Relations between the convexity of a set and the differentiability of its support function

It is known that, in finite dimensions, the support function of a compact convex set with nonempty interior is differentiable excepting the origin if and only if the set is strictly convex. In this paper, we realize a thorough study of the relations between the differentiability of the support function on the interior of its domain and the convexity of the set, mainly for unbounded sets. Then, we revisit some results related to the differentiability of the cost function associated to a production function.     

The differentiability of real functions on normed linear space using generalized subgradients

The modification of the Clarke generalized subdifferential due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Gâteaux differentiability of any real function can be deduced from the Gâteaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative. For any distance function on a space with uniformly Gâteaux differentiable norm, the Clarke and Michel-Penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for Gâteaux differentiability. However, on a Banach space with rotund dual, the Fréchet differentiability of a distance function implies that it is a convex function. A mean value theorem for the modified generalized subdifferential has implications for Gâteaux differentiability.     

Note on the differentiability of arithmetic Fourier series arising from Eisenstein series

The aim of this note is to present results concerning the differentiability of some Fourier series arising from Eisenstein series. Sine series exhibit different behaviours with respect to differentiability than the series with cosine function. The precise results are given for the series related to Eisenstein series of weight 2, whereas for the series arising from Eisenstein series of higher weight we conjecture the results.     

Heterogeneous cost sharing, the directional serial rule

The directional serial rule is introduced as a natural serial extension, generalizing the Moulin–Shenker cost sharing rule to heterogeneous cost sharing models. It is the unique regular rule compatible with the radial serial principle. In particular, this shows the incompatibility of the serial principle with differentiability of a cost sharing rule as a function of the individual demands.I would like to thank the editor and the referee for their comments which have been most useful.     

Solving the maximum problem for the terminal norm of a linear control system

We consider the maximization problem for the terminal norm of a linear control system. We seek for the extremal controls and improve them. Based on a sufficient optimality condition, we construct procedures that use the differentiability of the support function of the set of attainability.     

关于多元函数可微性的一个注记

研究了多元函数的可微性,给出了多元函数连续、可微的一些条件.     

Mass Transportation on Sub-Riemannian Manifolds

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier–McCann’s theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows us to write a weak form of the Monge–Ampère equation.     

Shape Optimal Design Problem with Convective and Radiative Heat Transfer: Analysis and Implementation

We present a study of an optimal design problem for a coupled system, governed by a steady-state potential flow equation and a thermal equation taking into account radiative phenomena with multiple reflections. The state equation is a nonlinear integro-differential system. We seek to minimize a cost function, depending on the temperature, with respect to the domain of the equations. First, we consider an optimal design problem in an abstract framework and, with the help of the classical adjoint state method, give an expression of the cost function differential. Then, we apply this result in the two-dimensional case to the nonlinear integro-differential system considered. We prove the differentiability of the cost function, introduce the adjoint state equation, and give an expression of its exact differential. Then, we discretize the equations by a finite-element method and use a gradient-type algorithm to decrease the cost function. We present numerical results as applied to the automotive industry.     

On differentiability of ruin functions under Markov-modulated models

This paper analyzes the continuity and differentiability of several classes of ruin functions under Markov-modulated insurance risk models with a barrier and threshold dividend strategy, respectively. Many ruin related functions in the literature, such as the expectation and the Laplace transform of the Gerber–Shiu discounted penalty function at ruin, of the total discounted dividends until ruin, and of the time-integrated discounted penalty and/or reward function of the risk process, etc, are special cases of the functions considered in this paper. Continuity and differentiability of these functions in the corresponding dual models are also studied.     

Higher order differentiability properties of the composition and of the inversion operator

This paper contains theorems of r-th order Fréchet differentiability, with r≥1, for the autonomous composition operator and for the inversion operator in Schauder spaces. The optimality of the differentiability theorems for the composition is indicated by means of an ‘inverse result’. A main point of this paper is that (higher order) ‘sharp’ differentiability theorems for the composition operator can be proved by approximating the operator by composition operators whose superposing function is a polynomial, an idea which may be employed in other function space settings.     

An optimal control problem for two-dimensional Schrödinger equation

In this paper we consider an optimal control problem controlled by three functions which are in the coefficients of a two-dimensional Schrödinger equation. After proving the existence and uniqueness of the optimal solution, we get the Frechet differentiability of the cost functional using Hamilton-Pontryagin function. Then we state a necessary condition to an optimal solution in the variational inequality form using the gradient.     

Dirichlet函数与Riemann函数的性质及应用

探讨Dirichlet函数和Riemann函数的连续性、可导性和可积性,并由此分别构造出在一点连续、多点连续、一点可导、多点可导的函数.     

