线性空间上凸函数的可微性与次可微性

讨论了定义在线性空间上的广义实值凸函数的可做性和次可微性．     

Mixed Functionals of Convex Bodies

We define a class of real functions on tuples of convex bodies. They are a common generalization of mixed volumes and of certain functionals which have been studied in translative integral geometry. For polytopes, these functionals have various explicit representations in terms of volumes of lower-dimensional faces. For the mentioned functionals from integral geometry, these representations generalize a result of Weil and answer a question posed by Janson. Received January 5, 1999, and in revised form March 12, 1999. Online publication May 16, 2000.     

Modulus of Strong Proximinality and Continuity of Metric Projection

In this paper we initiate a quantitative study of strong proximinality. We define a quantity ϵ(x, t) which we call as modulus of strong proximinality and show that the metric projection onto a strongly proximinal subspace Y of a Banach space X is continuous at x if and only if ϵ(x, t) is continuous at x whenever t > 0. The best possible estimate of ϵ(x, t) characterizes spaces with 1 \frac121 \frac{1}{2} ball property. Estimates of ϵ(x, t) are obtained for subspaces of uniformly convex spaces and of strongly proximinal subspaces of finite codimension in C(K).     

Maximization of Generalized Convex Functionals in Locally Convex Spaces

The major part of the investigation is related to the problem of maximizing an upper semicontinuous quasiconvex functional f over a compact (possibly nonconvex) subset K of a real Hausdorff locally convex space E. A theorem by Bereanu (Ref. 1) says that the condition f is quasiconvex (quasiconcave) on K is sufficient for the existence of maximum (minimum) point of f over K among the extreme points of K. But, as we prove by a counterexample, this is not true in general. On the further condition that the convex hull of the set of extreme points of K is closed, we show that it is sufficient to claim that f is induced-quasiconvex on K to achieve an equivalent conclusion. This new concept of quasiconvexity, which we define by requiring that each lower-level set of f can be represented as the intersection of K with some convex set, is suitable for functionals with a nonconvex domain. Under essentially the same conditions, we prove that an induced-quasiconvex functional f is directionally monotone in the sense that, for each y K, the functional f is increasing along a line segment starting at y and running to some extreme point of K. In order to guarantee the existence of maximum points on the relative boundary r K of K, it suffices to make weaker demands on the function f and the space E. By introducing a weaker kind of directional monotonicity, we are able to obtain the following result: If f is i.s.d.-increasing i.e., for each y y K, there is a half-line emanating from y such that f is increasing along this half-line, then f attains its maximum at rK , even if E is a topological linear Hausdorff space (infinite-dimensional and not necessarily locally convex). We state further a practical method of proving i.s.d.-monotonicity for functions in finite-dimensional spaces and we discuss also some aspects of classification.     

Local Uniform Strong Proximinality of the Space of Continuous Vector-Valued Functions

Let X be a Banach space, S be a compact Hausdorff space and Y be a U-proximinal subspace of X. We prove that C(S,Y) is locally uniformly strongly proximinal in C(S,X) and the corresponding metric projection map is Hausdorff metric continuous.     

An Identity Relating Moments of Functionals of Convex Hulls

Denote by $K_n$ the convex hull of $n$ independent random points distributed uniformly in a convex body $K$ in $\R^d$, by $V_n$ the volume of $K_n$, by $D_n$ the volume of $K\backslash K_n$, and by $N_n$ the number of vertices of $K_n$. A well-known identity due to Efron relates the expected volume ${\it ED}_n$---and thus ${\it EV}_n$---to the expected number ${\it EN}_{n+1}$. This identity is extended from expected values to higher moments. The planar case of the arising identity for the variances provides in a simple way the corrected version of a central limit theorem for $D_n$ by Cabo and Groeneboom ($K$ being a convex polygon) and an improvement of a central limit theorem for $D_n$ by Hsing ($K$ being a circular disk). Estimates of $\var D_n$ ($K$ being a two-dimensional smooth convex body) and $\var N_n$ ($K$ being a $d$-dimensional smooth convex body, $d\geq 4$) are obtained. The identity for moments of arbitrary order shows that the distribution of $N_n$ determines ${\it EV}_{n-1}, {\it EV}_{n-2}^2,\dots, {\it EV}_{d+1}^{n-d-1}$. Reversely it is proved that these $n-d-1$ moments determine the distribution of $N_n$ entirely. The resulting formula for the probability that $N_n=k\ (k=d+1,\dots , n)$ appears to be new for $k\geq d+2$ and yields an answer to a question raised by Baryshnikov. For $k=d+1$ the formula reduces to an identity which has been repeatedly pointed out.     

关于Banach空间中凸泛函的广义次梯度不等式

本文在前人^[1,2]的基础之上，以凸泛函的次梯度不等式为工具，将Jensen不等式推广到Banach空间中的凸泛函，导出了Banach空间中的Bochner积分型的广义Jensen不等式，给出其在Banach空间概率论中某些应用，从而推广了文献[3—6]的工作．     

Singular Abreu Equations and Minimizers of Convex Functionals with a Convexity Constraint

We study the solvability of second boundary value problems of fourth-order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and the monopolist's problem in economics. The right-hand sides of our Abreu-type equations are quasilinear expressions of second order; they are highly singular and a priori just measures. However, our analysis in particular shows that minimizers of the 2D Rochet-Choné model perturbed by a strictly convex lower-order term, under a convexity constraint, can be approximated in the uniform norm by solutions of the second boundary value problems of singular Abreu equations. © 2019 Wiley Periodicals, Inc.     

