scholarly journals Calmness and Calculus: Two Basic Patterns

Author(s):  
Matúš Benko ◽  
Patrick Mehlitz

AbstractWe establish two types of estimates for generalized derivatives of set-valued mappings which carry the essence of two basic patterns observed throughout the pile of calculus rules. These estimates also illustrate the role of the essential assumptions that accompany these two patters, namely calmness on the one hand and (fuzzy) inner calmness* on the other. Afterwards, we study the relationship between and sufficient conditions for the various notions of (inner) calmness. The aforementioned estimates are applied in order to recover several prominent calculus rules for tangents and normals as well as generalized derivatives of marginal functions and compositions as well as Cartesian products of set-valued mappings under mild conditions. We believe that our enhanced approach puts the overall generalized calculus into some other light. Some applications of our findings are presented which exemplary address necessary optimality conditions for minimax optimization problems as well as the calculus related to the recently introduced semismoothness* property.

Author(s):  
Patrick Mehlitz ◽  
Leonid I. Minchenko

AbstractThe presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.


2021 ◽  
Vol Volume 2 (Original research articles) ◽  
Author(s):  
Matúš Benko ◽  
Patrick Mehlitz

Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel programming, evaluated multiobjective optimization, or nonlinear optimization problems with slack variables. In order to deal with implicit variables, they are often interpreted as explicit ones. Here, we first point out that this is a light-headed approach which induces artificial locally optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type necessary optimality conditions which correspond to treating the implicit variables as explicit ones on the one hand, or using them only implicitly to model the constraints on the other. A detailed comparison of the obtained stationarity conditions as well as the associated underlying constraint qualifications will be provided. Overall, we proceed in a fairly general setting relying on modern tools of variational analysis. Finally, we apply our findings to different well-known problem classes of mathematical optimization in order to visualize the obtained theory. Comment: 34 pages


2018 ◽  
Vol 24 (1) ◽  
pp. 45-54
Author(s):  
Aleksandra Stasiak

Abstract Using the definitions of μ-th order lower and upper directional derivatives of vector-valued functions, introduced in Rahmo and Studniarski (J. Math. Anal. Appl. 393 (2012), 212–221), we provide some necessary and sufficient conditions for strict local Pareto minimizers of order μ for optimization problems where the partial order is introduced by a pointed polyhedral cone with non-empty interior.


Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 112
Author(s):  
Savin Treanţă

In this paper, we investigate the Lagrange dynamics generated by a class of isoperimetric constrained controlled optimization problems involving second-order partial derivatives and boundary conditions. More precisely, we derive necessary optimality conditions for the considered class of variational control problems governed by path-independent curvilinear integral functionals. Moreover, the theoretical results presented in the paper are accompanied by an illustrative example. Furthermore, an algorithm is proposed to emphasize the steps to be followed to solve a control problem such as the one studied in this paper.


1980 ◽  
Vol 32 (2) ◽  
pp. 257-280 ◽  
Author(s):  
R. T. Rockafellar

Studies of optimization problems and certain kinds of differential equations have led in recent years to the development of a generalized theory of differentiation quite distinct in spirit and range of application from the one based on L. Schwartz's “distributions.” This theory associates with an extended-real-valued function ƒ on a linear topological space E and a point x ∈ E certain elements of the dual space E* called subgradients or generalized gradients of ƒ at x. These form a set ∂ƒ(x) that is always convex and weak*-closed (possibly empty). The multifunction ∂ƒ: x →∂ƒ(x) is the sub differential of ƒ.Rules that relate ∂ƒ to generalized directional derivatives of ƒ, or allow ∂ƒ to be expressed or estimated in terms of the subdifferentials of other functions (whenƒ = ƒ1 + ƒ2,ƒ = g o A, etc.), comprise the sub differential calculus.


