an initial feasible point but most of the proposed algorithm do not have the superlinearly convergent property and can not guarantee that the iteration point is feasible after finite iterations in this paper we present a new superlinearly convergent algorithm with arbitrary initial point at each iteration a master direction is obtained by solving one direction finding subproblem dfs and an auxiliary direction is yielded by an explicit formula after finite iterations the iteration point goes into the feasible set and the master direction is a feasible direction of descent since a new generalized projection technique is contained in the auxiliary direction formula under some mild assumptions without the strict complementarity the global convergence and superlinear convergence of the algorithm can be obtained

sylvester s question the probability that n points are in convex position for a convex body k in the plane let p n k denote the probability that n random independent and uniform points from k are in convex position that is none of them lies in the convex hull of the others here we determine the asymptotic behavior of p n k by showing that as n goes to infinity n 2 n root p n k tends to a finite and positive limit

on normal approximation rates for certain sums of dependent random variables abstract let x1 xn be dependent random variables and set e ni 1xi and 2 var ni 1xi in most of the applications of stein s method for normal approximations the error rate p ni 1xi w w is of the order of 1 2 this rate was improved by stein 1986 and others in some special cases in this paper it is shown that for certain bounded random variables a simple refinement of error term calculations in stein s method leads to improved rates

a generalized super memory gradient projection method of strongly sub feasible directions with strong convergence for nonlinear inequality constrained optimization in this work combining the properties of the generalized super memory gradient projection methods with the ideas of the strongly sub feasible directions methods we present a new algorithm with strong convergence for nonlinear inequality constrained optimization at each iteration the proposed algorithm can sufficiently use the information of the previous t steps iterations to generate a new iterative point particularly the intervals of parameters in the super memory gradient projection direction are adjustable the main properties of the new algorithm are described as follows i the improving super memory gradient projection direction is a combination of the generalized gradient projection and the t steps super memory gradients which include both the previous t steps search directions and gradients moreover only the gradients associated with a generalized active constrained set are dealt with rather than the gradients of all constraints ii the initial point can be chosen arbitrarily and at each iteration the number of the functions satisfying the inequality constraints is nondecreasing especially once a feasible iteration is obtained then the subsequent iterations are also feasible iii under suitable assumptions it possesses global and strong convergence finally some preliminary numerical results show that the proposed algorithm is promising

a superlinearly convergent method of feasible directions method of feasible directions mfd is one of the most important methods for solving engineering optimization problems there exist two types of mfd one has at most a linear convergence rate and the other has a superlinear convergence rate the former type includes zoutendijk s mfd topkis veinott s mfd pironneau polak s mfd cawood kostreva s norm relaxed mfd and a modified mfd the latter includes panier tits fsqp approaches however the price of the superlinearly convergent mfd for achieving superlinearity is rather high computational cost per single iteration this paper presents an algorithm based on the modified mfd the theoretical analysis shows that this algorithm is locally superlinearly convergent this algorithm has attained the same convergence rate as fsqp while it only needs to solve two subproblems instead of three subproblems in fsqp hence this algorithm should be computationally more efficient than fsqp

a superlinearly convergent norm relaxed method of quasi strongly sub feasible direction for inequality constrained minimax problems in this paper nonlinear minimax problems with inequality constraints are discussed combined the norm relaxed sqp method with the idea of strongly sub feasible directions method a new method of quasi strongly sub feasible directions mqssfd with arbitrary initial point for the discussed problems is presented at each iteration of the proposed algorithm an improved search direction is obtained by solving a quadratic program qp which always has a solution and a high order correction direction is yielded via a system of linear equations sle to avoid the maratos effect after finite iterations the iteration point always get into the feasible set by introducing a new non monotone curve search under some mild conditions including the weak mangasarian fromovitz constraint qualification mfcq the proposed algorithm possesses global convergence and the superlinear convergence is obtained without the strict complementarity finally some elementary numerical experiments are implemented and reported

