Representations of Affine Multifunctions by Affine Selections

The paper deals with affine selections of affine (both convex and concave) multifunctions acting between finite-dimensional real normed spaces. It is proved that each affine multifunction with compact values possesses an exhaustive family of affine selections and, consequently, can be represented by its affine selections. Moreover, a convex multifunction with compact values possesses an exhaustive family of affine selections if and only if it is affine. Thus the existence of an exhaustive family of affine selections is the characteristic feature of affine multifunctions which differs them from other convex multifunctions with compact values. Besides a necessary and sufficient condition for a concave multifunction to be affine on a given convex subset is also proved. Finally it is shown that each affine multifunction with compact values can be represented as the closed convex hull of its exposed affine selections and as the convex hull of its extreme affine selections. These statements extend the Straszewicz theorem and the Krein–Milman theorem to affine multifunctions. Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

Robust combinatorial optimization with variable budgeted uncertainty

We introduce a new model for robust combinatorial optimization where the uncertain parameters belong to the image of multifunctions of the problem variables. In particular, we study the variable budgeted uncertainty, an extension of the budgeted uncertainty introduced by Bertsimas and Sim. Variable budgeted uncertainty can provide the same probabilistic guarantee as the budgeted uncertainty while being less conservative for vectors with few non-zero components. The feasibility set of the resulting optimization problem is in general non-convex so that we propose a mixed-integer programming reformulation for the problem, based on the dualization technique often used in robust linear programming. We show how to extend these results to non-binary variables and to more general multifunctions involving uncertainty set defined by conic constraints that are affine in the problem variables. We present a computational comparison of the budgeted uncertainty and the variable budgeted uncertainty on the robust knapsack problem. The experiments show a reduction of the price of robustness by an average factor of 18 %.     

Some topological problems in multifunctions theory

The problem of upper semicontinuity of graph-closed multifunctions is considered. Also, several recent results on extension of multifunctions are presented.     

On the Parametrized Integral of a Multifunction: The Unbounded Case

Integration of set-valued maps (alias multifunctions) depending on a parameter is revisited. Results of Artstein, and of Saint-Pierre and Sajid are extended to the case of set-valued maps whose values may be unbounded. In the general case, this is achieved assuming that the transition probabilities involved in the integration procedure are absolutely continuous with respect to some fixed probability measure. However, when the integrating probability measure does not depend on the parameter this hypothesis is shown to be unnecessary. On the other hand, an alternative proof of a result of Saint-Pierre and Sajid is provided for convex compact-valued multifunctions. An application is given to the control of chattering systems. It is an extension of a result of Artstein to the case of set-valued maps with unbounded values. The proof of the main results is simple and essentially relies on measurable selections arguments.      

Implicit mapping theorem for extended metric regularity in metric spaces

In this paper we prove an extension of the contraction mapping principle for set-valued mappings stated by A.L. Dontchev and W.W. Hager dealing with more general assumptions containing modulus instead of pseudo-contractive multifunctions. Using this result we show that the update Graves theorem, in company with the stability of metric regularity under perturbations can be extended to a much broader framework of set-valued mappings acting in abstract spaces.     

Hahn–Banach extension theorems for multifunctions revisited

Several generalizations of the Hahn–Banach extension theorem to K-convex multifunctions were stated recently in the literature. In this note we provide an easy direct proof for the multifunction version of the Hahn–Banach–Kantorovich theorem and show that in a quite general situation it can be obtained from existing results. Then we derive the Yang extension theorem using a similar proof as well as a stronger version of it using a classical separation theorem. Moreover, we give counterexamples to several extension theorems stated in the literature. Dedicated to Jean-Paul Penot with the occasion of his retirement.     

On multifunctions

In this paper, we introduce and study γ-continuous multifunctions as a generalization of quasi-continuous multifunctions due to Popa in 1985 and precontinuous multifunctions due to Popa in 1988. Some characterizations and several properties concerning upper (lower) γ-continuous multifunctions are obtained. The relationships between upper (lower) γ-continuous multifunctions and some known concepts are also discussed.     

On a Class of Weak Nonhomogeneous Affine Bi-frames for Reducing Subspaces of L~2(R~d)

For refinable function-based affine bi-frames, nonhomogeneous ones admit fast algorithms and have extension principles as homogeneous ones. But all extension principles are based on some restrictions on refinable functions. So it is natural to ask what are expected from general refinable functions. In this paper, we introduce the notion of weak nonhomogeneous affine bi-frame(WNABF).Under the setting of reducing subspaces of L~2(R~d), we characterize WNABFs and obtain a mixed oblique extension principle for WNABFs based on general refinable functions.     

An application of continuous fuzzy multifunctions

In this paper, the upper and lower δ-continuous multifunctions in fuzzy setting have been presented as a strong form and an application of fuzzy continuous multifunctions. Certain characterizations and several properties of these fuzzy multifunctions along with their mutual relationships are obtained. Attempts are also made to correlate this new class with the corresponding known types of fuzzy multifunctions. Also, applicability of the above new concepts to superstrings and space time could be probably possible in the near future.     

