proximal point approach and approximation of variational inequalities a general approach to analyze convergence of the proximal like methods for variational inequalities with set valued maximal monotone operators is developed it is oriented to methods coupling successive approximation of the variational inequality with the proximal point algorithm as well as to related methods using regularization on a subspace and weak regularization this approach also covers so called multistep regularization methods in which the number of proximal iterations in the approximated problems is controlled by a criterion characterizing these iterations as to be effective the conditions on convergence require control of the exactness of the approximation only in a certain region of the original space conditions ensuring linear convergence of the methods are established

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

a min max theorem on tournaments we present a structural characterization of all tournaments t v a such that for any nonnegative integral weight function defined on v the maximum size of a feedback vertex set packing is equal to the minimum weight of a triangle in t we also answer a question of frank by showing that it is np complete to decide whether the vertex set of a given tournament can be partitioned into two feedback vertex sets in addition we give exact and approximation algorithms for the feedback vertex set packing problem on tournaments

how to reduce the average complexity of convex hull finding algorithms abstract let x 1 x n be a sequence of independent r d valued random vectors with a common density the following class of convex hull finding algorithms is considered find the extrema in a finite number of carefully chosen directions eliminate the x i s that belong to the interior of the polyhedron formed by these extrema apply an 0 n worst case complexity algorithm to find the convex hull of the remaining points we give weak sufficient conditions that imply that the overall average complexity is 0 n we also show that for the standard normal density the average complexity is 0 n whenever n n log n

degree constrained 2 partitions of graphs a k1 k2 partition of a graph g is a vertex partition v1 v2 of g satisfying that g vi ki for i 1 2 we determine for all positive integers k1 k2 the complexity of deciding whether a given graph has a k1 k2 partition we also address the problem of finding a function g k1 k2 such that the k1 k2 partition problem is n p complete for the class of graphs of minimum degree less than g k1 k2 and polynomial for all graphs with minimum degree at least g k1 k2 we prove that g 1 k k for k 3 that g 2 2 3 and that g 2 3 if it exists has value 4 or 5

asymptotic convergence analysis of the proximal point algorithm the asymptotic convergence of the proximal point algorithm ppa for the solution of equations of type 0 in tz where t is a multivalued maximal monotone operator in a real hilbert space is analyzed when 0 in tz has a nonempty solution set bar z convergence rates are shown to depend on how rapidly t 1 grows away from bar z in a neighbourhood of 0 when this growth is bounded by a power function with exponent s then for a sequence z k generated by the ppa z k bar z converges to zero like o k s 2 linearly superlinearly or in a finite number of steps according to whether s in 0 1 s 1 s in 1 infty or s infty

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

finding good 2 partitions of digraphs i hereditary properties we study the complexity of deciding whether a given digraph d has a vertex partition into two disjoint subdigraphs with given structural properties let h and e denote the following two sets of natural properties of digraphs h acyclic complete arcless oriented no 2 cycle semicomplete symmetric tournament and e strongly connected connected minimum out degree at least 1 minimum in degree at least 1 minimum semi degree at least 1 minimum degree at least 1 having an out branching having an in branching in this paper we determine the complexity of deciding for any fixed pair of positive integers k 1 k 2 whether a given digraph has a vertex partition into two digraphs d 1 d 2 such that v d i k i and d i has property p i for i 1 2 when p 1 h and p 2 h e we also classify the complexity of the same problems when restricted to strongly connected digraphs the complexity of the 2 partition problems where both p 1 and p 2 are in e is determined in the companion paper 2

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

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

optimal control of the convection diffusion equation using stabilized finite element methods in this paper we analyze the discretization of optimal control problems governed by convection diffusion equations which are subject to pointwise control constraints we present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case moreover the in general different approaches optimize then discretize and discretize then optimize coincide for the proposed discretization scheme this allows for a symmetric optimality system at the discrete level and optimal order of convergence

the parameterized complexity landscape of finding 2 partitions of digraphs abstract given a network modeled by a directed graph d v a it is natural to ask whether we can partition the vertex set of d into two disjoint subsets v 1 v 2 called a 2 partition such that the digraphs d v 1 d v 2 induced by each of these has one of the two properties of interest this question gives rise to a rich realm of combinatorial problems the complexity of many such problems was determined in 2 3 we analyze a subset of those problems from the viewpoint of parameterized complexity and present a complete dichotomy of basic natural properties more precisely given a directed graph d v a and two non negative integers k 1 and k 2 we seek a 2 partition v 1 v 2 of the vertex set v such that v 1 k 1 v 2 k 2 and each of the subdigraphs induced by v 1 and v 2 has a structural property as defined by the problem at hand for example d v 1 is acyclic and d v 2 is strongly connected specifically we consider the following eight structural properties being strongly connected being connected having an out branching having an in branching having minimum degree at least one having minimum semi degree at least one being acyclic and being complete

penalization in non classical convex programming via variational convergence a penalty method for convex functions which cannot necessarily be extended outside their effective domains by an everywhere finite convex function is proposed and combined with the proximal method proofs of convergence rely on variational convergence theory