2021 ◽  
pp. 1-11
Author(s):  
Jian Wang ◽  
Yuanguo Zhu

Uncertain delay differential equation is a class of functional differential equations driven by Liu process. It is an important model to describe the evolution process of uncertain dynamical system. In this paper, on the one hand, the analytic expression of a class of linear uncertain delay differential equations are investigated. On the other hand, the new sufficient conditions for uncertain delay differential equations being stable in measure and in mean are presented by using retarded-type Gronwall inequality. Several examples show that our stability conditions are superior to the existing results.


2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raffaela Capitanelli ◽  
Maria Agostina Vivaldi

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.


2017 ◽  
Vol 27 (1) ◽  
pp. 77-89
Author(s):  
Adam Kowalewski

AbstractVarious optimization problems for linear parabolic systems with multiple constant time lags are considered. In this paper, we consider an optimal distributed control problem for a linear complex parabolic system in which different multiple constant time lags appear both in the state equation and in the Neumann boundary condition. Sufficient conditions for the existence of a unique solution of the parabolic time lag equation with the Neumann boundary condition are proved. The time horizon T is fixed. Making use of the Lions scheme [13], necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functional with pointwise observation of the state and constrained control are derived. The example of application is also provided.


PEDIATRICS ◽  
1962 ◽  
Vol 29 (5) ◽  
pp. 714-728
Author(s):  
H. Ghadimi ◽  
M. W. Partington ◽  
A. Hunter

A 3-year-old girl with fair hair and blue eyes came under observation because of speech retardation. The patient was given a phenylalanine-free diet, since persistently positive urine tests with ferric chloride and reagent strips (Phenistix) were strongly suggestive of phenylketonuria. On further investigation she was found to be suffering from a totally different and not heretofore recognized condition, the salient features of which were an abnormally high concentration of histidine in the blood and an excessive output of histidine in the urine. The child's sister, one year older, presented the same metabolic anomaly. In each case both the concentration of histidine in plasma and the daily output were directly related to the amount of protein in the diet; but even at their lowest levels they greatly exceeded those of normal children of the same age. An oral load of histidine was followed by an increase of histidine in plasma much higher and more prolonged than that observed in controls, and by the excretion of a much larger fraction of the ingested dose. The urine of each sister contained not only excessive amounts of histidine but also notable quantities of imidazole-pyruvic, imidazole-acetic, and imidazole-lactic acid, histidine derivatives of which normal urine contains only traces. It is concluded that in the two sisters the normally predominant pathway of histidine catabolism, which leads through urocanic acid to glutamic acid, was partially or completely blocked. Since, even after histidine loading, the urines never contained detectable amounts of urocanic acid, the block must precede the formation of that substance. The condition presented is therefore due primarily to a deficiency or total lack of histidine-alpha-deaminase; the enzyme which converts histidine to urocanic acid. As a result of this defect histidine is forced to take the alternative but less efficient pathway which begins with its transamination to imidazole-pyruvic acid. This substance is the one responsible for positive reactions in the ferric chloride and Phenistix tests. All urine specimens examined, whether from the patient or from normal controls, contained considerable quantities of an unidentifiable imidazole compound ("X"), which does not appear to have been previously reported. The existence of this substance may call for some modification of current concepts of histidine metabolism in man.


2018 ◽  
Vol 24 (3) ◽  
pp. 1059-1074
Author(s):  
Michel H. Geoffroy ◽  
Yvesner Marcelin

We introduce a class of positively homogeneous set-valued mappings, called inner prederivatives, serving as first order approximants to set-valued mappings. We prove an inverse mapping theorem involving such prederivatives and study their stability with respect to variational perturbations. Then, taking advantage of their properties we establish necessary optimality conditions for the existence of several kind of minimizers in set-valued optimization. As an application of these last results, we consider the problem of finding optimal allocations in welfare economics. Finally, to emphasize the interest of our approach, we compare the notion of inner prederivative to the related concepts of set-valued differentiation commonly used in the literature.


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