Shift ergodicity for stationary Markov processes

In this paper shift ergodicity and related topics are studied for certain stationary processes. We first present a simple proof of the conclusion that every stationary Markov process is a generalized convex combination of stationary ergodic Markov processes. A direct consequence is that a stationary distribution of a Markov process is extremal if and only if the corresponding stationary Markov process is time ergodic and every stationary distribution is a generalized convex combination of such extremal ones. We then consider space ergodicity for spin flip particle systems. We prove space shift ergodicity and mixing for certain extremal invariant measures for a class of spin systems, in which most of the typical models, such as the Voter Models and the Contact Models, are included. As a consequence of these results we see that for such systems, under each of those extremal invariant measures, the space and time means of an observable coincide, an important phenomenon in statistical physics. Our results provide partial answers to certain interesting problems in spin systems.     

Extremal Polygons with Minimal Perimeter

In this paper we will investigate an isoperimetric type problem in lattices. If K is a bounded O-symmetric (centrally symmetric with respect to the origin) convex body in En of volume v(K) = 2n det L which does not contain non-zero lattice points in its interior, we say that K is extremal with respect to the given lattice L. There are two variations of the isoperimetric problem for this class of polyhedra. The first one is: Which bodies have minimal surface area in the class of extremal bodies for a fixed n-dimensional lattice? And the second one is: Which bodies have minimal surface area in the class of extremal bodies with volume 1 of dimension n? We characterize the solutions of these two problems in the plane. There is a consequence of these results, the solutions of the above problems in the plane give the solution of the lattice-like covering problem: Determine those centrally symmetric convex bodies whose translated copies (with respect to a fixed lattice L) cover the space and have minimal surface area.

     

一些函数族的一般极值问题

本文主要讨论了带限制条件的正实部解析函数族及纯凸像函数族的一般极值问题．首先我们得了两类带限制条件的正实部函数族的支撑点的表达式．其次，我们讨论了亚纯凸像函数族的极值问题，得到了亚纯凸像函数族上Ｆｒｃｈｅｔ可导泛函所对应的极函数的最好形式．     

On bags and bugs

Usual graph classes, such as complete graphs, paths, cycles and stars, frequently appear as extremal graphs in graph theory problems. Here we want to turn the reader's attention to two novel, simply defined, graph classes that appear as extremal graphs in several graph theory problems. We call them bags and bugs. As examples of problems where bags and bugs appear, we show that balanced bugs maximize the index of graphs with fixed number of vertices and diameter ?2, while odd bags maximize the index of graphs with fixed number of vertices and radius ?3.     

Efficient estimates of the solutions of perturbed control problems

In this paper, we obtain estimates of the solutions for a sequence of strongly convex extremal problems. As applications of our abstract results, we consider optimal control problems with various types of perturbations. We estimate the solutions of problems with perturbations in the state equation and in the control constraining set. A singularly perturbed problem and a problem with perturbed time delay parameter are studied.     

Asymptotic behavior of extremal solutions and structure of extremal norms of linear differential inclusions of order three

Asymptotic properties of extremal solutions of linear inclusions of order three with zero Lyapunov exponent are investigated. Under certain conditions it is shown that all extremal solutions of such inclusions tend to the same (up to a multiplicative factor) solution, which is central symmetric. The structure of the convex set of extremal norm is studied. A number of extremal points of this set are described.     

Inducibility of directed paths

A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more tractable. Here we resolve this problem in the setting of oriented graphs without transitive triangles.     

Multiobjective problems of convex geometry

Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body $ \mathfrak{x} $ \mathfrak{x} , we try to maximize the volume of $ \mathfrak{x} $ \mathfrak{x} and minimize the width of $ \mathfrak{x} $ \mathfrak{x} simultaneously. These problems are addressed along the lines of multiple criteria decision making. We describe the Pareto-optimal solutions of isoperimetric-type vector optimization problems on using the techniques of the space of convex sets, linear majorization, and mixed volumes.     

Special Pair of Dual Parametric Nonlinear Optimization Problems with Common Set of Optimal Points

On the base of a given strictly convex function defined on the Euclidean space E n ( n S 2) we can-without the assumption that it is differentiable - introduce some manifolds in topologic sense. Such manifolds are sets of all optimal points of a certain parametric non-linear optimization problem. This paper presents above all certain generalization of some results of [F. No ? i ) ka and L. Grygarová (1991). Some topological questions connected with strictly convex functions. Optimization , 22 , 177-191. Akademie Verlag, Berlin] and [L. Grygarová (1988). Über Lösungsmengen spezieller konvexer parametrischer Optimierungsaufgaben . Optimization 19 , 215-228. Akademie Verlag Berlin], under less strict assumptions. The main results are presented in Sections 3 and 4, in Section 3 the geometrical characterization of the set of optimal points of a certain parametric minimization problem is presented; in Section 4 we study a maximization non-linear parametric problem assigned to it. It seems that it is a certain pair of parametric optimization problems with the same set of their optimal points, so that this pair of problems can be denoted as a pair of dual parametric non-linear optimization problems. This paper presents, most of all in Section 2, a number of interesting geometric facts about strictly convex functions. From the point of view of non-smooth analysis the present article is a certain complement to Chapter 4.3 of the book [B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer (1982). Nonlinear Parametric Optimization . Akademie Verlag, Berlin] where a convex parametric minimization problem is considered under more general and stronger conditions (but without any assumptions concerning strict convexity and without geometrical aspects).     