On the Semi-norm of Radial Basis Function Interpolants

Radial basis function interpolation has attracted a lot of interest in recent years. For popular choices, for example thin plate splines, this problem has a variational formulation, i.e. the interpolant minimizes a semi-norm on a certain space of radial functions. This gives rise to a function space, called the native space. Every function in this space has the property that the semi-norm of an arbitrary interpolant to this function is uniformly bounded. In applications it is of interest whether a sufficiently smooth function belongs to the native space. In this paper we give sufficient conditions on the differentiability of a function with compact support, in the case of cubic, linear and thin plate splines. In the case of multiquadrics and Gaussian functions, it is shown that the only compactly supported function that satisfies these conditions is identically zero.     

An unconstrained smooth minimization reformulation of the second-order cone complementarity problem

A popular approach to solving the nonlinear complementarity problem (NCP) is to reformulate it as the global minimization of a certain merit function over ℝⁿ. A popular choice of the merit function is the squared norm of the Fischer-Burmeister function, shown to be smooth over ℝⁿ and, for monotone NCP, each stationary point is a solution of the NCP. This merit function and its analysis were subsequently extended to the semidefinite complementarity problem (SDCP), although only differentiability, not continuous differentiability, was established. In this paper, we extend this merit function and its analysis, including continuous differentiability, to the second-order cone complementarity problem (SOCCP). Although SOCCP is reducible to a SDCP, the reduction does not allow for easy translation of the analysis from SDCP to SOCCP. Instead, our analysis exploits properties of the Jordan product and spectral factorization associated with the second-order cone. We also report preliminary numerical experience with solving DIMACS second-order cone programs using a limited-memory BFGS method to minimize the merit function. In honor of Terry Rockafellar on his 70th birthday     

Differentiability of the value function in continuous-time economic models

In this paper we provide some sufficient conditions for the differentiability of the value function in a class of infinite-horizon continuous-time models of convex optimization arising in economics. We dispense with the assumption of interior optimal paths. This assumption is quite unnatural in constrained optimization, and is usually hard to check in applications. The differentiability of the value function is used to prove Bellman’s equation as well as the existence and continuity of the optimal feedback policy. We also establish the uniqueness of the vector of dual variables. These results become useful for the characterization and computation of optimal solutions.     

Optimal mass transportation in the Heisenberg group

In this paper we consider the problem of optimal transportation of absolutely continuous masses in the Heisenberg group H_n, in the case when the cost function is either the square of the Carnot-Carathéodory distance or the square of the Korányi norm. In both cases we show existence and uniqueness of an optimal transport map. In the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function. In the latter case the proof requires some fine properties of BV functions in the Heisenberg group.     

A note on approximation of optimal solutions of free problems of the calculus of variations

M. Lavrentiev, and later B. Manià, have given examples of free problems of the Calculus of Variations in which it is impossible to find a piecewiseC ¹ function, satisfying the same boundary conditions as the absolutely continuous optimal solution, and which simultaneously approximates both the optimal cost and, uniformly, the optimal trajectory. We investigate this question of approximation for free problems inE ⁿ. Dispensing with differentiability assumptions on the integrand of the cost functional, we present sufficient conditions under which this approximation problem may be solved thus extending earlier work of Manià, Tonelli, and others for free problems in the plane.     

随机保费模型下绝对破产概率的可微性以及渐近性（英文）

本文研究了具有随机保费收入的风险模型的Gerber-Shiu罚金函数的可微性以及渐近性质,随机保费收入通过一个复合泊松过程刻画.本文得到了Gerber-Shiu函数所满足的积分微分方程,给出了Gerber-Shiu罚金函数二次可微与三次可微的充分条件.当所讨论的罚金函数是三次可微的时候,前述积分微分方程可以转化为一般的常微分方程.利用常微分方程的标准方法,当个体随机保费和随机理赔都是指数分布的时候,得到了绝对破产概率在初始盈余趋向于无穷大时的渐近性质.     

Irreducible function bases for simple fluids and liquid crystal films: A new derivation

In a recent paper (Steigmanm, ZAMP, 2002) the irreducible function bases for simple fluids and liquid crystal films have been derived from the known complete and irreducible representations for isotropic and hemitropic scalar-valued functions of second-order tensors. The method used is to analyze the local or differential structures of the representations for isotropic or hemitropic scalar-valued functions, further restricted by symmetry under the unimodular group. Therefore, the differentiability of these representations, as well as the resulting irreducible function bases, is assumed. It is noted that the basic equations of simple fluids and liquid crystal films involve derivatives so that differentiability is not a serious restriction from the viewpoint of applications. However, a more general derivation of the irreducible function bases, without the differentiability assumption, is of theoretical interest and is presented here.     

Drag minimization for Navier–Stokes Flow

This paper investigates the drag minimization in a two‐dimensional flow which is governed by a nonhomogeneous Navier–Stokes equations. Two approaches are utilized to derive shape gradient of the cost functional. The first one is to use the shape derivative of the fluid state and its associated adjoint state; the second one is to utilize the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. Finally, a gradient type algorithm is effectively formulated and implemented for the mentioned drag minimization problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009