stability results for the volume of random simplices it is known that for a convex body k in r d of volume one the expected volume of random simplices in k is minimized if k is an ellipsoid and for d 2 maximized if k is a triangle here we provide corresponding stability estimates

generalized monotone line search sqp algorithm for constrained minimax problems in this article non linear minimax problems with general constraints are discussed by means of solving one quadratic programming an improved direction is yielded and a second order correction direction can also be at hand via one system of linear equations so a new algorithm for solving the discussed problems is presented in connection with a special merit function the generalized monotone line search is used to yield the step size at each iteration under mild conditions we can ensure global and superlinear convergence finally some numerical experiments are operated to test our algorithm and the results demonstrate that it is promising

limit theorems for functionals of convex hulls in 4 a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of a convex polygon is derived this is done by approximating the process of vertices of the convex hull by the process of extreme points of a poisson point process and by considering the latter process of extreme points as a markov process for a particular parametrization we show that this method can also be applied to derive limit theorems for the boundary length and for the area of the convex hull this extents results of renyi and sulanke 1963 and buchta 1984 and shows that the boundary length and the area have a strikingly different probabilistic behavior

sequential quadratically constrained quadratic programming norm relaxed algorithm of strongly sub feasible directions abstract in this paper we present a sequential quadratically constrained quadratic programming sqcqp norm relaxed algorithm of strongly sub feasible directions for the solution of inequality constrained optimization problems by introducing a new unified line search and making use of the idea of strongly sub feasible direction method the proposed algorithm can well combine the phase of finding a feasible point by finite iterations and the phase of a feasible descent norm relaxed sqcqp algorithm moreover the former phase can preserve the sub feasibility of the current iteration and control the increase of the objective function at each iteration only a consistent convex quadratically constrained quadratic programming problem needs to be solved to obtain a search direction without any other correctional directions the global superlinear and a certain quadratic convergence which is between 1 step and 2 step quadratic convergence properties are proved under reasonable assumptions finally some preliminary numerical results show that the proposed algorithm is also encouraging

a superlinearly convergent norm relaxed sqp method of strongly sub feasible directions for constrained optimization without strict complementarity in this paper a kind of optimization problems with nonlinear inequality constraints is discussed combined the ideas of norm relaxed sqp method and strongly sub feasible direction method as well as a pivoting operation a new fast algorithm with arbitrary initial point for the discussed problem is presented at each iteration of the algorithm an improved direction is obtained by solving only one direction finding subproblem which possesses small scale and always has an optimal solution and to avoid the maratos effect another correction direction is yielded by a simple explicit formula since the line search technique can automatically combine the initialization and optimization processes after finite iterations the iteration points always get into the feasible set the proposed algorithm is proved to be globally convergent and superlinearly convergent under mild conditions without the strict complementarity finally some numerical tests are reported

variance asymptotics and scaling limits for random polytopes abstract let k be a convex set in r d and let k be the convex hull of a homogeneous poisson point process p of intensity on k when k is a simple polytope we establish scaling limits as for the boundary of k in a vicinity of a vertex of k and we give variance asymptotics for the volume and k face functional of k k 0 1 d 1 resolving an open question posed in 17 the scaling limit of the boundary of k and the variance asymptotics are described in terms of a germ grain model consisting of cone like grains pinned to the extreme points of a poisson point process on r d 1 r having intensity d e d h d h d v

a generalization of the norm relaxed method of feasible directions this paper generalizes the norm relaxed method of feasible directions which has roots in the classical method of feasible directions mfd this generalized algorithm introduces some parameters which can be adjusted to speed up the convergence an investigation of its global convergence is included and the numerical performance is shown on 15 test problems its global convergence is guaranteed under a rather mild and standard assumption the numerical tests show that this new algorithm converges faster than the previous norm relaxed mfd for most test problems by selecting a set of suitable parameters even though it only has the same guaranteed convergence rate as the previous norm relaxed mfd

