Constraint decomposition algorithms in global optimization

Many global optimization problems can be formulated in the form min{c(x, y): x X, y Y, (x, y) Z, y G} where X, Y are polytopes in ^p , ⁿ , respectively, Z is a closed convex set in ^p+n, while G is the complement of an open convex set in ⁿ . The function c: ^p+n is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ⁿ . Computational experiments show that the resulting algorithms work well for problems with smalln.     

The obstacle problem for an elastoplastic body

The obstacle problem for elastoplastic bodies is considered within the framework of general existence results for unilateral problems recently presented by Baiocchiet al. Two models of plasticity are considered: one is based on a displacement-plastic strain formulation and the second, a specialization of the first, is the standard Hencky model. Existence theorems are given for the Neumann problem for a body constrained to lie on or above the half-space {x³:x ³0}. For hardening materials the displacements are sought in the Sobolev spaceH ¹(, ³) while for perfectly plastic materials they are sought in BD(), the space of functions of bounded deformation. Conditions for the existence of solutions are given in terms of compatibility and safe load conditions on applied loads.This work was initiated while B. D. Reddy was on leave at the Istituto di Analisi Numerica del CNR, Pavia, Italy. The hospitality of that institution is gratefully acknowledged, as is the award of an FRD sabbatical grant. F. Tomarelli was partially supported by IAN and GNAFA of CNR, and by Ministero della Pubblica Istruzione.     

A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

Summary In this paper, we shall be concerned with the solution of constrained convex minimization problems. The constrained convex minimization problems are proposed to be transformable into a convex-additively decomposed and almost separable form, e.g. by decomposition of the objective functional and the restrictions. Unconstrained dual problems are generated by using Fenchel-Rockafellar duality. This decomposition-dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable. Moreover, the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return-formulas. In quadratic programming the decomposition-dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated byH-matrices. Numerical tests for solving obstacle problems in ¹ discretized by using piecewise quadratic finite elements and in ² by using the five-point difference approximation are presented.     

On Two-Point Boundary Conditions in Optimal Control Problems

Let x=g(t,x(t),u(t)) be the governing equation of an optimal control problem with two-point boundary conditions h ₀(x(a))+h ₁(x(b)) = 0, where x: [a,b] ⁿ is continuous, u: [a,b] ^k-n is piecewise continuous and left continuous, h₀,h₁: ^{n q} are continuously differentiable, and g:[a,b]× ^{k n} is continuous. The paper finds functions _i C¹([a,b]× ⁿ ) such that (x(t),u(t)) is a solution of the governing equation if and only if

Minimal Surfaces in a Cone

We prove the convex hull property for properly immersed minimal hypersurfaces in a cone of ⁿ . We deal with the existence of new barriers for the maximum principle application in noncompact truncated tetrahedral domains of ³, describing the space of such domainsadmitting barriers of this kind. Nonexistence results for nonflatminimal surfaces whose boundary lies in opposite faces of a tetrahedraldomain are obtained. Finally, new simple closed subsets of ³ whichhave the property of intersecting any properly immersed minimal surfaceare shown.     

On Non-Negative Elliptic-Type Differential Inequalities in ℝ n , Vanishing Almost Everywhere on Some Open Subset in ℝ n

It is proved in this article that any generalized solution of a sufficiently general class of elliptic-type differential inequalities in  ⁿ that is non-negative almost everywhere in  ⁿ and vanishes almost everywhere on an open set ⁿ is trivial in  ⁿ .     

Quasi-Newton methods for saddlepoints

The well-known quadratically convergent methods of the Huang type (Refs. 1 and 2) to maximize or minimize a functionf: ⁿ are generalized to find saddlepoints off. Furthermore, a procedure is derived which homes in on saddlepoints with prescribed inertia, i.e., with a given number of positive and negative eigenvalues in the Hessian matrix off. Examples are presented to show that saddlepoints with different inertia can be calculated from the same starting vector.     

Random continued fractions and inverse Gaussian distribution on a symmetric cone

In this paper we introduce the inverse Gaussian and Wishart distributions on the cone of real (n, n) symmetric positive definite matricesH _n ⁺ () and more generally on an irreducible symmetric coneC. Then we study the convergence of random continued fractions onH _n ⁺ () andC by means of real Lagrangians forH _n ⁺ () and by new algebraic identities on symmetric cones forC. Finally we get a characterization of the inverse Gaussian distribution onH _n ⁺ () andC.     

Reducible Dupin submanifolds

Pinkall's standard constructions for obtaining a Dupin hypersurface W in ^N from a Dupin hypersurface M in ⁿ , N>n, are studied in the context of Lie sphere geometry. It is shown that a compact Dupin hypersurface W in ^N with g distinct principal curvatures at each point is reducible to a compact Dupin hypersurface M in ⁿ if and only if g=2.This research was supported by NSF Grant No. DMS 87-06015.     

The (n + 1)2 m -ray algorithm: a new simplicial algorithm for the variational inequality problem on ℝ + m ×S n

In this paper we propose a variable dimension simplicial algorithm for solving the variational inequality problem on the cross product of the nonnegative orthant ₊ ^m of them-dimensional Euclidean space ^m and then-dimensional unit simplexS ⁿ of ⁿ⁺¹. Starting from an arbitrary point (u, v) є ₊ ^m ×S ⁿ, the algorithm generates a piecewise linear path in ₊ ^m ×S ⁿ. The path is traced by making alternately linear programming pivot operations and replacement steps in an appropriate simplicial subdivision of ₊ ^m ×S ⁿ. The algorithm differs from the thus far known algorithm in the number of directions in which it may leave the starting point. More precisely, the algorithm has (n+1)2 ^m rays to leave the starting point whereas the existing algorithm hasn+m+1 rays. A convergence condition is presented and the accuracy estimation of an approximate solution generated is also given.     

