Stability and continuation of solutions to obstacle problems

AbstractIn this paper we will give a summary of some of our results which we have obtained recently. We mainly consider the question whether solutions to variational inequalities with an eigenvalue parameter are stable in the sense defined in Section 1. More precisely, we ask whether a solution to the variational inequality yields a strict local minimum of an associated energy functional defined on a closed convex subset of a real Hilbert space. This nonlinearity of the space of admissible vectors implies a new and interesting stability behavior of the solutions which is not present in the case of equations.Moreover, it is noteworthy that optimal regularity properties of the solutions to the variational inequality are needed for the stability criterion which we will describe in Section 2. Applications to the beam and plate are considered in Sections 4 and 5. In the case of a plate, numerical computations are crucial because it is impossible to find an analytical expression for a branch of solutions to the variational inequality which is not also a solution to the free problem. Closely connected to the question of stability of a given solution to a variational inequality is the question of the continuation of this solution, which we will discuss in Section 3.In Section 6 a survey will be given on the methods used for the computation of stability bounds. This includes in particular a short introduction to continuation algorithms for both equations and variational inequalities.Frequent references will be made to the literature of direct relevance to the material presented. A few additional related research papers or monographs have been included in the bibliography (Courant and Hilbert (1962/1968), Fichera (1972), Funk (1962), Glowinski et al. (1981), Kikuchi and Oden (1988), Landau and Lifschitz (1970), Lions (1971) and Lions and Stampacchia (1967))

High accuracy semidefinite programming bounds for kissing numbers

The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing number based on semidefinite programming. This paper is a report on high accuracy calculations of these upper bounds for n <= 24. The bound for n = 16 implies a conjecture of Conway and Sloane: There is no 16-dimensional periodic point set with average theta series 1 + 7680q^3 + 4320q^4 + 276480q^5 + 61440q^6 + ..

Second order optimality conditions and their role in PDE control

If f : Rn R is twice continuously differentiable, f’(u) = 0 and f’’(u) is positive definite, then u is a local minimizer of f. This paper surveys the extension of this well known second order suffcient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled order sufficient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled? It turns out that infinite dimensions cause new difficulties that do not occur in finite dimensions. We will be faced with the surprising fact that the space, where f’’(u) exists can be useless to ensure positive definiteness of the quadratic form v f’’(u)v2. In this context, the famous two-norm discrepancy, its consequences, and techniques for overcoming this difficulty are explained. To keep the presentation simple, the theory is developed for problems in function spaces with simple box constraints of the form a = u = ß. The theory of second order conditions in the control of partial differential equations is presented exemplarily for the nonlinear heat equation. Different types of critical cones are introduced, where the positivity of f’’(u) must be required. Their form depends on whether a so-called Tikhonov regularization term is part of the functional f or not. In this context, the paper contains also new results that lead to quadratic growth conditions in the strong sense. As a first application of second-order sufficient conditions, the stability of optimal solutions with respect to perturbations of the data of the control problem is discussed. Second, their use in analyzing the discretization of control problems by finite elements is studied. A survey on further related topics, open questions, and relevant literature concludes the paper.The first author was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2011-22711, the second author by DFG in the framework of the Collaborative Research Center SFB 910, project B6

Parallel Multisplittings for Constrained Optimization

. The philosophy of multisplitting methods is the replacement of a large-scale linear or nonlinear problem by a set of smaller subproblems, each of which can be solved locally and independently in parallel by taking advantage of well-tested sequential algorithms. Because of this formulation most compute-intensive operations can be calculated independently and the algorithms are highly parallel. In continuation of our earlier work we utilize a new parameter-free formulation of linearly constrained convex minimization problems to obtain a parallel algorithm of multisplitting type. Numerical results both serial and parallel are reported which demonstrate its efficiency and which also show that it compares favorably to our earlier parameter-dependent approach. Key words. Parallel algorithms, multisplitting, constrained optimization. Computing Reviews. D.1.3, G.1.6 1. Introduction We consider the general convex minimization problem minf(x) subject to h(x) = 0; g(x) 0: (1) Here, f : R n ..

Semidefinite code bounds based on quadruple distances

Let A(n,d) be the maximum number of 0, 1 words of length n , any two having Hamming distance at least d. It is proved that A(20,8)=256, which implies that the quadruply shortened Golay code is optimal. Moreover, it is shown that A(18,6) ≤ 673, A(19,6) ≤ 1237, A(20,6) ≤ 2279, A(23,6) ≤ 13674, A(19,8) ≤ 135, A(25,8) ≤ 5421, A(26,8) ≤ 9275, A(27,8) ≤ 17099, A(21,10) ≤ 47, A(22,10) ≤ 84, A(24,10) ≤ 268, A(25,10) ≤ 466, A(26,10) ≤ 836, A(27,10) ≤ 1585, A(28,10) ≤ 2817, A(25,12) ≤ 55, and A(26,12) ≤ 96. The method is based on the positive semidefiniteness of matrices derived from quadruples of words. This can be put as constraint in a semidefinite program, whose optimum value is an upper bound for A(n,d). The order of the matrices involved is huge. However, the semidefinite program is highly symmetric, by which its feasible region can be restricted to the algebra of matrices invariant under this symmetry. By block diagonalizing this algebra, the order of the matrices will be reduced so as to make the program solvable with semidefinite programming software in the above range of values of n and d