central limit theorems for random polytopes in a smooth convex set let k be a smooth convex set with volume one in choose n random points in k independently according to the uniform distribution the convex hull of these points denoted by kn is called a random polytope we prove that several key functionals of kn satisfy the central limit theorem as n tends to infinity

supporting points processes and some of their applications we introduce a stochastic point process of s supporting points and prove that upon rescaling it converges to a gaussian field the notion of s supporting points specializes for adequately chosen s to pareto or more generally cone extremal points or to vertices of convex hulls or to centers of generalized voronoi tessellations in the models of large scale structure of the universe based on burgers equation the central limit theorems proven here imply i a the asymptotic normality for the number of convex hull vertices in large poisson sample from a simple polyhedra or for the number of pareto vector extremal points in poisson samples with independent coordinates

the convex hull of the integer points in a large ball where fk p denotes the number of k dimensional faces of the polytope p the limit as r of the average of r 2 3f0 pr on an interval r r h is determined by balog and deshoullier bd and turns out to be 3 453 h must be large our main result extends 1 1 to any d 2 and to any fk pr with k 0 d 1 theorem 1 for every d 2 there are constants c1 d and c2 d such that for all k 0 d 1

cones generated by random points on half spheres and convex hulls of poisson point processes let u 1 u 2 ldots be random points sampled uniformly and independently from the d dimensional upper half sphere we show that as n rightarrow infty the f vector of the d 1 dimensional convex cone c n generated by u 1 ldots u n weakly converges to a certain limiting random vector without any normalization we also show convergence of all moments of the f vector of c n and identify the limiting constants for the expectations we prove that the expected grassmann angles of c n can be expressed through the expected f vector this yields convergence of expected grassmann angles and conic intrinsic volumes and answers thereby a question of barany et al random struct algorithms 50 1 3 22 2017 https doi org 10 1002 rsa 20644 our approach is based on the observation that the random cone c n weakly converges after a suitable rescaling to a random cone whose intersection with the tangent hyperplane of the half sphere at its north pole is the convex hull of the poisson point process with power law intensity function proportional to vert x vert d gamma where gamma 1 we compute the expected number of facets the expected intrinsic volumes and the expected t functional of this random convex hull for arbitrary gamma 0

on the variance of random polytopes abstract a random polytope is the convex hull of uniformly distributed random points in a convex body k a general lower bound on the variance of the volume and f vector of random polytopes is proved also an upper bound in the case when k is a polytope is given for polytopes as for smooth convex bodies the upper and lower bounds are of the same order of magnitude the results imply a law of large numbers for the volume and f vector of random polytopes when k is a polytope

equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points a general formula is proved which relates the equiaffine inner parallel curves of a plane convex body and the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points this formula is applied to improve some well known results in geometric probability for example an estimate which was established for a special case by l c g rogers is obtained with the best possible bound and an asymptotic formula due to a renyi and r sulanke is extended to an asymptotic expansion

a new sqp method of feasible directions for nonlinear programming in this paper a new sequential quadratic programming sqp method of feasible directions is proposed and analyzed for nonlinear programming where a feasible direction of descent can be derived from solving only one qp subproblem in particular this method can produce automatically a revised direction with the explicit expression which can avoid maratos effect without solving qp subproblem the theoretical analysis shows that global and superlinear convergence can be induced in the end numerical experiment is given to illustrate the effectiveness of the method

random points on the boundary of smooth convex bodies the convex hull of n independent random points chosen on the boundary of a convex body k c r d according to a given density function is a random polytope the expectation of its i th intrinsic volume for i 1 d is investigated in the case that the boundary of k is sufficiently smooth asymptotic expansions for these expected intrinsic volumes as n oo are derived

affine perimeter and limit shape it is proved here that as n infinity almost all convex 1 n z 2 lattice polygons lying in a given convex compact set k subset of r 2 are very close to a fixed convex set k 0 the distinguishing property of k 0 is that its affine perimeter is the largest among all convex sets contained in k

