scholarly journals Regularisations of convex functions and slicewise suprema

1994 ◽  
Vol 50 (3) ◽  
pp. 481-499
Author(s):  
S. Simons
Keyword(s):  
Banach Space ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Lower Envelope ◽  
Semicontinuous Function ◽  
Original Function ◽  
Scalar Multiple ◽  
Proper Convex ◽  
Sublinear Functional ◽  
Convex Lower Semicontinuous Function

For a number of years, there has been interest in the regularisation of a given proper convex lower semicontinuous function on a Banach space, defined to be the episum (=inf-convolution) of the function with a scalar multiple of the norm. There is an obvious geometric way of characterising this regularisation as the lower envelope of cones lying above the graph of the original function. In this paper, we consider the more interesting problem of characterising the regularisation in terms of approximations from below, expressing the regularisation as the upper envelope of certain subtangents to the graph of the original function. We shall show that such an approximation is sometimes (but not always) valid. Further, we shall give an extension of the whole procedure in which the scalar multiple of the norm is replaced by a more general sublinear functional. As a by-product of our analysis, we are led to the consideration of two senses stronger than the pointwise sense in which a function on a Banach space can be expressed as the upper envelope of a family of functions. These new senses of suprema lead to some questions in Banach space theorey.

scholarly journals Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 21
Author(s):  
Yasunori Kimura ◽  
Keisuke Shindo
Keyword(s):  
Asymptotic Behavior ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Convergent Sequence ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Geodesic Space ◽  
Proper Convex ◽  
Convex Lower Semicontinuous Function ◽  
Sequence Of Functions

The asymptotic behavior of resolvents of a proper convex lower semicontinuous function is studied in the various settings of spaces. In this paper, we consider the asymptotic behavior of the resolvents of a sequence of functions defined in a complete geodesic space. To obtain the result, we assume the Mosco convergence of the sets of minimizers of these functions.

Higher Connectedness Properties of Support Points and Functionals of Convex Sets

2013 ◽  
Vol 65 (6) ◽  
pp. 1236-1254
Author(s):  
Carlo Alberto De Bernardi
Keyword(s):  
Convex Sets ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Infinite Dimensional ◽  
Support Points ◽  
Bounded Set ◽  
Infinite Dimensional Banach Space ◽  
Proper Convex ◽  
Convex Lower Semicontinuous Function

AbstractWe prove that the set of all support points of a nonempty closed convex bounded set C in a real infinite-dimensional Banach space X is AR(σ-compact) and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals of C and for the domain, the graph, and the range of the subdifferential map of a proper convex lower semicontinuous function on X.

scholarly journals Smooth variational principles in Radon-Nikodým spaces

1999 ◽  
Vol 60 (1) ◽  
pp. 109-118 ◽  
Author(s):  
Robert Deville ◽  
Abdelhakim Maaden
Keyword(s):  
Banach Space ◽  
Real Function ◽  
Variational Principles ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Weakly Continuous ◽  
Fréchet Differentiable

We prove that if f is a real valued lower semicontinuous function on a Banach space X, for which there exist a > 0 and b ∈ ℝ such that f(x) ≥ 2a∥x∥ + b, x ∈ X, and if X has the Radon-Nikody´m property, then for every Ε > 0 there exists a real function φ X such that φ is Fréchet differentiable, ∥φ∥∞ < Ε, ∥φ′∥∞ < Ε, φ′ is weakly continuous and f + φ attains a minimum on X. In addition, if we assume that the norm in X is β-smooth, we can take the function φ = g1 + g2 where g1 is radial and β-smooth, g2 is Fréchet differentiable, ∥g1∥∞ < Ε, ∥g2∥∞ < Ε, ∥g′1∥∞ < Ε, ∥g′1∥∞ < Ε, g′2 is weakly continuous and f + g1 + g2 attains a minimum on X.

A Note on Minimal Usco Maps

1991 ◽  
Vol 34 (3) ◽  
pp. 412-416 ◽  
Author(s):  
Andrei Verona ◽  
Maria Elena Verona
Keyword(s):  
Banach Space ◽  
Short Proof ◽  
Lower Semicontinuous ◽  
Baire Space ◽  
Asplund Space ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Strictly Convex ◽  
Minimal Usco Map ◽  
Weak Asplund Space

AbstractWe prove that the composition of a minimal usco map, defined on a Baire space, with a lower semicontinuous function is single valued and usco at each point of a dense G$ subset of its domain. This extends earlier results of Kenderov and Fitzpatrick. As a first consequence, we prove that a Banach space, with the property that there exists a strictly convex, weak* lower semicontinuous function on its dual, is a weak Asplund space. As a second consequence, we present a short proof of the fact that a Banach space with separable dual is an Asplund space.

scholarly journals Stronger maximal monotonicity properties of linear operators

1999 ◽  
Vol 60 (1) ◽  
pp. 163-174 ◽  
Author(s):  
H.H. Bauschke ◽  
S. Simons
Keyword(s):  
Banach Space ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Real Banach Space ◽  
Lower Semicontinuous ◽  
Maximal Monotone ◽  
Linear Mapping ◽  
Linear Operators ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function

