A technical note on "The KKT Optimality Conditions For Optimization Problem With Interval-Valued Objective Function On Hadamard Manifolds"

In this technical note, we disprove S.-L. Chen's \cite{chen} assertion that the generalized Hukuhara directional differentiability of an interval-valued function defined on a Hadamard manifold is fully identical to the directional differentiability of its endpoint functions by providing a counterexample. Moreover, we address imprecisions in several results proved by using the aforementioned assertion in \cite{chen}. We provide correct statements of the imprecise results and construct examples in their support

Differentiability in Banach Spaces

There are three chapters in this work of which the first two contain differentiability results for continuous convex functions on Banach spaces. The final chapter contains differentiability results for Lipschitz isomorphisms of ℓ2. The aim of chapter 1 is to improve on a result of I. Ekeland and G. Lebourg [EL] who show that a Banach space E that admits a Lipschitz Fréchet smooth bump function is an Asplund space. It is shown that if E admits a continuous lower Fréchet smooth bump function then E is an Asplund space. Chapter 2 contains partial results towards showing that there are Gâteaux differentiability spaces that are not weak Asplund spaces. Suppose that K is a totally ordered first countable Hausdorff compact space. A topology Tw is defined on C(K) called the wedge topology, and it is shown that if every subdifferential of a continuous convex function f on C(K) contains a measure of finite support then f is Gâteaux differentiable on a τw residual set. Chapter 3 contains three examples of Lipschitz isomorphisms of ℓ2 to itself for which the derivative fails to be surjective; in the first example the Gâteaux derivative is not surjective at one point, in the second example the weak limit of limt→0(f(th) -(0))/t is zero for all h ∈ ℓ2, and in the third example the Gateaux derivative is not surjective at all points of the cube {x ∈ ℓ2 : |xi| < 2=i for all i} which is mapped affinely into a hyperplane

Wulff shapes and a characterization of simplices via a Bezout type inequality

Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumes $$V(L_1,\dots,L_{n})V_n(K)\leq V(L_1,K[{n-1}])V(L_2,\dots, L_{n},K).$$ We show that the above inequality characterizes simplices, i.e. if $K$ is a convex body satisfying the inequality for all convex bodies $L_1, \dots, L_n \subset {\mathbb R}^n$, then $K$ must be an $n$-dimensional simplex. The main idea of the proof is to study perturbations given by Wulff shapes. In particular, we prove a new theorem on differentiability of the support function of the Wulff shape, which is of independent interest. In addition, we study the Bezout inequality for mixed volumes introduced in arXiv:1507.00765 . We introduce the class of weakly decomposable convex bodies which is strictly larger than the set of all polytopes that are non-simplices. We show that the Bezout inequality in arXiv:1507.00765 characterizes weakly indecomposable convex bodies

Equivalence of the Traditional and Non-Standard Definitions of Concepts from Real Analysis

ACL2(r) is a variant of ACL2 that supports the irrational real and complex numbers. Its logical foundation is based on internal set theory (IST), an axiomatic formalization of non-standard analysis (NSA). Familiar ideas from analysis, such as continuity, differentiability, and integrability, are defined quite differently in NSA-some would argue the NSA definitions are more intuitive. In previous work, we have adopted the NSA definitions in ACL2(r), and simply taken as granted that these are equivalent to the traditional analysis notions, e.g., to the familiar epsilon-delta definitions. However, we argue in this paper that there are circumstances when the more traditional definitions are advantageous in the setting of ACL2(r), precisely because the traditional notions are classical, so they are unencumbered by IST limitations on inference rules such as induction or the use of pseudo-lambda terms in functional instantiation. To address this concern, we describe a formal proof in ACL2(r) of the equivalence of the traditional and non-standards definitions of these notions.Comment: In Proceedings ACL2 2014, arXiv:1406.123

Asymptotic Normality of Support Vector Machine Variants and Other Regularized Kernel Methods

In nonparametric classification and regression problems, regularized kernel methods, in particular support vector machines, attract much attention in theoretical and in applied statistics. In an abstract sense, regularized kernel methods (simply called SVMs here) can be seen as regularized M-estimators for a parameter in a (typically infinite dimensional) reproducing kernel Hilbert space. For smooth loss functions, it is shown that the difference between the estimator, i.e.\ the empirical SVM, and the theoretical SVM is asymptotically normal with rate $\sqrt{n}$. That is, the standardized difference converges weakly to a Gaussian process in the reproducing kernel Hilbert space. As common in real applications, the choice of the regularization parameter may depend on the data. The proof is done by an application of the functional delta-method and by showing that the SVM-functional is suitably Hadamard-differentiable