stable vortex solutions to the ginzburg landau equation with a variable coefficient in a disk abstract this paper deals with stable solutions with a single vortex to the ginzburg landau equation having a variable coefficient subject to the neumann boundary condition in a planar disk the equation has a positive parameter say which will play an important role for the stability of the solution we consider the equation with a radially symmetric coefficient in the disk and suppose that the coefficient is monotone increasing in a radial direction then the equation possesses a pair of solutions with a single vortex for large although these solutions for the constant coefficient are unstable they can be stable for a suitable variable coefficient and large the purpose of this article is to give a sufficient condition for the coefficient to allow those solutions being stable for any sufficiently large as an application we show an example of the coefficient enjoying the condition which has an arbitrarily small total variation

quadratically constraint quadratical algorithm model for nonlinear minimax problems abstract in this paper a quadratically approximate algorithm framework for solving general constrained minimax problems is presented the framework contains the idea of the sequential quadratic programming method the sequential quadratically constrained quadratic programming method norm relaxed method and strong sub feasible method the global convergence of the algorithm framework is obtained under the mangasarian fromovitz constraint qualification mfcq and the conditions for superlinear convergence of the algorithm framework are presented under the mfcq the constant rank constraint qualification crcq as well as the strong second order sufficiency conditions ssosc and quadratic convergence rate is obtained under the mfcq and ssosc

a method combining norm relaxed qp subproblems with systems of linear equations for constrained optimization based on the ideas of norm relaxed sequential quadratic programming sqp method and the strongly sub feasible direction method we propose a new sqp algorithm for the solution of nonlinear inequality constrained optimization unlike the previous work at each iteration the norm relaxed quadratic programming subproblem nrqps in our algorithm only consists of the constraints corresponding to an estimate of the active set and the high order correction direction used to avoid the maratos effect is obtained by solving a system of linear equations sle which also only consists of such a subset of constraints and gradients moreover the line search technique can effectively combine the initialization process with the optimization process and therefore if the starting point is not feasible the iteration points always get into the feasible set after a finite number of iterations the global convergence is proved under the mangasarian fromovitz constraint qualification mfcq and the superlinear convergence is obtained without assuming the strict complementarity finally the numerical experiments show that the proposed algorithm is effective and promising for the test problems

feasible generalized monotone line search sqp algorithm for nonlinear minimax problems with inequality constraints in this paper the nonlinear minimax problems with inequality constraints are discussed and a sequential quadratic programming sqp algorithm with a generalized monotone line search is presented at each iteration a feasible direction of descent is obtained by solving a quadratic programming qp to avoid the maratos effect a high order correction direction is achieved by solving another qp as a result the proposed algorithm has global and superlinear convergence especially the global convergence is obtained under a weak mangasarian fromovitz constraint qualification mfcq instead of the linearly independent constraint qualification licq at last its numerical effectiveness is demonstrated with test examples

random polytopes in a convex polytope independence of shape and concentration of vertices write d for the set of all convex bodies convex compact sets with nonempty interior in d define o g l d as the set of those k e 5 b d with vol k 1 fix k e g i d and choose points x l x e k randomly independently and according to the uniform distribution on k then k c o n v x l xn is a random polytope in k write e k n for the expectation of the random variable v o l k k n e k n shows how well k approximates k in volume on the average groemer grl proved that among all convex bodies k e o g l d the ellipsoids are approximated worst i e

flag numbers and floating bodies abstract we investigate weighted floating bodies of polytopes we show that the weighted volume depends on the complete flags of the polytope this connection is obtained by introducing flag simplices which translate between the metric and combinatorial structure our results are applied in spherical and hyperbolic space this leads to new asymptotic results for polytopes in these spaces we also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in euclidean space

limit theorems for convex hulls it is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally after rescaling to a strongly mixing markov process as the sample size tends to infinity the structure of the limiting markov process is determined explicitly and from this a central limit theorem for the number of vertices of the convex hull is derived similar results are given for uniform samples from the unit disk