The subdifferential mapping associated with a proper, convex lower semicontinuous function on a real Banach space is always a special kind of maximal monotone operator. Specifically, it is always “strongly maximal monotone” and of “type (ANA)”. In an attempt to find maximal monotone operators that do not satisfy these properties, we investigate (possibly discontinuous) maximal monotone linear operators from a subspace of a (possibly nonreflexive) real Banach space into its dual. Such a linear mapping is always “strongly maximal monotone”, but we are only able to prove that is of “type (ANA)” when it is continuous or surjective — the situation in general is unclear. In fact, every surjective linear maximal monotone operator is of “type (NA)”, a more restrictive condition than “type (ANA)”, while the zero operator, which is both continuous and linear and also a subdifferential, is never of “type (NA)” if the underlying space is not reflexive. We examine some examples based on the properties of derivatives.

scholarly journals Subdifferentials are locally maximal monotone

1993 ◽  
Vol 47 (3) ◽  
pp. 465-471 ◽  
Author(s):  
S. Simons
Keyword(s):  
Lower Semicontinuous ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Affirmative Answer ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
New Class ◽  
Locally Maximal ◽  
Convex Lower Semicontinuous Function

In a recent paper, Fitzpatrick and Phelps introduced a new class of operators on a Banach space, the locally maximal monotone operators, and showed that this kind of operator can be approximated by a sequence of nicer maximal monotone operators. We give here an affirmative answer to a question posed in this paper: is the subdifferential of a proper convex lower semicontinuous function necessarily locally maximal monotone? Since a locally maximal operator is maximal monotone, our result represents a strengthening of Rockafellar's maximal monotonicity theorem.

Minimizing non-convex multiple integrals: a density result

2000 ◽  
Vol 130 (4) ◽  
pp. 721-741 ◽  
Author(s):  
P. Celada ◽  
S. Perrotta
Keyword(s):  
Lower Semicontinuous ◽  
Convergent Sequence ◽  
Variational Problems ◽  
Continuous Functions ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Open Set ◽  
Multiple Integrals ◽  
Image Position ◽  
Convex Lower Semicontinuous Function

We consider variational problems of the form where Ω is a bounded open set in RN, f : RN → R is a possibly non-convex lower semicontinuous function with p-growth at infinity for some 1 < p < ∞, and the boundary datum u0 is any function in W1, p (Ω). Assuming that the convex envelope f** of f is affine on each connected component of the set {f** < f}, we prove the existence of solutions to ( P) for every continuous function g such that (i) g has no strict local minima and (ii) every convergent sequence of extremum points of g eventually belongs to an interval where g is constant, thus showing that the set of continuous functions g that yield existence to (P) is dense in the space of continuous functions on R.

scholarly journals $L^{p}-$Variational solutions of multivalued backward stochastic differential equations

2021 ◽  
Author(s):  
Lucian Maticiuc ◽  
Aurel Rascanu
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Existence And Uniqueness ◽  
Lower Semicontinuous ◽  
Backward Stochastic Differential Equation ◽  
Stopping Time ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Variational Solutions ◽  
Convex Lower Semicontinuous Function

We prove the existence and uniqueness of the $L^{p}-$variational solution, with $p>1,$ of the fo\-llo\-wing multivalued backward stochastic differential equation with $p-$integrable data: \[ \left\{ \begin{array}[c]{l} -dY_{t}+\partial_{y}\Psi(t,Y_{t})dQ_{t}\ni H(t,Y_{t},Z_{t})dQ_{t}-Z_{t}dB_{t},\;0\leq t<\tau,\\[0.2cm] Y_{\tau}=\eta, \end{array} \right. \] where $\tau$ is a stopping time, $Q$ is a progressively measurable increasing continuous stochastic process and $\partial_{y}\Psi$ is the subdifferential of the convex lower semicontinuous function $y\mapsto\Psi(t,y).$

A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function

2018 ◽  
Vol 24 (2) ◽  
pp. 463-477 ◽  
Author(s):  
Radu Ioan Boţ ◽  
Ernö Robert Csetnek
Keyword(s):  
Dynamical System ◽  
Smooth Function ◽  
Convergence Rates ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Proximal Point ◽  
Nonconvex Function ◽  
Proper Convex ◽  
Point Operator

We address the minimization of the sum of a proper, convex and lower semicontinuous function with a (possibly nonconvex) smooth function from the perspective of an implicit dynamical system of forward-backward type. The latter is formulated by means of the gradient of the smooth function and of the proximal point operator of the nonsmooth one. The trajectory generated by the dynamical system is proved to asymptotically converge to a critical point of the objective, provided a regularization of the latter satisfies the Kurdyka−Łojasiewicz property. Convergence rates for the trajectory in terms of the Łojasiewicz exponent of the regularized objective function are also provided.

The Proximal Subgradient Formula in Banach Space

1988 ◽  
Vol 31 (3) ◽  
pp. 353-361 ◽  
Author(s):  
Philip D. Loewen
Keyword(s):  
Banach Space ◽  
Lower Semicontinuous ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Generalized Gradient ◽  
Infinite Dimensional ◽  
Fréchet Differentiable ◽  
The One ◽  
The Given ◽  
Normal Formula

AbstractThe proximal subgradient formula is a refinement due to Rockafellar of Clarke's fundamental proximal normal formula. It expresses Clarke's generalized gradient of a lower semicontinuous function in terms of analytically simpler proximal subgradients. We use the infinite-dimensional proximal normal formula recently given by Borwein and Strojwas to derive a new version of the proximal subgradient formula in a reflexive Banach space X with Frechet differentiable and locally uniformly convex norm. Our result improves on the one given by Borwein and Strojwas by referring only to the given norm on X.

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