Right Derivatives Research Articles

Some recent results in parametric semi-infinite programming

In this tutorial paper we present some recent developments in parametric semi-infinite programming. We mainly focus our attention to two problems: The determination of one-sided derivatives of the value-function with a special application to the problem of finding zeros of that function and, secondly, the problem of following a path of solutions. For the latter, a continuation method is presented which can be shown to be applicable to an open and dense class of problems

Theory of algebraic invariants

Preface Introduction Part I. The Elements of Invariant Theory: 1. The forms 2. The linear transformation 3. The concept of an invariant 4. Properties of invariants and covariants 5. The operation symbols D and D 6. The smallest system of conditions for the determination of the invariants and covariants 7. The number of invariants of degree g 8. The invariants and covariants of degree two and three 9. Simultaneous invariants and covariants 10. Covariants of covariants 11. The invariants and covariants as functions of the roots 12. The invariants and covariants as functions of the one-sided derivatives 13. The symbolic representation of invariants and covariants Part II. The Theory of Invariant Fields: 14. Proof of the finitenesss of the full invariant system via representation by root differences 15. A generalizable proof for the finiteness of the full invariant system 16. The system of invariants I I1, I2, ..., Ik 17. The vanishing of the invariants 18. The ternary nullform 19. The finiteness of the number of irreducible syzygies and of the syzygy chain 20. The inflection point problem for plane curves of order three 21. The generalization of invariant theory 22. Observations about new types of coordinates.

EFFECT OF GRID ORTHOGONALITY ON THE SOLUTION ACCURACY OF THE TWO-DIMENSIONAL CONVECTION-DIFFUSION EQUATION

Abstract The effects of grid orthogonality and smoothness on the accuracy of finite-difference solutions of the two-dimensional Laplace, convection-diffusion, and Navier-Stokes equations are studied analytically and numerically. The examples include flow past an airfoil and in a branching channel. It is concluded that orthogonality has little impact on accuracy in general, provided the angle between grid lines is not too small. Rather, accuracy is very sensitive to the clustering of points in the regions of rapid variation of the solution, and orthogonality may in fact have an adverse effect on the quality of the solution when it leads to a coarser resolution of these regions. These conclusions also extend to orthogonality at boundaries (either physical or computational), where Neumann conditions are implemented by one-sided derivatives.

One-sided derivatives for the value function in convex parametric programming

In [9] it is shown that the one-sided derivatives of parametric linear semi-infinite programs can be expressed in terms of one-sided cluster points of solutions and Lagrange-multipliers of the perturbed problem. In this paper convex programs on Banach-spaces are studied. We generalize the results in [9] to this case. The analysis is based on convex duality theory [2].

One-sided derivative of min max and saddle points with respect to a parameter*

The object of this paper is to present new theorems on the differentiability of a minimum, maximum, Min Max, or saddle point with respect to a parameter t ⩾ 0 at t = 0 under relaxed assumptions. Those technical results have a wide spectrum of applications: Control Theory, Shape Sensitivity Analysis, Game Theory, etc... In non-differentiable situations they provide an interesting description of the non-differentiability

Reality and Differential Calculus on the q -Euclidean Space

By introducing the left and right derivatives, we establish a real structure for the covariant differential calculus on the N -dimensional quantum Euclidean space. The method is also applicable to the quantum Minkowski space.

Border-collision bifurcations including “period two to period three” for piecewise smooth systems

We examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter μ. In the simplest case there is a surface Γ in phase space along which the map has no derivative (or has two one-sided derivatives). Γ is the border of two regions in which the map is smooth. As the parameter μ is varied, a fixed point E μ may collide with the border Γ, and we may assume that this collision occurs at μ = 0. A variety of bifurcations occur frequently in such situations, but never or almost never occur in smooth systems. In particular E μ may cross the border and so will exist for μ<0 and for μ > 0 but it may be a saddle in one case, say μ<0, and it may be a repellor for μ > 0. For μ<0 there can be a stable period two orbit which shrinks to the point E 0 as μ→0, and for μ > 0 there may be a stable period 3 orbit which similarly shrinks to E 0 as μ→0. Hence one observes the following stable periodic orbits: a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit. We also see analogously “stable period 2 to stable period p orbit bifurcations”, with p = 5,11,52, or period 2 to quasi-periodic or even to a chaotic attractor. We believe this phenomenon will be seen in many applications.

On semidifferentiable functions

We prove that, under suitable assumptions, the two general schemes of generalized derivatives (K-derivatives, introduced in [3], and G-derivatives introduced in [4]) are equivalent. Afterwards we analyze some properties of the G-semidifferentiable functions which are important for treating extremum problems

General b-spline hermite interpolation

This paper presents a formula for the coefficients of the B-spline function of degree 2 m+1 whose values and (possibly distinct one-sided) derivatives of orders 1,…, m are prescribed at the knots.

