Local Convexity on Smooth Manifolds

Some properties of the spaces of paths are studied in order to define and characterize the local convexity of sets belonging to smooth manifolds and the local convexity of functions defined on local convex sets of smooth manifolds. This paper is dedicated to the memory of Guido Stampacchia. This research was supported in part by the Hungarian Scientific Research Fund, Grants OTKA-T043276 and OTKA-T043241, and by CNR, Rome, Italy.     

Stable Solutions of Elliptic Equations on Riemannian Manifolds

This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new weighted Poincaré inequality which allows us to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold.     

Invex-convexlike functions and duality

We define a class of invex-convexlike functions, which contains all convex, pseudoconvex, invex, and convexlike functions, and prove that the Kuhn-Tucker sufficient optimality condition and the Wolfe duality hold for problems involving such functions. Applications in control theory are given.The author is grateful to Professor W. Stadler and the referees for many valuable remarks and suggestions, which have enabled him to improve considerably the paper.     

On the functions with pseudoconvex sublevel sets and optimality conditions

A new class of generalized convex functions, called the functions with pseudoconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets.     

A NEW FORMULA WITHOUT BOUNDARY INTEGRALS ON A STEIN MANIFOLD

A new Koppelman-Leray-Norguet formula of (p-1,q) differential forms for a strictly pseudoconvex polyhedron with not necessarily smooth boundary on a Stein manifold is obtained, and an integral representation for the solution of (?)-equation on this domain which does not involve integrals on boundary is given, so one can avoid complex estimates of boundary integrals.     

Monotone Variable–Metric Algorithm for Linearly Constrained Nonlinear Programming

A new method for linearly constrained nonlinear programming is proposed. This method follows affine scaling paths defined by systems of ordinary differential equations and it is fully parallelizable. The convergence of the method is proved for a nondegenerate problem with pseudoconvex objective function. In practice, the algorithm works also under more general assumptions on the objective function. Numerical results obtained with this computational method on several test problems are shown.     

Classes de Nevanlinna dans certains domaines strictement pseudoconvexes non lisses

We define Nevanlinna classes in strictly pseudoconvex domains that are not necessarily smooth. If the boundary is defined by a Morse function we give sufficient conditions of Blaschke type for an analytic set to be the zero set of a function in a given Nevanlinna class. In particular, in the case of a smooth boundary we obtain new and rather explicit proofs of the theorems of Henkin-Skoda and Dautov-Henkin.     

Optimality Conditions and Characterizations of the Solution Sets in Generalized Convex Problems and Variational Inequalities

We derive necessary and sufficient conditions for optimality of a problem with a pseudoconvex objective function, provided that a finite number of solutions are known. In particular, we see that the gradient of the objective function at every minimizer is a product of some positive function and the gradient of the objective function at another fixed minimizer. We apply this condition to provide several complete characterizations of the solution sets of set-constrained and inequality-constrained nonlinear programming problems with pseudoconvex and second-order pseudoconvex objective functions in terms of a known solution. Additionally, we characterize the solution sets of the Stampacchia and Minty variational inequalities with a pseudomonotone-star map, provided that some solution is known.     

Optimality Conditions for Isolated Local Minimum of Nonsmooth Multiobjective Problems

This article is devoted to the study of Fritz John and strong Kuhn-Tucker necessary conditions for properly efficient solutions, efficient solutions and isolated efficient solutions of a nonsmooth multiobjective optimization problem involving inequality and equality constraints and a set constraints in terms of the lower Hadamard directional derivative. Sufficient conditions for the existence of such solutions are also provided where the involved functions have pseudoconvex sublevel sets. Our results are based on the concept of pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets are a class of generalized convex functions that include quasiconvex functions.     

Analysis and Geometry on Worm Domains

In this primarily expository article, we study the analysis of the Diederich-Fornæss worm domain in complex Euclidean space. We review its importance as a domain with nontrivial Nebenhülle, and as a counterexample to a number of basic questions in complex geometric analysis. Then we discuss its more recent significance in the theory of partial differential equations: the worm is the first smoothly bounded, pseudoconvex domain to exhibit global non-regularity for the \(\overline{\partial}\)-Neumann problem. We take this opportunity to prove a few new facts. Next, we turn to specific properties of the Bergman kernel for the worm domain. An asymptotic expansion for this kernel is considered, and applications to function theory and analysis on the worm are provided.     

A STOCHASTIC REPRESENTATION FOR THE LEVEL SET EQUATIONS

ABSTRACT

A Feynman-Kac representation is proved for geometric partial differential equations. This representation is in terms of a stochastic target problem. In this problem the controller tries to steer a controlled process into a given target by judicial choices of controls. The sublevel sets of the unique level set solution of the geometric equation is shown to coincide with the reachability sets of the target problem whose target is the sublevel set of the final data.     

