Calculus Rules on the Approximate Second-Order Directional Derivative of a Convex Function

1984 ◽  
Vol 22 (3) ◽  
pp. 381-404 ◽  
Author(s):  
J.-B. Hiriart-Urruty
Keyword(s):  
Convex Function ◽  
Directional Derivative ◽  
Second Order ◽  
Calculus Rules ◽  
Second Order Directional Derivative

scholarly journals Generalised hessian, max function and weak convexity

1996 ◽  
Vol 53 (1) ◽  
pp. 21-32 ◽  
Author(s):  
X. Q. Yang
Keyword(s):  
Hilbert Space ◽  
Convex Function ◽  
Directional Derivative ◽  
Second Order ◽  
Weakly Convex ◽  
Second Order Directional Derivative

In this paper, a second-order characterisation of η-convex C1, 1 functions is derived in a Hilbert space using a generalised second-order directional derivative. Using this result it is then shown that every C1, 1 function is locally weakly convex, that is, every C1, 1 real-valued function f can be represented as f (x) = h (x) − η‖x‖2 on a neighbourhood of x where h is a convex function and η > 0. Moreover, a characterisation of the generalised second-order directional derivative for max functions is given.

scholarly journals Second-Order Optimality Conditions: An extension to Hadamard Manifolds

2020 ◽  
Author(s):  
Gabriel Ruiz-Garzón ◽  
Jaime Ruiz-Zapatero ◽  
Rafaela Osuna-Gómez ◽  
Antonio Rufián-Lizana
Keyword(s):  
Optimality Conditions ◽  
Optimization Problems ◽  
Directional Derivative ◽  
Second Order ◽  
Stationary Points ◽  
Sufficient Optimality Conditions ◽  
General Notion ◽  
Hadamard Manifolds ◽  
Scalar Optimization ◽  
Second Order Directional Derivative

This work is intended to lead a study of necessary and sufficient optimality conditions for scalar optimization problems on Hadamard manifolds. In the context of this geometry, we obtain and present new function types characterized by the property of having all their second-order stationary points to be global minimums. In order to do so, we extend the concept convexity in Euclidean space to a more general notion of invexity on Hadamard manifolds. This is done employing notions of second-order directional derivative, second-order pseudoinvexity functions and the second-order Karush-Kuhn-Tucker-pseudoinvexity problem. Thus, we prove that every second-order stationary point is a global minimum if and only if the problem is either second-order pseudoinvex or second-order KKT-pseudoinvex depending on whether the problem regards unconstrained or constrained scalar optimization respectively. This result has not been presented in the literature before. Finally, examples of these new characterizations are provided in the context of \textit{"Higgs Boson like"} potentials among others.

scholarly journals Smoothing approximations to nonsmooth optimization problems

1995 ◽  
Vol 36 (3) ◽  
pp. 274-285 ◽  
Author(s):  
X.Q. Yang
Keyword(s):  
Optimization Problems ◽  
Directional Derivative ◽  
Optimal Solution ◽  
Necessary Optimality Conditions ◽  
Second Order ◽  
Smooth Approximation ◽  
Minimization Problems ◽  
Second Order Directional Derivative ◽  
Nonsmooth Minimization ◽  
Original Objective

AbstractWe study certain types of composite nonsmooth minimization problems by introducing a general smooth approximation method. Under various conditions we derive bounds on error estimates of the functional values of original objective function at an approximate optimal solution and at the optimal solution. Finally, we obtain second-order necessary optimality conditions for the smooth approximation prob lems using a recently introduced generalized second-order directional derivative.

Infrared Small-target Detection Using The Maximum of Second-order Directional Derivative

2016 ◽  
Vol 37 (9) ◽  
pp. 1142-1151
Author(s):  
赵爱罡 ZHAO Ai-gang ◽  
王宏力 WANG Hong-li ◽  
杨小冈 YANG Xiao-gang ◽  
陆敬辉 LU Jing-hui ◽  
姜伟 JIANG Wei ◽  
...  
Keyword(s):  
Target Detection ◽  
Directional Derivative ◽  
Second Order ◽  
Small Target ◽  
Small Target Detection ◽  
Second Order Directional Derivative ◽  
Infrared Small Target
Sign in / Sign up

Export Citation Format

Share Document