Embedded minimal annuli solving an exterior problem

Given a compact, strictly convex body in ³ and a closed Jordan curve ³ satisfying several additional assumptions, the existence of a parametric, annulus type minimal surface is proved, which parametrizes along one boundary component, has a free boundary onX along the other boundary component, and which stays in ³. As a consequence of this and a reasoning developed by W. H. Meeks and S. -T. Yau we find an embedded minimal surface with these properties. Another application is the existence of an embedded minimal surface with a flat end, free boundary onX and controlled topology.This article was processed by the author using the LATEX style filepljourlm from Springer-Verlag.     

Downward Sets and their separation and approximation properties

We develop a theory of downward subsets of the space ^I, where I is a finite index set. Downward sets arise as the set of all solutions of a system of inequalities x^I,f_t(x)0 (tT), where T is an arbitrary index set and each f _t (tT) is an increasing function defined on ^I. These sets play an important role in some parts of mathematical economics and game theory. We examine some functions related to a downward set (the distance to this set and the plus-Minkowski gauge of this set, which we introduce here) and study lattices of closed downward sets and of corresponding distance functions. We discuss two kinds of duality for downward sets, based on multiplicative and additive min-type functions, respectively, and corresponding separation properties, and we give some characterizations of best approximations by downward sets. Some links between the multiplicative and additive cases are established.     

Ouverts d"harmonicité pour les fonctions séparément harmoniques

Let D ^N , G ^M be two open sets, E D and F G two compact sets which satisfy the condition (H) (that is a harmonic condition similar to Leja"s condition). We find an open set ^N+M such that each separately harmonic function f : X : = (D× F) (E × G) (i.e.: for all x in E, f(x,.) is harmonic on G; for all y in F, f(., y) is harmonic on D) extends to a harmonic function on .     

Fibred links and a construction of real singularities via complex geometry

This note gives a method for constructing real analytic maps from ²ⁿ into ², with an isolated critical point at 0 ²ⁿ , for alln>1. This provides infinite families of real singularities which fiber a la Milnor.Research partially supported by CONACYT, Mexico, grant 1206-E92103.     

Conditions for separately subharmonic functions to be subharmonic

Letu be a function on ^m × ⁿ , wherem2 andn2, such thatu(x, .) is subharmonic on ⁿ for each fixedx in ^m andu(.,y) is subharmonic on ^m for each fixedy in ⁿ . We give a local integrability condition which ensures the subharmonicity ofu on ^m × ⁿ , and we show that this condition is close to being sharp. In particular, the local integrability of (log⁺ u ⁺) ^m+n–2+ is enough to secure the subharmonicity ofu if >0, but not if <0.     

Bochner-Riesz means of functions in weak-L p

The Bochner-Riesz means of order 0 for suitable test functions on ^N are defined via the Fourier transform by . We show that the means of the critical index , do not mapL ^p,( ^N ) intoL ^p,( ^N ), but they map radial functions ofL ^p,( ^N ) intoL ^p,( ^N ). Moreover, iff is radial and in theL ^p,( ^N ) closure of test functions,S _R f(x) converges, asR+, tof(x) in norm and for almost everyx in ^N . We also observe that the means of the function|x| ^–N/p, which belongs toL ^p,( ^N ) but not to the closure of test functions, converge for nox.     

Outer approximation algorithm for nondifferentiable optimization problems

It is known that the problem of minimizing a convex functionf(x) over a compact subsetX of ⁿ can be expressed as minimizing max{g(x, y)|y X}, whereg is a support function forf[f(x) g(x, y), for ally X andf(x)=g(x, x)]. Standard outer-approximation theory can then be employed to obtain outer-approximation algorithms with procedures for dropping previous cuts. It is shown here how this methodology can be extended to nonconvex nondifferentiable functions.This research was supported by the Science and Engineering Research Council, UK, and by the National Science Foundation under Grant No. ECS-79-13148.     

Gewichtete Volumsmittelwerte von Simplices,welche zufällig in einem konvexen Körper des ℝ gewählt werden

For any set ofn+1 pointsx ₁, ...,x _n+1F we denote byv(C(x ₁,...,x _n+1)) then-dimensional oriented volume of the convex hullC(x ₁,...,x _n+1) of these points. With a fixed symmetric functionf: >> strictly monotone increasing on the nonnegative real line, we consider the real functional RODEL on the set of all convex bodiesK of ⁿ with absolute volume |v(K)|=1 and assert, that it takes its minimal value on the ellipsoids with absolute volume 1.     

On recurrence

Summary LetT be a non-singular ergodic automorphism of a Lebesgue space (X,L,) and letf: X be a measurable function. We define the notion of recurrence of such a functionf and introduce the recurrence setR(f)={:f– is recurrent}. If , then R()={0}, but in general recurrence sets can be very complicated. We prove various conditions for a number to lie in R(f) and, more generally, forR(f) to be non-empty. The results in this paper have applications to the theory of random walks with stationary increments.     

Generalized solutions to semilinear hyperbolic systems

In this article semilinear hyperbolic first order systems in two variables are considered, whose nonlinearity satisfies a global Lipschitz condition. It is shown that these systems admit unique global solutions in the Colombeau algebraG(²). In particular, this provides unique generalized solutions for arbitrary distributions as initial data. The solution inG(²) is shown to be consistent with the locally integrable or the distributional solutions, when they exist.