The Mergelyan property for weakly pseudoconvex domains

We show that pseudoconvex domains have the Mergelyan property if the Levi form is degenerate on a sufficiently small set in the boundary. This includes the case when the weakly pseudoconvex points all lie on a smooth curve.     

Measure-Theoretic Properties of Level Sets of Distance Functions

We consider the level sets of distance functions from the point of view of geometric measure theory. This lays the foundation for further research that can be applied, among other uses, to the derivation of a shape calculus based on the level-set method. Particular focus is put on the \((n-1)\)-dimensional Hausdorff measure of these level sets. We show that, starting from a bounded set, all sub-level sets of its distance function have finite perimeter. Furthermore, if a uniform-density condition is satisfied for the initial set, one can even show an upper bound for the perimeter that is uniform for all level sets. Our results are similar to existing results in the literature, with the important distinction that they hold for all level sets and not just almost all. We also present an example demonstrating that our results are sharp in the sense that no uniform upper bound can exist if our uniform-density condition is not satisfied. This is even true if the initial set is otherwise very regular (i.e., a bounded Caccioppoli set with smooth boundary).     

Minimizing pseudoconvex functions on convex compact sets

An algorithm is presented which minimizes continuously differentiable pseudoconvex functions on convex compact sets which are characterized by their support functions. If the function can be minimized exactly on affine sets in a finite number of operations and the constraint set is a polytope, the algorithm has finite convergence. Numerical results are reported which illustrate the performance of the algorithm when applied to a specific search direction problem. The algorithm differs from existing algorithms in that it has proven convergence when applied to any convex compact set, and not just polytopal sets.This research was supported by the National Science Foundation Grant ECS-85-17362, the Air Force Office Scientific Research Grant 86-0116, the Office of Naval Research Contract N00014-86-K-0295, the California State MICRO program, and the Semiconductor Research Corporation Contract SRC-82-11-008.     

Optimization and Variational Inequalities with Pseudoconvex Functions

In the present paper we consider a pseudoconvex (in an extended sense) function f using higher order Dini directional derivatives. A Variational Inequality, which is a refinement of the Stampacchia Variational Inequality, is defined. We prove that the solution set of this problem coincides with the set of global minimizers of f if and only if f is pseudoconvex. We introduce a notion of pseudomonotone Dini directional derivatives (in an extended sense). It is applied to prove that the solution sets of the Stampacchia Variational Inequality and Minty Variational Inequality coincide if and only if the function is pseudoconvex. At last, we obtain several characterizations of the solution set of a program with a pseudoconvex objective function.     

Quasiconvex,pseudoconvex, and strictly pseudoconvex quadratic functions

The purpose of this paper is twofold. Firstly, criteria for quasiconvex and pseudoconvex quadratic functions in nonnegative variables of Cottle, Ferland, and Martos are derived by specializing criteria proved by the author. We do not make use of the concept of positive subdefinite matrices. Instead, we are specializing criteria that were derived for quadratic functions on arbitrary convex sets to the special case of quadratic functions in nonnegative variables. The second purpose of this paper is to present several new criteria involving also strictly pseudoconvex quadratic functions.The author wishes to thank Professor R. W. Cottle for helpful discussions.     

An algorithm for constrained convex optimization

This paper presents a feasible direction algorithm for the minimization of a pseudoconvex function over a smooth, compact, convex set. We establish that each cluster point of the generated sequence is an optimal solution of the problem without introducing anti-jamming procedures. Each iteration of the algorithm involves as subproblems only one line search for a zero of a continuously differentiable convex function and one univariate function minimization on a compact interval.     

On the pseudolinearity of quadratic fractional functions

The main result is a new characterization of the pseudolinearity of quadratic fractional functions. This research was supported in part by the Hungarian Scientific Research Fund, Grant No. OTKA-T043276 and OTKA-K60480.     

THE EXSISTENCE OF $\mu -HOLOMORPHIC$ SEPARATING FUNCTION ON BOUNDED SMOOTH DOMAINS

The Complex analysis of strongly pseudoconvex domains in C^n is rather well known.In this paper it is proved that for a bounded smoothly domain $\omega$ there is a new complex structure on it under which $\omega$ will locally become a strongly convex even though the point on b&\omega& is not a pseudoconvex point from the view of the original complex structure.Particularly if $\omega$ is a weakly pseudoconvex domain,the $\mu$ cam be ,ade siffocoemt;u c;pse tp tje progoma; cp,[;ex strictire.Therefore a lot of properties of strongly pseudoconvex domains will become true on weakly pseudoconvex domains,or general domains.For example,it is proved that there is a $\mu-holomorphic$ separatin function which is holomorphic under the new complex structure.