Metric Regularity of the Sum of Multifunctions and Applications

The metric regularity of multifunctions plays a crucial role in modern variational analysis and optimization. This property is a key to study the stability of solutions of generalized equations. Many practical problems lead to generalized equations associated to the sum of multifunctions. This paper is devoted to study the metric regularity of the sum of multifunctions. As the sum of closed multifunctions is not necessarily closed, almost all known results in the literature on the metric regularity for one multifunction (which is assumed usually to be closed) fail to imply regularity properties of the sum of multifunctions. To avoid this difficulty, we use an approach based on the metric regularity of so-called epigraphical multifunctions and the theory of error bounds to study the metric regularity of the sum of two multifunctions, as well as some related important properties of variational systems. Firstly, we establish the metric regularity of the sum of a regular multifunction and a pseudo-Lipschitz multifunction with a suitable Lipschitz modulus. These results subsume some recent results by Durea and Strugariu. Secondly, we derive coderivative characterizations of the metric regularity of epigraphical multifunctions associated with the sum of multifunctions. Applications to the study of the behavior of solutions of variational systems are reported.     

Jensen measures and analytic multifunctions

This paper describes plurisubharmonic convexity and hulls, and also analytic multifunctions in terms of Jensen measures. In particular, this allows us to get a new proof of Słodkowski's theorem stating that multifunctions are analytic if and only if their graphs are pseudoconcave. We also show that multifunctions with plurisubharmonically convex fibers are analytic if and only if their graphs locally belong to plurisubharmonic hulls of their boundaries. In the last section we prove that minimal analytic multifunctions satisfy the maximum principle and give a criterion for the existence of holomorphic selections in the graphs of analytic multifunctions. The author was partially supported by an NSF Grant.     

Existence theorems for generalized vector variational inequalities with a variable ordering relation

In this paper we study the solvability of the generalized vector variational inequality problem, the GVVI problem, with a variable ordering relation in reflexive Banach spaces. The existence results of strong solutions of GVVIs for monotone multifunctions are established with the use of the KKM-Fan theorem. We also investigate the GVVI problems without monotonicity assumptions and obtain the corresponding results of weak solutions by applying the Brouwer fixed point theorem. These results are also the extension and improvement of some recent results in the literature.     

On upper and lower δ-precontinuous multifunctions

In this paper we introduce and study δ-precontinuous multifunctions as a generalization of precontinuous multifunctions due to Popa [Problemy Mat. 10 (1988) 9]. Some characterizations and several properties concerning upper (lower) δ-precontinuous multifunctions are obtained. The relationships between upper (lower) δ-precontinuous multifunctions and some known concepts are also discussed.     

非线性控制系统的梯度扩张的能达性

研究了仿射非线性控制系统的梯度扩张系统.利用非线性控制系统的微分几何理论,通过计算梯度扩张系统的输出函数沿着输入向量场和系统向量场的李导数,讨论仿射非线性控制系统的梯度扩张系统的能达性分布,研究了非线性控制系统和它的梯度扩张系统的能达性之间的关系,证明了如果梯度扩张系统是能达的,则原非线性控制系统也是能达的.     

Affine Module Groups and Their Automorphisms

An affine module group is a semidirect extension of an additive module group by its automorphism group. Maximal Abelian normal subgroups of an affine group are described. It is proved that operator isomorphisms of affine groups are induced by module automorphisms. Automorphisms of an affine group which do not leave a module fixed are treated. And conditions are specified for a module to be non-characteristic in its affine group.     

THE NORMALITY OF ALGEBROID MULTIFUNCTIONS AND THEIR COEFFICIENT FUNCTIONS

In this paper, we investigate the normality relationship between algebroid multifunctions and their coefficient functions. We prove that the normality of a k-valued entire algebroid multifunctions family is equivalent to their coefficient functions in some conditions.Furthermore, we obtain some new normality criteria for algebroid multifunctions families based on these results. We also provide some examples to expound that some restricted conditions of our main results are necessary.     

Affine Invariant Detection: Edge Maps, Anisotropic Diffusion, and Active Contours

In this paper we undertake a systematic investigation of affine invariant object detection and image denoising. Edge detection is first presented from the point of view of the affine invariant scale-space obtained by curvature based motion of the image level-sets. In this case, affine invariant maps are derived as a weighted difference of images at different scales. We then introduce the affine gradient as an affine invariant differential function of lowest possible order with qualitative behavior similar to the Euclidean gradient magnitude. These edge detectors are the basis for the extension of the affine invariant scale-space to a complete affine flow for image denoising and simplification, and to define affine invariant active contours for object detection and edge integration. The active contours are obtained as a gradient flow in a conformally Euclidean space defined by the image on which the object is to be detected. That is, we show that objects can be segmented in an affine invariant manner by computing a path of minimal weighted affine distance, the weight being given by functions of the affine edge detectors. The gradient path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology. Based on the same theory of affine invariant gradient flows we show that the affine geometric heat flow is minimizing, in an affine invariant form, the area enclosed by the curve.     

On a class of weak nonhomogeneous affine bi-frames for reducing subspaces of <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>)

For refinable function-based affine bi-frames, nonhomogeneous ones admit fast algorithms and have extension principles as homogeneous ones. But all extension principles are based on some restrictions on refinable functions. So it is natural to ask what are expected from general refinable functions. In this paper, we introduce the notion of weak nonhomogeneous affine bi-frame (WNABF). Under the setting of reducing subspaces of L ²(R ^d ), we characterize WNABFs and obtain a mixed oblique extension principle for WNABFs based on general refinable functions.     

On affine scaling and semi-infinite programming

We consider an extension of the affine scaling algorithm for linear programming problems with free variables to problems having infinitely many constraints, and explore the relationship between this algorithm and the finite affine scaling method applied to a discretization of the problem.This material is based on research supported by Air Force Office of Scientific Research Grant AFOSR 89-0410.     

Upper and lower pre-strong na continuous multifunctions

The aim of this paper is to introduce a new class of multifunctions namely pre-strong na continuous multifunctions and to obtain some characterizations and properties of it.