proximal point method and elliptic regularization abstract a generalized proximal point method for solving variational inequalities with maximal monotone operators is developed it admits a successive approximation of the feasible set and of a symmetric component of the operator as well as an inexact solving of the regularized problems the conditions on the approximation are coordinated with the properties of finite element methods for solving problems in mathematical physics the choice of the regularizing functional exploits a possible reserve of monotonicity of the operator in the variational inequality for the minimal surface problem and related variational inequalities as well as for the convection diffusion problem the studied method extends the principle of elliptic regularization a special convergence analysis shows a more qualitative convergence of the method applied to these problems than it follows from the general theory of proximal point methods also applications to some variational inequalities from the elasticity theory are investigated

finding good 2 partitions of digraphs ii enumerable properties we continue the study initiated in 3 of the complexity of deciding whether a given digraph d has a vertex partition into two disjoint subdigraphs with given structural properties and given minimum cardinality let e be the following set of properties of digraphs e strongly connected connected minimum out degree at least 1 minimum in degree at least 1 minimum semi degree at least 1 minimum degree at least 1 having an out branching having an in branching in this paper we determine for all choices of p 1 p 2 from e and all pairs of fixed positive integers k 1 k 2 the complexity of deciding whether a digraph has a vertex partition into two digraphs d 1 d 2 such that d i has property p i and v d i k i i 1 2 we also classify the complexity of the same problems when restricted to strongly connected digraphs the complexity of the analogous problems when p 1 h and p 2 h e where h acyclic complete arc less oriented no 2 cycle semicomplete symmetric tournament were completely characterized in 3

construction of inertial manifolds by elliptic regularization abstract in many cases an inertial manifold 2 m for an infinite dimensional dissipative dynamical system can be represented as the graph of a smooth function from a finite dimensional hilbert space hp to another hilbert space hq the invariance property of m means that can be written as the solution of a first order partial differential equation d p g1 p p a p g2 p p 0 over hp where g1 and g2 are nonlinear functions which depend on the original dynamical system and a is a suitably stable linear operator in this paper we use a method introduced by sacker r j sacker a new approach to the perturbation theory of invariant surface comm pure appl math 18 1965 717 732 for the study of finite dimensional dynamical systems to find inertial manifolds in the infinite dimensional setting this method involves replacing the first order equation for by the regularized elliptic equation e d p g1 p p a p g2 p p with suitable boundary conditions it is shown that if a satisfies a spectral gap condition then the solutions e of the elliptic equation converge to a weak solution of 0 as e 0 furthermore m graph is an invariant manifold for the given dynamical system

out degree reducing partitions of digraphs abstract let k be a fixed integer we determine the complexity of finding a p partition v 1 v p of the vertex set of a given digraph such that the maximum out degree of each of the digraphs induced by v i 1 i p is at least k smaller than the maximum out degree of d we show that this problem is polynomial time solvable when p 2 k and np complete otherwise the result for k 1 and p 2 answers a question posed in 3 we also determine for all fixed non negative integers k 1 k 2 p the complexity of deciding whether a given digraph of maximum out degree p has a 2 partition v 1 v 2 such that the digraph induced by v i has maximum out degree at most k i for i 2 it follows from this characterization that the problem of deciding whether a digraph has a 2 partition v 1 v 2 such that each vertex v v i has at least as many neighbours in the set v 3 i as in v i for i 1 2 is np complete this solves a problem from 6 on majority colourings

on the parameterized complexity of 2 partitions abstract we give an fpt algorithm for deciding whether the vertex set of a digraph d can be partitioned into two disjoint sets v 1 v 2 such that the digraph d v 1 induced by v 1 has a vertex that can reach all other vertices by directed paths the digraph d v 2 has no vertex of in degree zero and v i k i where k 1 k 2 are part of the input this settles an open problem from 1 4

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

proximal schemes for parabolic optimal control problems with sparsity promoting cost functionals abstractfast first order proximal methods that solve linear and bilinear parabolic optimal control problems with a sparsity cost functional are discussed weak convergence of these methods is proved and for benchmarking purposes the proposed inexact proximal schemes are compared to an inexact semi smooth newton method results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes and to validate the theoretical estimates

degree constrained 2 partitions of semicomplete digraphs abstract a 2 partition of a digraph d is a partition v 1 v 2 of v d into two disjoint non empty sets v 1 and v 2 such that v 1 v 2 v d a semicomplete digraph is a digraph with no pair of non adjacent vertices we consider the complexity of deciding whether a given semicomplete digraph has a 2 partition such that each part of the partition induces a semicomplete digraph with some specified property in 4 and 5 bang jensen cohen and havet determined the complexity of 120 such 2 partition problems for general digraphs several of these problems are np complete for general digraphs and thus it is natural to ask whether this is still the case for well structured classes of digraphs such as semicomplete digraphs this is the main topic of the paper more specifically we consider 2 partition problems where the set of properties are minimum out minimum in or minimum semi degree among other results we prove the following for all integers k 1 k 2 1 and k 1 k 2 3 it is np complete to decide whether a given digraph d has a 2 partition v 1 v 2 such that d v i has out degree at least k i for i 1 2 for every fixed choice of integers k 1 k 2 1 there exists a polynomial algorithm for deciding whether a given digraph of independence number at most has a 2 partition v 1 v 2 such that d v i has out degree at least k i for i 1 2 for every fixed integer k 1 there exists a polynomial algorithm for deciding whether a given semicomplete digraph has a 2 partition v 1 v 2 such that d v 1 has out degree at least one and d v 2 has in degree at least k it is np complete to decide whether a given semicomplete digraph d has a 2 partition v 1 v 2 such that d v i is a strong tournament