This paper deals with second-order optimality conditions for a quasilinear elliptic control problem nonlinear coefficient in the principal part that is countably $PC^2$ (continuous and $C^2$ apart from many points). We prove control-to-state operator continuously differentiable even though non-smooth. enables us to establish "no-gap" necessary sufficient terms of an abstract curvature functiona...

This paper focuses on second-order necessary optimality conditions for constrained optimization problems Banach spaces. For in the classical setting, where objective function is $$C^2$$ -smooth, we show that strengthened are valid if constraint set generalized polyhedral convex. a new just assumed to be $$C^1$$ -smooth and convex, establish sharp based Fréchet subdifferential of tangent set. Th...

We prove that first-order flexes of smooth surfaces in Euclidean 3-space, which are tangent to the set all nonrigid surfaces, can be extended second-order flexes.

In this paper, we study stochastic optimization of two-level composition functions without Lipschitz continuous gradient. The smoothness property is generalized by the notion relative which provokes Bregman gradient method. We propose three algorithms for possible relatively smooth compositional scenarios and provide their sample complexities to achieve an $$\epsilon $$ -approximate stationary ...

We studied smooth-pursuit eye movements elicited by first- and second-order motion stimuli. Stimuli were random dot fields whose contrast was modulated by a Gaussian window with a space constant of 0.5 degrees. For the first-order stimuli, the random dots simply moved across the screen at the same speed as the window; for the second-order stimuli the window moved across stationary or randomly f...

Series representations are obtained for the entire class of measurable, second order stochastic processes defined on any interval of the real line. They include as particular cases all earlier representations; they suggest a notion of "smoothness" that generalizes well known continuity notions; and they decompose the stochastic process into two orthogonal parts, the smooth part and a strongly d...