About contingent epiderivatives

In this paper we analyze some questions concerning the contingent epiderivative. When the ordering cone is not necessarily pointed we introduce the notion of family of contingent epiderivatives. We also investigate the relationship between the contingent derivative and the generalized epiderivative. Furthermore, we give existence theorems with respect to Daniell cones. As a particular case, we study set-valued maps that take values in Rn.     

Effect of microencapsulation using soy protein isolate and gum arabic as wall material on red raspberry anthocyanin stability,characterization, and simulated gastrointestinal conditions

This study involves microencapsulation of anthocyanin extract (AC) from red raspberry using a freeze drying technique involving ultrasonication of soy protein isolate (SPI), gum arabic (AG), and their combination. Analyses included microcapsule properties, encapsulation efficiency, antioxidant capacity using the DPPH assay, Fourier-transform infrared spectroscopy (FTIR), color, thermal stability, stability during storage, morphology, and simulated gastrointestinal conditions. Encapsulation efficiency (EE) ranged from 93.05% to 98.87% for all microcapsules. The morphology of microcapsules by scanning electron microscopy (SEM) indicated a broken glass shape, with mean size of microcapsules ranging from 21.07 ± 2.71 μm to 48.19 ± 2.33 μm using Mastersizer analyzer. A good relationship obtained between zeta potential (ZP) and FTIR spectrum indicated that there is a chemical cross linking interaction between wall material and core material. All microcapsules enhanced the thermal stability of AC in the temperature range 80–114 °C. Furthermore, AC retention (up to 48%) during storage at 37 °C for 60 days revealed that treatment H combination of SPI and AG was best. In addition, SPI and AG combination presented good release behavior under simulated gastrointestinal conditions compared with unencapsulated anthocyanin.     

A note on the compactness of the index set in convex optimization. Application to metric regularity

We consider the linear quadratic optimal control problem in the infinite time horizon case for a class of discrete-time systems controlled by a continuous inputs. We show that, under certain hypothesis, the Hilbert uniqueness method can be used to determine the optimal control.     

A smoothing method for mathematical programs with equilibrium constraints

Received May 3, 1996 / Revised version received November 19, 1997 Published online January 20, 1999     

Optimally Conditions for Pareto Nonsmooth Nonconvex Programming in Banach Spaces

In this paper, we consider a vector optimization problem where all functions involved are defined on Banach spaces. We obtain necessary and sufficient criteria for optimality in the form of Karush–Kuhn–Tucker conditions. We also introduce a nonsmooth dual problem and provide duality theorems.     

A study of tilt-stable optimality and sufficient conditions

Recent results by Eberhard et al. (2006) [4] and Eberhard and Wenczel (2009) [3] on the interaction of single- and double-envelope operations of nonsmooth functions and their interaction with second-order derivations have been used to study tilt-stability of local minima. This continues the study begun by Poliquin and Rockafellar (1998) [1] but now, armed with new tools we are able to make some new observations. We observe that tilt-stability entails a local density within the graph of the proximal subderivative of strict local minima order two of the tilted function. Indeed, it also entails the strict local minimality (order two) of the tilt-stable local minimum itself. For prox-regular, subdifferentially continuous functions this density property characterises tilt stability.     

Error bounds for analytic systems and their applications

Using a 1958 result of Lojasiewicz, we establish an error bound for analytic systems consisting of equalities and inequalities defined by real analytic functions. In particular, we show that over any bounded region, the distance from any vectorx in the region to the solution set of an analytic system is bounded by a residual function, raised to a certain power, evaluated atx. For quadratic systems satisfying certain nonnegativity assumptions, we show that this exponent is equal to 1/2. We apply the error bounds to the Karush—Kuhn—Tucker system of a variational inequality, the affine variational inequality, the linear and nonlinear complementarity problem, and the 0–1 integer feasibility problem, and obtain new error bound results for these problems. The latter results extend previous work for polynomial systems and explain why a certain square-root term is needed in an error bound for the (monotone) linear complementarity problem.The research of this author is based on work supported by the Natural Sciences and Engineering Research Council of Canada under grant OPG0090391.The research of this author is based on work supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739 and by the Office of Naval Research under grant 4116687-01.     

一类二阶非线性微分系统有界性与收敛性的充分必要条件

本文给出了如下一类二阶非线性微分系统 解的有界性的一些新的充分条件,获得了（Ｓ’）的所有解有界或收敛到零的充分必要 条件．所获的结果能够应用于Ｌｉｅｎａｒｄ类型方程 它改进与扩展了［１,２］等文献关于同样问题的一些重要结果．     

Asymptotic approximation for the solution to a semilinear parabolic problem in a thin star‐shaped junction

A semilinear parabolic problem is considered in a thin 3‐D star‐shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter The purpose is to study the asymptotic behavior of the solution u_ε as ε→0, ie, when the star‐shaped junction is transformed in a graph. In addition, the passage to the limit is accompanied by special intensity factors and in nonlinear perturbed Robin boundary conditions. We establish qualitatively different cases in the asymptotic behavior of the solution depending on the value of the parameters {α_i}and {β_i}. Using the multiscale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε→0. Namely, in each case, we derive the limit problem (ε=0)on the graph with the corresponding Kirchhoff transmission conditions (untypical in some cases) at the vertex, define other terms of the asymptotic approximation and prove appropriate asymptotic estimates that justify these coupling conditions at the vertex, and show the impact of the local geometric heterogeneity of the node and physical processes in the node on some properties of the solution.     

Amplitude equation for the stochastic reaction‐diffusion equations with random Neumann boundary conditions

In this paper, we consider a quite general class of reaction‐diffusion equations with cubic nonlinearities and with random Neumann boundary conditions. We derive rigorously amplitude equations, using the natural separation of time‐scales near a change of stability and investigate whether additive degenerate noise and random boundary conditions can lead to stabilization of the solution of the stochastic partial differential equation or not. The nonlinear heat equation (Ginzburg–Landau equation) is used to illustrate our result. Copyright © 2015 John Wiley & Sons, Ltd.