Implicit Functions and Solution Mappings

Book review: Implicit Functions and Solution Mappings:A View from Variational Analysis. Second Edition. By A. L. Dontchev and R. T. Rockafellar. Springer, New York, 2014

This book is the first comprehensive modern treatment of implicit functions. It consists of a preface followed by six chapters. Each chapter starts with a useful preamble and concludes with a careful and instructive commentary. The book includes a good set of references, a notation guide, and a somewhat brief index. This book is recommended to all practitioners and graduate students interested in modern optimization theory or control theory.

Implicit Functions and Solution Mappings

Metric fixed point theory is becoming increasingly significant across various fields, including data science and iterative methods for solving optimization problems. This paper aims to introduce new fixed point theorems for set-valued mappings under novel regularity conditions, such as orbital regularity and orbital pseudo-Lipschitzness. Instead of traditional metric spaces, we adopt the framework of quasi-metric spaces, motivated by the need to address problems in spaces that are not necessarily metric, such as function spaces of homogeneous type. We also explore the stability of the set of fixed points under variations of the set-valued mapping. Additionally, we provide estimates for the distances from a given point to the set of fixed points and between two sets of fixed points. Building on these findings, we extend the discussion to similar problems involving fixed, coincidence, and cyclic/double fixed points within this framework. Our results generalize recent findings from the literature, including those in Ait Mansour M, Bahraoui MA, El Bekkali A. [Metric regularity and Lyusternik-Graves theorem via approximate fixed points of set-valued maps in noncomplete metric spaces. Set-Valued Var Anal. 2022;30(1):233–256. doi: 10.1007/s11228-020-00553-1], Dontchev AL, Rockafellar RT. [Implicit functions and solution mappings. a view from variational analysis. Dordrecht: Springer; 2009. Springer Monographs in Mathematics], Ioffe AD. [Variational analysis of regular mappings. Springer, Cham; 2017. Springer Monographs in Mathematics; theory and applications. doi: 10.1007/978-3-319-64277-2], Lim TC. [On fixed point stability for set-valued contractive mappings with applications to generalized differential equations. J Math Anal Appl. 1985;110(2):436–441. doi: 10.1016/0022-247X(85)90306-3] and Tron NH. [Coincidence and fixed points of set-valued mappings via regularity in metric spaces. Set-Valued Var Anal. 2023;31(2):22. Paper No. 17. doi: 10.1007/s11228-023-00680-5].

Let g : E → F be an analytic function between two Hilbert spaces E and F. We study the set g(B(x, e)) ⊂ E, the image under g of the closed ball about x∈ E with radius e . When g(x) expresses the solution of an equation depending on x , then the elements of g(B(x,e )) are e -pseudosolutions. Our aim is to investigate the size of the set g(B(x,e )) . We derive upper and lower bounds of the following form: g(x) + Dg (x) ( B(0, c 1 e N)) g(B(x,e )) g(x) +Dg (x) ( B(0, c 2 e ) ), where Dg (x) denotes the derivative of g at x . We consider both the case where g is given explicitly and the case where g is given implicitly. We apply our results to the implicit function associated with the evaluation map, namely the solution map, and to the polynomial eigenvalue problem. Our results are stated in terms of an invariant γ which has been extensively used by various authors in the study of Newton's method. The main tool used here is an implicit γ theorem, which estimates the γ of an implicit function in terms of the γ of the function defining it.

Implicit analytic solutions for a nonlinear fractional partial differential beam equation

Implicit functions and solutions of equations in groups

Relation between an implicit function and solutions of a singularly perturbed equation

McCormick's classical relaxation technique constructs closed-form convex and concave relaxations of compositions of simple intrinsic functions. These relaxations have several properties which make them useful for lower bounding problems in global optimization: they can be evaluated automatically, accurately, and computationally inexpensively, and they converge rapidly to the relaxed function as the underlying domain is reduced in size. They may also be adapted to yield relaxations of certain implicit functions and differential equation solutions. However, McCormick's relaxations may be nonsmooth, and this nonsmoothness can create theoretical and computational obstacles when relaxations are to be deployed. This article presents a continuously differentiable variant of McCormick's original relaxations in the multivariate McCormick framework of Tsoukalas and Mitsos. Gradients of the new differentiable relaxations may be computed efficiently using the standard forward or reverse modes of automatic differentiation. Extensions to differentiable relaxations of implicit functions and solutions of parametric ordinary differential equations are discussed. A C++ implementation based on the library MC++ is described and applied to a case study in nonsmooth nonconvex optimization.

The implicit function theorem asserts that there exists a ball of nonzero radius within which one can express a certain subset of variables, in a system of equations, as functions of the remaining variables. We derive a lower bound for the radius of this ball in the case of Lipschitz maps. Under a sign-preserving condition, we prove that an implicit function exists in the case of a set of inequalities. Also in this case, we state an estimate for the size of the domain. An application to the local Lipschitz behavior of solution maps is discussed.

An implicit wave equation model for the shallow water equations

This paper deals with a kind of nonconvex optimistic bilevel optimization programs. In some process of dealing this kind of bilevel programs, difficulties are essentially moved to estimating for coderivative of the solution map. To deal with these difficulties, we use value function of the lower level problem and its modifications as implicit functions to describe the solution map. By applying techniques in variational analysis, we give estimates for coderivative of the solution map. Then, we will show the applications in optimality conditions for these bilevel programs which we derived by using the extremal principle.

Solutions mappings in the classical setting of the implicit function theorem concern problems in the form of parameterized equations. The concept can go far beyond that, however. In any situation where some kind of problem in x depends on a parameter p, there is the mapping S that assigns to each p the corresponding set of solutions x.

Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs

In this paper we develop a stability theory for broad classes of parametric generalized equations and variational inequalities in finite dimensions. These objects have a wide range of applications in optimization, nonlinear analysis, mathematical economics, etc. Our main concern is Lipschitzian stability of multivalued solution maps depending on parameters. We employ a new approach of nonsmooth analysis based on the generalized differentiation of multivalued and nonsmooth operators. This approach allows us to obtain effective sufficient conditions as well as necessary and sufficient conditions for a natural Lipschitzian behavior of solution maps. In particular, we prove new criteria for the existence of Lipschitzian multivalued and single-valued implicit functions.