Generalized Directional Derivatives and Subgradients of Nonconvex Functions

1980 ◽  
Vol 32 (2) ◽  
pp. 257-280 ◽  
Author(s):  
R. T. Rockafellar
Keyword(s):  
Differential Calculus ◽  
Dual Space ◽  
Optimization Problems ◽  
Linear Topological Space ◽  
Directional Derivatives ◽  
Generalized Gradients ◽  
Nonconvex Functions ◽  
Generalized Directional Derivatives ◽  
The One ◽  
Derivatives Of

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.

Directionally Lipschitzian Mappings on Baire Spaces

1984 ◽  
Vol 36 (1) ◽  
pp. 95-130 ◽  
Author(s):  
J. M. Borwein ◽  
H. M. Stròjwas
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Optimization Problems ◽  
Directional Derivatives ◽  
Primary Interest ◽  
Lipschitzian Functions ◽  
Lipschitzian Mappings ◽  
Generalized Directional Derivatives ◽  
Derivatives Of ◽  
Locally Convex

Studies of optimization problems have led in recent years to definitions of several types of generalized directional derivatives. Those derivatives of primary interest in this paper were introduced and investigated by F. M. Clarke ([5], [6], [7], [8]), J. B. Hiriart-Urruty ([12]), Lebourg ([16], [17]), R. T. Rockafellar ([23], [24], [26], [27]), Penot ([21], [22]) among others.In an attempt to explore in more detail relationships between various types of generalized directional derivatives we discovered some unexpected results which were not observed by the above mentioned authors. We are able to give simple conditions which characterize directionally Lipschitzian functions defined on a Baire metrizable locally convex topological vector space.

Generalized Gradients, Lipschitz behavior and Directional Derivatives

1985 ◽  
Vol 37 (6) ◽  
pp. 1074-1084 ◽  
Author(s):  
Jay S. Treiman
Keyword(s):  
Optimization Problems ◽  
Lower Semicontinuous ◽  
Directional Derivative ◽  
Directional Derivatives ◽  
Value Functions ◽  
Generalized Gradients ◽  
Closed Set ◽  
Nonconvex Functions ◽  
Closed Sets ◽  
Optimal Value

In the study of optimization problems it is necessary to consider functions that are not differentiable. This has led to the consideration of generalized gradients and a corresponding calculus for certain classes of functions. Rockafellar [16] and others have developed a very strong and elegant theory of subgradients for convex functions. This convex theory gives point-wise criteria for the existence of extrema in optimization problems.There are however many optimization problems that involve functions which are neither differentiable nor convex. Such functions arise in many settings including optimal value functions [15]. In order to deal with such problems Clarke [3] defined a type of subgradient for nonconvex functions. This definition was initially for Lipschitz functions on R”. Clarke extended this definition to include lower semicontinuous (l.s.c.) functions on Banach spaces through the use of a directional derivative, the distance function from a closed set and tangent and normal cones to closed sets.

Higher-order conditions for strict local Pareto minima for problems with partial order introduced by a polyhedral cone

2018 ◽  
Vol 24 (1) ◽  
pp. 45-54
Author(s):  
Aleksandra Stasiak
Keyword(s):  
Partial Order ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Polyhedral Cone ◽  
Directional Derivatives ◽  
Pareto Minima ◽  
Necessary And Sufficient ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Derivatives Of

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.

scholarly journals Calmness and Calculus: Two Basic Patterns

2021 ◽  
Author(s):  
Matúš Benko ◽  
Patrick Mehlitz
Keyword(s):  
Optimization Problems ◽  
Sufficient Conditions ◽  
Necessary Optimality Conditions ◽  
Generalized Derivatives ◽  
Minimax Optimization ◽  
Cartesian Products ◽  
Calculus Rules ◽  
Set Valued Mappings ◽  
The One ◽  
Derivatives Of

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.

INBORN ERROR OF HISTIDINE METABOLISM

PEDIATRICS ◽  
1962 ◽  
Vol 29 (5) ◽  
pp. 714-728
Author(s):  
H. Ghadimi ◽  
M. W. Partington ◽  
A. Hunter
Keyword(s):  
Ferric Chloride ◽  
Urocanic Acid ◽  
High Concentration ◽  
Normal Children ◽  
Total Lack ◽  
Normal Urine ◽  
One Year ◽  
The One ◽  
Blue Eyes ◽  
Derivatives Of

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.

