Analysis of Symmetric Matrix Valued Functions

For any symmetric function f: ? ⁿ  → ? ⁿ , one can define a corresponding function on the space of n × n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Fréchet differentiability, and continuous differentiability.     

Analytic approximation and differentiability of joint chance constraints

ABSTRACT

An inner–outer approximation approach was recently developed to solve single chance constrained optimization (SCCOPT) problems. In this paper, we extend this approach to address joint chance constrained optimization (JCCOPT) problems. Using an inner–outer approximation, two smooth parametric optimization problems are defined whose feasible sets converge to the feasible set of JCCOPT from inside and outside, respectively. Any optimal solution of the inner approximation problem is a priori feasible to the JCCOPT. As the approximation parameter tends to zero, a subsequence of the solutions of the inner and outer problems, respectively, converge asymptotically to an optimal solution of the JCCOPT. As a main result, the continuous differentiability of the probability function of a joint chance constraint is obtained by examining the uniform convergence of the gradients of the parametric approximations.     

Best L2 Local Approximation on Two Small Intervals

In this article, we introduce the τ condition, which is weaker than the L² differentiability. If a function satisfies the τ condition on two points of ?, we prove the existence and characterization of the best local polynomial approximation on these points.     

New Constrained Optimization Reformulation of Complementarity Problems

We suggest a reformulation of the complementarity problem CP(F) as a minimization problem with nonnegativity constraints. This reformulation is based on a particular unconstrained minimization reformulation of CP(F) introduced by Geiger and Kanzow as well as Facchinei and Soares. This allows us to use nonnegativity constraints for all the variables or only a subset of the variables on which the function F depends. Appropriate regularity conditions ensure that a stationary point of the new reformulation is a solution of the complementarity problem. In particular, stationary points with negative components can be avoided in contrast to the reformulation as unconstrained minimization problem. This advantage will be demonstrated for a class of complementarity problems which arise when the Karush–Kuhn–Tucker conditions of a convex inequality constrained optimization problem are considered.     

A note on non-support points, negligible sets, Gâteaux differentiability and Lipschitz embeddings

This paper first presents a characterization of three classes of negligible closed convex sets (i.e., Gauss null sets, Aronszajn null sets and cube null sets) in terms of non-support points; then gives a generalization of Gâteaux differentiability theorems of Lipschitz mapping from open sets to those closed convex sets admitting non-support points; and as their application, finally shows that a closed convex set in a separable Banach space X can be Lipschitz embedded into a Banach space Y with the Radon–Nikodym property if and only if the closure of its linear span is linearly isomorphic to a closed subspace of Y.     

Numerical estimation of the piecewise constant Robin coefficient by identifying its discontinuous points

We propose a new numerical method for estimating the piecewise constant Robin coefficient in two-dimensional elliptic equation from boundary measurements. The Robin inverse problem is recast into a minimization of an output least-square formulation. A technique based on determining the discontinuous points of the unknown coefficient is suggested, and we investigate the differentiability of the solution and the objective functional with respect to the discontinuous points. Then we apply the Gauss-Newton method for reconstructing the shape of the unknown Robin coefficient. Numerical examples illustrate its efficiency and stability.     

Gâteaux differentiability of the dual gap function of a variational inequality

In terms of the mapping involved in a variational inequality, we characterize the Gâteaux differentiability of the dual gap function G and present several sufficient conditions for its directional derivative expression, including one weaker than that of Danskin [J.M. Danskin, The theory of max–min, with applications, SIAM Journal on Applied Mathematics 14 (1966) 641–664]. When the solution set of a variational inequality problem is contained in that of its dual problem, the Gâteaux differentiability of G on the latter turns out to be equivalent to the conditions appearing in the authors’ recent results about the weakly sharp solutions of the variational inequality problem.     

The product of a Gâteaux differentiability space and a separable space is a Gâteaux differentiability space

This paper shows that the product of a Gâteaux differentiability space and a separable Banach space is again a Gâteaux differentiability space.

     

Sensitivity analysis for parametric control problems with control-state constraints

Parametric nonlinear control problems subject to vector-valued mixed control-state constraints are investigated. The model perturbations are implemented by a parameter p of a Banach-space P. We prove solution differentiability in the sense that the optimal solution and the associated adjoint multiplier function are differentiable functions of the parameter. The main assumptions for solution differentiability are composed by regularity conditions and recently developed second-order sufficient conditions (SSC). The analysis generalizes the approach in [16, 20] and establishes a link between (1) shooting techniques for solving the associated boundary value problem (BVP) and (2) SSC. We shall make use of sensitivity results from finite-dimensional parametric programming and exploit the relationships between the variational system associated to BVP and its corresponding Riccati equation.Solution differentiability is the theoretical backbone for any numerical sensitivity analysis. A numerical example with a vector-valued control is presented that illustrates sensitivity analysis in detail.     

