One-sided Directional Derivative Research Articles

Detecting invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps

We show how to formally identify chaotic attractors in continuous, piecewise-linear maps on . For such a map f, this is achieved by constructing three objects. First, is trapping region for f. Second, is a finite set of words that encodes the forward orbits of all points in . Finally, is an invariant expanding cone for derivatives of compositions of f formed by the words in . The existence of , , and C implies f has a topological attractor with a positive Lyapunov exponent. We develop an algorithm that identifies these objects for two-dimensional homeomorphisms comprised of two affine pieces. The main effort is in the explicit construction of and C. Their existence is equated to a set of computable conditions in a general way. This results in a computer-assisted proof of chaos throughout a relatively large region of parameter space. We also observe how the failure of C to be expanding can coincide with a bifurcation of f. Lyapunov exponents are evaluated using one-sided directional derivatives so that forward orbits that intersect a switching manifold (where f is not differentiable) can be included in the analysis.

Topological derivatives via one-sided derivative of parametrized minima and minimax

PurposeThe object of the paper is to illustrate how to obtain the topological derivative as a semidifferential in a general and practical mathematical setting for d-dimensional perturbations of a bounded open domain in the n-dimensional Euclidean space.Design/methodology/approachThe underlying methodology uses mathematical notions and powerful tools with ready to check assumptions and ready to use formulas via theorems on the one-sided derivative of parametrized minima and minimax.FindingsThe theory and the examples indicate that the methodology applies to a wide range of problems: (1) compliance and (2) state constrained objective functions where the coupled state/adjoint state equations appear without a posteriori substitution of the adjoint state.Research limitations/implicationsDirect approach that considerably simplifies the analysis and computations.Originality/valueIt was known that the shape derivative was a differential. But the topological derivative is only a semidifferential, that is, a one-sided directional derivative, which is not linear with respect to the direction, and the directions are d-dimensional bounded measures.

Morse theory for the uniform energy

In this paper we develop a Morse theory for the uniform energy. We use the one-sided directional derivative of the distance function to study the minimizing properties of variations through closed geodesics. This derivative is then used to define a one-sided directional derivative for the uniform energy which allows us to identify gradient-like vectors at those points where the function is not differentiable. These vectors are used to restart the standard negative gradient flow of the Morse energy at its critical points. We illustrate this procedure on the flat torus and demonstrate that the restarted flow improves the minimizing properties of the associated closed geodesics.

Arc length associated with generalized distance functions

We propose a generalization of the traditional definition of arc length in a manifold. In our definition, the arc length is associated with a distance function d that satisfies the identity property but not necessarily the triangle inequality, non-negativity, definiteness and symmetry. A new class of directed arcs, which we call “ d-conservative” arcs, arises in an evident manner from our definition. These arcs satisfy a property of conservation of the d-distance along the arc. Each d-conservative arc has a d-length equal to the d-distance between its endpoints. If d satisfies the triangle inequality, the d-conservative arcs coincide with the arcs of minimum d-length. We prove that the d-length of an arc can be expressed as the integral of a function F along the arc, where F is the one-sided directional derivative of d. This last relation between d and F was proved by Busemann and Mayer (1941) [3] for the Finsler distances, which satisfy, among others, the triangle inequality and non-negativity, requirements that we do not need in our proof. We also prove that if the one-sided directional derivative F of a distance function d is continuous, then d satisfies the triangle inequality if, and only if, F is convex. We analyze an example of a non-positive definite distance function.

On the uniqueness of Groves mechanisms and the payoff equivalence principle

Consider a standard mechanism design setting with quasi-linear preferences and private valuations. From Holmström (1979), we know that if the valuations are smooth with respect to types then any efficient, dominant strategy mechanism is in the class of Groves mechanisms. Here I show that, given regular assumptions on the primitives of the design problem, a weaker condition that includes the case of non-smooth valuations is sufficient and necessary for the uniqueness of Groves mechanisms among all efficient, dominant strategy mechanisms. This condition, which imposes a restriction on the behavior of the one-sided directional derivatives of the valuation functions with respect to individual types, is also shown to be sufficient and necessary to obtain the Payoff Equivalence principle for dominant strategy mechanisms whose choice rules are affine maximizers.

Derivatives of Generalized Distance Functions and Existence of Generalized Nearest Points

The relationship between directional derivatives of generalized distance functions and the existence of generalized nearest points in Banach spaces is investigated. Let G be any nonempty closed subset in a compact locally uniformly convex Banach space. It is proved that if the one-sided directional derivative of the generalized distance function associated to G at x equals to 1 or −1, then the generalized nearest points to x from G exist. We also give a partial answer (Theorem 3.5) to the open problem put forward by S. Fitzpatrick (1989, Bull. Austral. Math. Soc.39, 233–238).

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.

One-sided directional derivative and applications

In this paper certain theorems establishing the existence of onesided directional derivative for some classes of functions are proved. These theorems are then applied to problems of best approximations.

Mean value theorem for convex functions

Closed proper convex functions have many properties in common with differentiable functions such as continuity and one-sided directional derivatives. In this paper it is shown that there exists a mean value theorem for such functions with the gradient vector in the differentiable case replaced by an element of the subdifferential in the convex case. Under compactness of the isoquant sets, (minus) the cost functions are convex and with the theorem above it is easy to show that under incomplete specialization and constant returns to scale the prices of international goods determine uniquely the prices of domestic goods if the isoquant sets are linearly independent.

Two convex counterexamples: A discontinuous envelope function and a nondifferentiable nearest-point mapping

Introduction. This paper gives counterexamples for two unrelated conjectures which pertain to convexity theory. Supposef is any function defined on a closed convex set C in finitedimensional Euclidean space. Its envelope function env f is defined as the pointwise supremum of all linear functions which f dominates everywhere. It is elementary and well known that env f is convex, and hence continuous in the relative interior of C. It is similarly known that env f is lower-semicontinuous on the boundary of C. Witsenhausen [14] uses and studies envelope functions. He suggested the following orally: CONJECTURE 1. Iff is continuous everywhere on C, then envf is also continuous everywhere on C. I provide a very simple counterexample in 3-space which shows that this is not true. Incidentally, though the proof is not on paper, I feel fairly certain that Witsenhausen's conjecture is correct in 2-space. If C is a closed convex set in finite-dimensional Euclidean space, then for any point x in the space there is a unique point P(x) in C which is nearest to x. The mapping P is called the projection onto C. (Many interesting results about this mapping are given in Phelps [9], [10].) It is easy to prove and well known that P is a contraction mapping, hence continuous. It is also easy to see that P need not have much in the way of differentiability properties, for if C is a polyhedron in 2-space, then it is easy to find points and directions at which P does not even have a two-sided directional derivative. The existence in general of a one-sided directional derivative, however, is not so easy to decide. This question does not seem to be raised in such natural places as Eggleston [4] and Valentine [11]. CONJECTURE 2. For all points and directions, P has a one-sided directional derivative. My mildly complicated counterexample is a set C (in 3-space), which is the convex hull of a countable infinity of points. I suspect that the conjecture is correct in 2-space. In n-space it would be interesting to know whether the set of points and directions at which the conjecture holds is dense in EnXSn-. If not, the convex set would have to be extremely jagged!

A Note on One-Sided Directional Derivatives

(1968). A Note on One-Sided Directional Derivatives. Mathematics Magazine: Vol. 41, No. 3, pp. 147-150.

A Note on One-Sided Directional Derivatives

A Note on One-Sided Directional Derivatives