On second-order directional derivatives of value functions

Optimization ◽  
2013 ◽  
Vol 64 (2) ◽  
pp. 389-407 ◽  
Author(s):  
L. Minchenko ◽  
A. Tarakanov
Keyword(s):  
Second Order ◽  
Directional Derivatives ◽  
Value Functions ◽  
Derivatives Of

scholarly journals An Extensive Operational Law for Monotone Functions of LR Fuzzy Intervals with Applications to Fuzzy Optimization

2021 ◽  
Author(s):  
Mingxuan Zhao ◽  
Yulin Han ◽  
Jian Zhou
Keyword(s):  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Fuzzy Optimization ◽  
Fuzzy Programming ◽  
Planning Problem ◽  
Monotone Functions ◽  
Solution Strategy ◽  
Deterministic Equivalent ◽  
The One ◽  
Short Time

Abstract The operational law put forward by Zhou et al. on strictly monotone functions with regard to regular LR fuzzy numbers makes a valuable push to the development of fuzzy set theory. However, its applicable conditions are confined to strictly monotone functions and regular LR fuzzy numbers, which restricts its application in practice to a certain degree. In this paper, we propose an extensive operational law that generalizes the one proposed by Zhou et al. to apply to monotone (but not necessarily strictly monotone) functions with regard to regular LR fuzzy intervals (LR-FIs), of which regular fuzzy numbers can be regarded as particular cases. By means of the extensive operational law, the inverse credibility distributions (ICDs) of monotone functions regarding regular LR-FIs can be calculated efficiently and effectively. Moreover, the extensive operational law has a wider range of applications, which can deal with the situations hard to be handled by the original operational law. Subsequently, based on the extensive operational law, the computational formulae for expected values (EVs) of LR-FIs and monotone functions with regard to regular LR-FIs are presented. Furthermore, the proposed operational law is also applied to dispose fuzzy optimization problems with regular LR-FIs, for which a solution strategy is provided, where the fuzzy programming is converted to a deterministic equivalent first and then a newly-devised solution algorithm is utilized. Finally, the proposed solution strategy is applied to a purchasing planning problem, whose performances are evaluated by comparing with the traditional fuzzy simulation-based genetic algorithm. Experimental results indicate that our method is much more efficient, yielding high-quality solutions within a short time.

scholarly journals Boolean differential equations: A common model for classes, lattices, and arbitrary sets of Boolean functions

2015 ◽  
Vol 28 (1) ◽  
pp. 51-76 ◽  
Author(s):  
Bernd Steinbach ◽  
Christian Posthoff
Keyword(s):  
Differential Equations ◽  
Boolean Functions ◽  
Differential Calculus ◽  
Short Introduction ◽  
The Core ◽  
Boolean Equation ◽  
General Difference ◽  
Derivatives Of

The Boolean Differential Calculus (BDC) significantly extends the Boolean Algebra because not only Boolean values 0 and 1, but also changes of Boolean values or Boolean functions can be described. A Boolean Differential Equation (BDe) is a Boolean equation that includes derivative operations of the Boolean Differential Calculus. This paper aims at the classification of BDEs, the characterization of the respective solutions, algorithms to calculate the solution of a BDe, and selected applications. We will show that not only classes and arbitrary sets of Boolean functions but also lattices of Boolean functions can be expressed by Boolean Differential equations. In order to reach this aim, we give a short introduction into the BDC, emphasize the general difference between the solutions of a Boolean equation and a BDE, explain the core algorithms to solve a BDe that is restricted to all vectorial derivatives of f (x) and optionally contains Boolean variables. We explain formulas for transforming other derivative operations to vectorial derivatives in order to solve more general BDEs. New fields of applications for BDEs are simple and generalized lattices of Boolean functions. We describe the construction, simplification and solution. The basic operations of XBOOLE are sufficient to solve BDEs. We demonstrate how a XBooLe-problem program (PRP) of the freely available XBooLe-Monitor quickly solves some BDes.

scholarly journals The Fourier transform of thick distributions

2020 ◽  
pp. 1-26
Author(s):  
Ricardo Estrada ◽  
Jasson Vindas ◽  
Yunyun Yang
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Canonical Projection ◽  
Test Functions ◽  
Tempered Distributions ◽  
Delta Functions ◽  
The Fourier Transform ◽  
The One ◽  
One Point Compactification ◽  
Logarithmic Type

We first construct a space [Formula: see text] whose elements are test functions defined in [Formula: see text] the one point compactification of [Formula: see text] that have a thick expansion at infinity of special logarithmic type, and its dual space [Formula: see text] the space of sl-thick distributions. We show that there is a canonical projection of [Formula: see text] onto [Formula: see text] We study several sl-thick distributions and consider operations in [Formula: see text] We define and study the Fourier transform of thick test functions of [Formula: see text] and thick tempered distributions of [Formula: see text] We construct isomorphisms [Formula: see text] [Formula: see text] that extend the Fourier transform of tempered distributions, namely, [Formula: see text] and [Formula: see text] where [Formula: see text] are the canonical projections of [Formula: see text] or [Formula: see text] onto [Formula: see text] We determine the Fourier transform of several finite part regularizations and of general thick delta functions.

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