Variational Analysis

Rockafellar, R. T.; Wets, Roger J.-B.

doi:10.1007/978-3-642-02431-3

Cited by 5,244 publications

(7,400 citation statements)

References 192 publications

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“…It can be shown that if h is locally Lipschitz aroundx then this definition coincides with the definition given in Section 6, so there is no danger of confusion, see [20,Theorem 9.13 (b) and Theorem 8.49].…”

mentioning

confidence: 85%

“…If h is infinite at x then the set∂h(x) is defined to be empty. It is not difficult to show that it is always a closed and convex set, see [20].…”

Section: Continuity Of the Hessian Theorem The Function G := F • β Imentioning

confidence: 99%

“…There is no danger of confusion because the subdifferentials coincide when h is a proper, convex function, see [20,Proposition 8.12]. Part (ii).…”

mentioning

confidence: 99%

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Nonsmooth Analysis of Lorentz Invariant Functions

Sendov¹

2007

SIAM J. Optim.

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Abstract. A real valued function g(x, t) on R n ×R is called Lorentz invariant if g(x, t) = g(U x, t) for all n×n orthogonal matrices U and all (x, t) in the domain of g. In other words, g is invariant under the linear orthogonal transformations preserving the Lorentz cone:It is easy to see that every Lorentz invariant function can be decomposed as g = f • β, where f : R 2 → R is a symmetric function and β is the root map of the hyperbolic polynomial p(x, t) = t 2 −x 2 1 −· · ·−x 2 n . We investigate variety of important variational and non-smooth properties of g and characterize them in terms of the symmetric function f .

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mentioning

confidence: 85%

“…If h is infinite at x then the set∂h(x) is defined to be empty. It is not difficult to show that it is always a closed and convex set, see [20].…”

Section: Continuity Of the Hessian Theorem The Function G := F • β Imentioning

confidence: 99%

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Nonsmooth Analysis of Lorentz Invariant Functions

Sendov¹

2007

SIAM J. Optim.

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“…The tools of convex analysis needed in this paper can be found, for instance, in the monographs [2,13]. Moreover, [9] (mentioned already above) starts with a compact, but very informative survey of convex analysis applied to elasticity.…”

Section: σ(T) A(t)) ∈ ∂ψ( (T) V (T)) V (T) ∈ ∂φ * (−A(t)) (12)mentioning

confidence: 99%

On the wellposedness of constitutive laws involving dissipation potentials

Desch

Grimmer

2001

Trans. Amer. Math. Soc.

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Abstract. We consider a material with memory whose constitutive law is formulated in terms of internal state variables using convex potentials for the free energy and the dissipation. Given the stress at a material point depending on time, existence of a strain and a set of inner variables satisfying the constitutive law is proved. We require strong coercivity assumptions on the potentials, but none of the potentials need be quadratic.As a technical tool we generalize the notion of an Orlicz space to a cone "normed" by a convex functional which is not necessarily balanced. Duality and reflexivity in such cones are investigated.

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“…In case of real-valued functions, the Clarke subdifferential is known to contain the so-called Fréchet (or regular [17]) subdifferential ∂f (x 0 ) := p ∈ R n : lim inf…”

Section: Introductionmentioning

confidence: 99%

Generalized Hessians of $C^{1,1}$-Functions and Second-Order Viscosity Subjets

Barbet¹,

Daniilidis²,

Soravia³

2010

SIAM J. Optim.

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Abstract. Given a C 1,1 -function f : U → R (where U ⊂ R n open) we deal with the question of whether or not at a given point x 0 ∈ U there exists a local minorant ϕ of f of class C 2 that satisfies ϕ(. This question is motivated by the second-order viscosity theory of the PDE, since for nonsmooth functions, an analogous result between subgradients and first-order viscosity subjets is known to hold in every separable Asplund space. In this work we show that the aforementioned second-order result holds true whenever Hf (x 0 ) has a minimum with respect to the semidefinite cone (thus in particular, in one dimension), but it fails in two dimensions even for piecewise polynomial functions. We extend this result by introducing a new notion of directional minimum of Hf (x 0 ).

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Variational Analysis

Cited by 5,244 publications

References 192 publications

Nonsmooth Analysis of Lorentz Invariant Functions

Nonsmooth Analysis of Lorentz Invariant Functions

On the wellposedness of constitutive laws involving dissipation potentials

Generalized Hessians of $C^{1,1}$-Functions and Second-Order Viscosity Subjets

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