Necessary conditions of higher order for semi-infinite programming

2005 ◽  
pp. 31-50 ◽  
Author(s):  
Hans-Joachim Kornstaedt
Keyword(s):  
Necessary Conditions ◽  
Higher Order ◽  
Infinite Programming

Oscillation and asymptotic behavior of a higher-order neutral delay difference equation with variable delays under Δ m

2021 ◽  
Vol 71 (1) ◽  
pp. 129-146
Author(s):  
Chittaranjan Behera ◽  
Radhanath Rath ◽  
Prayag Prasad Mishra
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Variable Delays ◽  
All Solutions ◽  
Delay Difference

Abstract In this article we obtain sufficient conditions for the oscillation of all solutions of the higher-order delay difference equation Δ m ( y n − ∑ j = 1 k p n j y n − m j ) + v n G ( y σ ( n ) ) − u n H ( y α ( n ) ) = f n , $$\begin{array}{} \displaystyle \Delta^{m}\big(y_n-\sum_{j=1}^k p_n^j y_{n-m_j}\big) + v_nG(y_{\sigma(n)})-u_nH(y_{\alpha(n)})=f_n\,, \end{array}$$ where m is a positive integer and Δ xn = x n+1 − xn . Also we obtain necessary conditions for a particular case of the above equation. We illustrate our results with examples for which it seems no result in the literature can be applied.

ON FIRST-ORDER NECESSARY CONDITIONS FOR OPTIMALITY OF REGULAR GENERALIZED SEMI-INFINITE PROGRAMMING

2012 ◽  
Vol 05 (04) ◽  
pp. 1250054
Author(s):  
Nader Kanzi
Keyword(s):  
Constraint Qualification ◽  
Necessary Conditions ◽  
Natural Extension ◽  
Clarke Subdifferential ◽  
Optimal Solutions ◽  
First Order ◽  
Locally Lipschitz ◽  
Necessary Conditions For Optimality ◽  
Infinite Programming ◽  
The Clarke Subdifferential

This paper is concerned with the optimality for generalized semi-infinite programming (GSIP) with nondifferentiable and nonconvex (but being regular in Clarke sense) constraint functions. The objective function is only locally Lipschitz. We consider a lower level constraint qualification which is based on the Clarke subdifferential. This constraint qualification is a natural extension of Mangasarian–Fromovitz one to the differentiable GSIP. The main results are Fritz-John type necessary conditions for optimal solutions.

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