Nonsmooth Analysis of Lorentz Invariant Functions

Abstract

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 .

Generalized subdifferentials of spectral functions over Euclidean Jordan algebras

“…Spectral functions over the algebra associated to the second order cone were initially studied by Fukushima, Luo and Tseng [11] and by Chen, Chen and Tseng [7]. In [7], there is a discussion of the Clarke subdifferential of locally Lipschitz spectral functions and Sendov [26] gave formulae for regular, approximating and horizon subdifferentials. Sendov also proved a formula for the Clarke subdifferential under the hypothesis of local lower semicontinuity.…”

“…Every Jordan algebra can be decomposed as a direct sum of simple algebras and simplicity is, in many cases, a harmless hypothesis. Previous work by Lewis [21] and Sendov [26] can be seen as containing results for specific cases of simple Euclidean Jordan algebras. However, because the generalized subdifferentials do not behave nicely with respect partial subdifferentiation, there are cases where we cannot extend results from simple to general Euclidean Jordan algebras in a straightforward way.…”

“…Let {x k } ⊆ L α,ǫ be a sequence converging to some x. Then, (26) implies that {λ(x k )} is a sequence contained in L ′ α,ǫ ′ . Since λ(•) is continuous and L ′ α,ǫ ′ is closed, we obtain that λ(x) ∈ L ′ α,ǫ ′ .…”

BFGS convergence to nonsmooth minimizers of convex functions

“…This function is C (2) convex and symmetric. The function f δ : R n → R defined by f δ (x) = g δ ( x ) is also C (2) convex, either as a consequence of [9] or via a straightforward direct calculation. The result now follows from Theorem 4.1.…”

Nonsmooth Variants of Powell's BFGS Convergence Theorem

“…This function is C (2) , convex and even. The function f δ : R n → R defined by f δ (x) = g δ ( x ) is also C (2) and convex, both as a consequence of [12] and via a straightforward direct calculation. The result now follows from Theorem 4.1.…”