Extremal properties of nondifferentiable convex functions on euclidean sets of combinations with repetitions

A general approach is suggested for studying extremal properties of nondifferentiable convex functions on Euclidean combinatorial sets. On the basis of this approach, by solving the linear optimization problem on a set of combinations with repetitions, we obtain estimates of minimum values of convex and strongly convex objective functions in optimization problems on sets of combinations with repetitions and establish sufficient conditions for the existence of the corresponding minima. Kharkov Institute of Radioelectronics, Kharkov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 680–691, June, 1994.     

Necessary optimality conditions for a set-valued fractional extremal programming problem under inclusion constraints

In this paper, we are concerned with a set-valued fractional extremal programming problem under inclusion constraints. Our approach consists of using the extremal principle (an approach initiated by Mordukhovich, which does not involve any convex approximations and convex separation arguments) for the study of necessary optimality conditions.     

Minimax detection of a signal for nondegenerate loss functions, and extremal convex problems

We study a wide class of minimax problems of signal detection under nonparametric alternatives and a modification of these problems for a special class of loss functions. Under rather general assumptions, we obtain the exact asymptotics (of Gaussian type) for the minimax error probabilities and the structure of asymptotically minimax tests. The methods are based on a reduction of the problems under consideration to extremal problems of minimization of a certain Hilbert norm on a convex set of sequences of probability measures on the real line. These extremal problems are investigated in a paper by I. A. Suslina for alternatives having the type of l_q-ellipsoids with l_p-balls removed. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 162–188.     

Exterior Monge-Ampère solutions

We discuss the Siciak-Zaharjuta extremal function of a real convex body in Cn, a solution of the homogeneous complex Monge-Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadratic curves, and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria. As a byproduct we encounter a new invariant of an exterior domain, the Robin indicatrix, which is in some cases the dual of the Kobayashi indicatrix for a bounded domain. Finally, we construct extremal curves for two non-convex bodies in R².     

Density of the Extremal Solutions for a Class of Second Order Boundary Value Problem

In this paper, we prove a theorem on the existence of extremal solutions to a second-order differential inclusion with boundary conditions, governed by the subdifferential of a convex function. We also show that the extremal solutions set is dense in the solutions set of the original problem.     

Lagrange multiplier rules for non-differentiable DC generalized semi-infinite programming problems

This paper aims to study a broad class of generalized semi-infinite programming problems with (upper and lower level) objectives given as the difference of two convex functions, and (lower level) constraints described by a finite number of convex inequalities and a set constraints. First, we are interested in some various lower level constraint qualifications for the problem. Then, the results are used to establish efficient upper estimate of certain subdifferential of value functions. Finally, we apply the obtained subdifferential estimates to derive necessary optimality conditions for the problem.     

Best uniform approximation in the metric space of continuous mappings with compact convex images

For the problem of the best uniform approximation of a continuous mapping with compact convex images by sets of other continuous mappings with compact convex images, we establish necessary and sufficient conditions and a criterion for an element to be extremal; the criterion obtained is a generalization of the classic Kolmogorov criterion for a polynomial of the best approximation.     

Stability estimates for solutions of control problems for the Maxwell equations with mixed boundary conditions

We consider extremal problems for the time-harmonic Maxwell equations with mixed boundary conditions for the electric field. Namely, the tangential component of the electric field is given on one part of the boundary, and an impedance boundary condition is posed on the other part. We prove the solvability of the original mixed boundary value problem and the extremal problem. We obtain sufficient conditions on the input data ensuring the stability of solutions of specific extremal problems under certain perturbations of both the performance functional and some functions occurring in the boundary value problem.     

A polynomial oracle-time algorithm for convex integer minimization

In this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures. We show that a greedy augmentation procedure that employs only directions from certain Graver bases needs only polynomially many augmentation steps to solve the given problem. We extend these results to convex N-fold integer minimization problems and to convex 2-stage stochastic integer minimization problems. Finally, we present some applications of convex N-fold integer minimization problems for which our approach provides polynomial time solution algorithms.     

Some Properties of Smooth Convex Functions and Newton’s Method

Doklady Mathematics - New properties of convex infinitely differentiable functions related to extremal problems are established. It is shown that, in a neighborhood of the solution, even if the...