Subdifferentiability and the Duality Gap

We point out a connection between sensitivity analysis and the fundamental theorem of linear programming by characterizing when a linear programming problem has no duality gap. The main result is that the value function is subdifferentiable at the primal constraint if and only if there exists an optimal dual solution and there is no duality gap. To illustrate the subtlety of the condition, we extend Kretschmer's gap example to construct (as the value function of a linear programming problem) a convex function which is subdifferentiable at a point but is not continuous there. We also apply the theorem to the continuum version of the assignment model.     

The Strong Continuity of Convex Functions

A convex function defined on an open convex set of a finite-dimensional space is known to be continuous at every point of this set. In fact, a convex function has a strengthened continuity property. The notion of strong continuity is introduced in this study to show that a convex function has this property. The proof is based on only the definition of convexity and Jensen’s inequality. The definition of strong continuity involves a constant (the constant of strong continuity). An unimprovable value of this constant is given in the case of convex functions. The constant of strong continuity depends, in particular, on the form of a norm introduced in the space of arguments of a convex function. The polyhedral norm is of particular interest. It is straightforward to calculate the constant of strong continuity when it is used. This requires a finite number of values of the convex function.     

Lower Subdifferentiability and Integration

We consider the question of integration of a multivalued operator T, that is the question of finding a function f such that Tf. If is the Fenchel–Moreau subdifferential, the above problem has been completely solved by Rockafellar, who introduced cyclic monotonicity as a necessary and sufficient condition. In this article we consider the case where f is quasiconvex and is the lower subdifferential ^<. This leads to the introduction of a property that is reminiscent to cyclic monotonicity. We also consider the question of the density of the domains of subdifferential operators.     

Subdifferentiability and Inf-Sup Theorems

Subdifferentiability criterions for non necessarily lower semicontinuous convex functions on general locally convex spaces or Fréchet spaces are used to derive inf-sup theorems. The importance of quasicontinuous convex functions is pointed out, and the usual compactness condition relaxed.     

Two-Scale Convergence of Integral Functionals with Convex, Periodic and Nonstandard Growth Integrands

We study the periodic homogenization for a family of functionals defined on Orlicz-Sobolev spaces. One fundamental in this topic is to extend the classical compactness results of the two-scale convergence method to this type of spaces.     

Strong Geodesic Convex Functions of Order m

Abstract

Strong geodesic convex function and strong monotone vector field of order m on Riemannian manifolds are established. A characterization of strong geodesic convex function of order m for the continuously differentiable functions is discussed. The relation between the solution of a new variational inequality problem and the strict minimizers of order m for a multiobjective programing problem is also established.     

Convex Modeling of Interactions With Strong Heredity

We consider the task of fitting a regression model involving interactions among a potentially large set of covariates, in which we wish to enforce strong heredity. We propose FAMILY, a very general framework for this task. Our proposal is a generalization of several existing methods, such as VANISH, hierNet, the all-pairs lasso, and the lasso using only main effects. It can be formulated as the solution to a convex optimization problem, which we solve using an efficient alternating directions method of multipliers (ADMM) algorithm. This algorithm has guaranteed convergence to the global optimum, can be easily specialized to any convex penalty function of interest, and allows for a straightforward extension to the setting of generalized linear models. We derive an unbiased estimator of the degrees of freedom of FAMILY, and explore its performance in a simulation study and on an HIV sequence dataset. Supplementary materials for this article are available online.     

Intervals and Convex Sets in Strong Product of Graphs

In this note we consider intervals and convex sets of strong product. Vertices of an arbitrary interval of ${G\boxtimes H}$ are classified with shortest path properties of one factor and a walk properties of a slightly modified second factor. The convex sets of the strong product are characterized by convexity of projections to both factors and three other local properties, one of them being 2-convexity.     

Strong Duality for Generalized Convex Optimization Problems

In this paper, strong duality for nearly-convex optimization problems is established. Three kinds of conjugate dual problems are associated to the primal optimization problem: the Lagrange dual, Fenchel dual, and Fenchel-Lagrange dual problems. The main result shows that, under suitable conditions, the optimal objective values of these four problems coincide. The first author was supported in part by Gottlieb Daimler and Karl Benz Stiftung 02-48/99. This research has been performed while the second author visited Chemnitz University of Technology under DAAD (Deutscher Akademischer Austauschdienst) Grant A/02/12866. Communicated by T. Rapcsák     

Strong Convergence in Stabilised Degenerate Convex Problems

Solutions to non–convex variational problems typically exhibit enforced finer and finer oscillations called microstructures such that the infimal energy is not attained. Those oscillations are physically meaningful, but finite element approximations typically experience dramatic difficulty in their reproduction. The relaxation of the non–convex minimisation problem by (semi–)convexification leads to a macroscopic model for the effective energy. The resulting discrete macroscopic problem is degenerate in the sense that it is convex but not strictly convex. This paper discusses a modified discretisation by adding a stabilisation term to the discrete energy. It will be announced that, for a wide class of problems, this stabilisation technique leads to strong H¹–convergence of the macroscipic variables even on unstructured triangulations. This is in contrast to the work [2] for quasi–uniform triangulations and enables the use of adaptive algorithms for the stabilised formulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)