An explicit non-expansive function whose subdifferential is the entire dual ball

Borwein, Jonathan M.; Sciffer, Scott

doi:10.1090/conm/514/10101

Cited by 3 publications

(6 citation statements)

References 14 publications

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“…for all points x ∈ X [13,10]. Similarly, one can show that nonconvex equilibrium results will frequently contain little or no non-trivial information [13].…”

Section: Achievements and Limitationsmentioning

confidence: 99%

Future Challenges for Variational Analysis

Borwein

2010

Variational Analysis and Generalized Differentiation in Optimization and Control

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135

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Modern non-smooth analysis is now roughly thirty-five years old. In this paper I shall attempt to analyse (briefly): where the subject stands today, where it should be going, and what it will take to get there? In summary, the conclusion is that the first order theory is rather impressive, as are many applications. The second order theory is by comparison somewhat underdeveloped and wanting of further advance. 1 It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully,but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.-Carl Friedrich Gauss (1777-1855). 2 1 Preliminaries and Precursors I intend to first discuss First-Order Theory, and then Higher-Order Theory-mainly second-order-and only mention passingly higher-order theory which really devolves to second-order theory. I'll finish by touching on Applications of Variational Analysis or VA both inside and outside Mathematics, mentioning both successes and limitations or failures. Each topic leads to open questions even in the convex case which I'll refer to as CA. Some issues are technical and specialized, others are

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“…for all points x ∈ X [13,10]. Similarly, one can show that nonconvex equilibrium results will frequently contain little or no non-trivial information [13].…”

Section: Achievements and Limitationsmentioning

confidence: 99%

Future Challenges for Variational Analysis

Borwein

2010

Variational Analysis and Generalized Differentiation in Optimization and Control

Self Cite

135

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“…The reason is that for smooth functions the Lipschitz norm || · || Lip coincides with the norm of uniform convergence of the derivatives and under this norm C 1 (K) is a Banach subspace of Lip(K). In this work we complement the results [7], [8], [9] by establishing a topology-independent result (Theorem 12(i)), namely, that the set of Clarke-saturated Lipschitz functions contains an infinite dimensional linear space of uncountable dimension; in particular it is lineable, according to the terminology of [16], and consequently algebraically large. Moreover, surprisingly, (Lip(K), || · || Lip ) contains a closed non-separable subspace of Clarke-saturated functions, hence this set is also spaceable.…”

Section: Introductionmentioning

confidence: 82%

“…where g k are given by (7) and {e i } i=1,...,d is the canonical basis of R d . Let us first show that the functions {G k } k∈N are "derivatives" of functions of Lip x 0 (U ).…”

Section: The Case D >mentioning

confidence: 99%

“…However, this result highly depends on the chosen metric, the reason being that wild functions with oscillating derivatives can be obtained as uniform limits of well-behaved ones (piecewise linear or quadratic). An explicit construction of such a wild function with maximal Clarke subdifferential is given in [7]. Therefore, in some generic sense, most Lipschitz functions are Clarke-saturated (see forthcoming Definition 1), but this genericity is strongly related to the chosen topology.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linear Structure of Functions with Maximal Clarke Subdifferential

Daniilidis¹,

Flores²

2019

SIAM J. Optim.

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It is hereby established that the set of Lipschitz functions f : U → R (U nonempty open subset of ℓ 1 d ) with maximal Clarke subdifferential contains a linear subspace of uncountable dimension (in particular, an isometric copy of ℓ ∞ (N)). This result goes in the line of a previous result of Borwein-Wang ([8], [9]). However, while the latter was based on Baire category theorem, our current approach is constructive and is not linked to the uniform convergence. In particular we establish lineability (and spaceability for the Lipschitz norm) of the above set inside the set of all Lipschitz continuous functions.

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