Semidifferentiable functions and necessary optimality conditions

In the last two decades, there has been an increasing interest in nonsmooth optimization, both from a theoretical viewpoint and because of several applications. Necessary optimality conditions, as well as other important topics, have received new attention (see, for instance, Refs. 1–11 and references therein). In recent papers (see, for instance, Refs. 12–18 and references therein), theorems of the alternative for generalized systems have been studied and their use in optimization has been exploited. As a consequence of this analysis, the concept of image of a constrained extremum problem has been developed; such a concept, whose introduction goes back to the work of Caratheodory, has only recently been recognized to be a powerful tool (Refs. 8, 10, 13, 14, 18–21). On the basis of these ideas, in the present paper we deal with a necessary condition for constrained extremum problems having a finite-dimensional image, while those having an infinite-dimensional one will be treated in a subsequent paper. The necessary condition is established within a class of semidifferentiable functions, which is introduced here and which embraces several classic types of functions (e.g., convex functions, differentiable functions, and even some discontinuous functions). The condition embodies the classic theorems of Lagrange, John, Karush, Kuhn-Tucker, and Euler.

Kuhn-Tucker curves for one-parametric semi-infinite programming

An algorithm for computing Kuhn-Tucker curves of one-parametric semi-infinite optimization problems p(t),t ∈R is presented. Starting with a Kuhn-Tucker point xfor a fixed tan underdetermined system of nonlinear equations is used to describe a piece of the Kuhn-Tucker curve (t(α)x(x)) by continuation methods. An active-set-strategy is applied to obtain a new system defining another part of that curve. In addition, a formula for one-sided derivatives of the value-function of p(t) is derived. Numerical examples from air-pollution control and Chebyshev approximation by splines with free knots are presented.

Fixed point theorems for semidifferentiable k-set-contractions in ordered Banach spaces

Soit X un espace de Banach reel, K un cone dans X, et «≤» un ordre dans X induit par K en definissant y≥x si et seulement si y−x∈K. La paire (X,K) est appelee un espace de Banach ordonne avec cone positif K si X est d'ordre donne induit par K. On etend certains theoremes de point fixe pour des contractions ensemble T:K→K qui sont differentiables de Frechet en 0 et/ou a l'infini

Some Remarks on Functions with One-Sided Derivatives

(1986). Some Remarks on Functions with One-Sided Derivatives. The American Mathematical Monthly: Vol. 93, No. 6, pp. 471-475.

Some Remarks on Functions with One-Sided Derivatives

Some Remarks on Functions with One-Sided Derivatives

Optimization of upper semidifferentiable functions

In this paper, we present an implementable algorithm to minimize a nonconvex, nondifferentiable function in ℝ m . The method generalizes Wolfe's algorithm for convex functions and Mifflin's algorithm for semismooth functions to a broader class of functions, so-called upper semidifferentiable. With this objective, we define a new enlargement of Clarke's generalized gradient that recovers, in special cases, the enlargement proposed by Goldstein. We analyze the convergence of the method and discuss some numerical experiments.

Boundary Behaviour of Level Curves of Harmonic Functions

Sufficient conditions are obtained for a function harmonic in the disk and continuous at a boundary point to have a level curve emanating from the boundary point into the disk. The one-sided derivative of the level curve is studied at the boundary point.

Boundary behaviour of level curves of harmonic functions

Sufficient conditions are obtained for a function harmonic in the disk and continuous at a boundary point to have a level curve emanating from the boundary point into the disk. The one-sided derivative of the level curve is studied at the boundary point.

Optimality conditions in nondifferentiable programming and their applications to best approximations

This paper studies constrained optimization problems in Banach spaces without usual differentiability and convexity assumptions on the functionals involved in the data. The aim is to give optimality conditions for the problems from which one can derive the characterization of best approximations. The objective and the inequality constraint functionals are assumed to have one-sided directional derivatives. First-order necessary conditions are given in terms of subdifferentials of the directional derivatives. The notion of “max-pseudoconvexity” weaker than pseudoconvexity is introduced for sufficiency. The optimality conditions are applied to linear and nonlinear Tchebycheff approximation problems to derive the characterization of best approximations.

Differentiability via one-sided directional derivatives

Let F F be a continuous mapping from an open subset D D of a separable Banach space X X into a Banach space Y Y . We show that if the one sided directional derivative D x + F ( a ) D_x^ + F(a) of F F at a a in the direction x x exists for each ( a , x ) (a,x) from a dense G δ {G_\delta } subset S S of an open set D × U ⊂ X × X D \times U \subset X \times X , then F F is Gâteaux differentiable on a dense G δ {G_\delta } subset of D D . Similar results are obtained for Fréchet differentiability when X X is finite-dimensional and for w ∗ {w^ * } -Gâteaux differentiability.

A second-order condition that strengthens Pontryagin's maximum principle

We derive a second-order necessary condition for optimal control problems defined by ordinary differential equations with endpoint restrictions. This condition, based on a second-order restricted minimization test, bears a somewhat similar relation to the Weierstrass E-condition (the Pontryagin maximum principle) as the Legendre and Jacobi conditions bear to the Euler-Lagrange equation. Specifically, in the context of relaxed controls, the E-condition for free endpoint problems asserts that if a function achieves its minimum over a convex set Q at some point q then its one-sided directional derivatives at q into Q are nonnegative. Our new condition, when applied to the special case of free endpoint problems, corresponds to the observation that if such a one-sided directional derivative at q is 0 then the corresponding second directional derivative is nonnegative. This new condition effectively supplements the Pontryagin maximum principle over the singular regimes of “weakly” normal extremals that are candidates for either a relaxed or an ordinary restricted minimum. Like some other second-order methods, this condition is global over the control set but, unlike the other tests, it is also global over time. A number of examples illustrate its use and behavior.