Linearization of a free boundary problem in corrosion detection

We consider a boundary identification problem arising in nondestructive testing of materials. The problem is to recover a part ΓI⊂∂Ω of the boundary of a bounded, planar domain Ω from one Cauchy data pair (u,∂u/∂ν) of a harmonic potential u in Ω collected on an accessible boundary subset ΓA⊂∂Ω. We prove Fréchet differentiability of a suitably defined forward map, and discuss local uniqueness and Lipschitz stability results for the linearized problem.     

函数的逼近及其在下半连续函数可微性中的应用

通过函数的下卷积函数列的逼近方法,在变分原理中从扰动最小值点集的“大小”入手,研究了下半连续函数的可微性。     

Fréchet differentiability for a damped sine-Gordon equation

We consider a damped sine-Gordon equation with a variable diffusion coefficient. The goal is to derive necessary conditions for the optimal set of parameters minimizing the objective function J. First, we show that the solution map is continuous under a weak assumption on the topology of the admissible set P. Then the solution map is shown to be weakly Gâteux differentiable on P, implying the Gâteux differentiability of the objective function. Finally we show the Fréchet differentiability of J. The optimal set of parameters is shown to satisfy a bang–bang control law.     

关于初值问题解的稳定性

本文利用解对初值和参数的可微性,提出一种不同于Liapunov直接法的方法,用它可以判定初值问题非驻定解的稳定性.而该非驻定解无需事先求出.     

Lateral boundary differentiability of solutions of parabolic equations in nondivergence form

The lateral boundary differentiability is shown for solutions of parabolic differential equations in nondivergence form under the assumptions that the parabolic boundary satisfies the exterior Dini condition and is punctually C¹$C^1$ differentiable one-sided in t-direction. The classical barrier technique, the maximum principle, the interior Harnack inequality and an iteration procedure are the main analytical tools.     

DIFFERENTIABILITY OF CONVEX FUNCTIONS ON SUBLINEAR TOPOLOGICAL SPACES AND VARIATIONAL PRINCIPLES IN LOCALLY CONVEX SPACES

This paper presents a type of variational principles for real valued w lower semicon-tinuous functions on certain subsets in duals of locally convex spaces, and resolve a problem concerning differentiability of convex functions on general Banach spaces. They are done through discussing differentiability of convex functions on nonlinear topological spaces and convexification of nonconvex functions on topological linear spaces.     

Numerical solution of fuzzy differential equations under generalized differentiability

In this paper, we interpret a fuzzy differential equation by using the strongly generalized differentiability concept. Utilizing the Generalized Characterization Theorem, we investigate the problem of finding a numerical approximation of solutions. Then we show that any suitable numerical method for ODEs can be applied to solve numerically fuzzy differential equations under generalized differentiability. The generalized Euler approximation method is implemented and its error analysis, which guarantees pointwise convergence, is given. The method’s applicability is illustrated by solving a linear first-order fuzzy differential equation.     

A note on the sufficient initial condition ensuring the convergence of directly extended 3-block ADMM for special semidefinite programming

ABSTRACT

In this note, we consider three types of problems, H-weighted nearest correlation matrix problem and two types of important doubly non-negative semidefinite programming, derived from the binary integer quadratic programming and maximum cut problem. The dual of these three types of problems is a 3-block separable convex optimization problem with a coupling linear equation constraint. It is known that, the directly extended 3-block alternating direction method of multipliers (ADMM3d) is more efficient than many of its variants for solving these convex optimization, but its convergence is not guaranteed. By choosing initial points properly, we obtain the convergence of ADMM3d for solving the dual of these three types of problems. Furthermore, we simplify the iterative scheme of ADMM3d and show the equivalence of ADMM3d to the 2-block semi-proximal ADMM for solving the dual's reformulation, under these initial conditions.     

Design sensitivity for arch structures with respect to midsurface shape under static loading

We consider in this paper an example of structural optimization in which the structure is a loaded arch and the design variable is the shape of the arch. We concentrate on differentiability of static response with respect to shape changes. After recalling the arch equation with its functional spaces and the optimization problem, we state a differentiability theorem and provide a detailed proof. Numerical use of this result is finally discussed.     

多元函数的可微性

给出了多元函数可微性的等价条件 ;利用这个条件可以简便有效地判别多元函数的可微性