Lower Semicontinuous Function Research Articles

Choose Your Path Wisely: Gradient Descent in a Bregman Distance Framework

We propose an extension of a special form of gradient descent---in the literature known as linearized Bregman iteration---to a larger class of nonconvex functions. We replace the classical (squared) two norm metric in the gradient descent setting with a generalized Bregman distance, based on a proper, convex, and lower semicontinuous function. The algorithm's global convergence is proven for functions that satisfy the Kurdyka--Łojasiewicz property. Examples illustrate that features of different scale are being introduced throughout the iteration, transitioning from coarse to fine. This coarse-to-fine approach with respect to scale allows us to recover solutions of nonconvex optimization problems that are superior to those obtained with conventional gradient descent, or even projected and proximal gradient descent. The effectiveness of the linearized Bregman iteration in combination with early stopping is illustrated for the applications of parallel magnetic resonance imaging, blind deconvolution, as well as image classification with neural networks.

Law-Invariant Functionals on General Spaces of Random Variables

We establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and law-invariant functionals. Our results extend well-known results from the literature to a large clas...

Lower semi-continuity for 𝒜-quasiconvex functionals under convex restrictions

We show weak lower semi-continuity of functionals assuming the new notion of a “convexly constrained” 𝒜-quasiconvex integrand. We assume 𝒜-quasiconvexity only for functions defined on a set K which is convex. Assuming this and sufficient integrability of the sequence we show that the functional is still (sequentially) weakly lower semi-continuous along weakly convergent “convexly constrained” 𝒜-free sequences. In a motivating example, the integrand is − det1/d−1 and the convex constraint is positive semi-definiteness of a matrix field.

Lp-Variational solutions of multivalued backward stochastic differential equations

We prove the existence and uniqueness of the Lp-variational solution, with p > 1, of the following multivalued backward stochastic differential equation with p-integrable data: {−dYt + ∂yΨ(t,Yt)dQt∋H(t,Yt,Zt)dQt−ZtdBt,0≤t<τ, Yτ = η, where τ is a stopping time, Q is a progressively measurable increasing continuous stochastic process and ∂yΨ is the subdifferential of the convex lower semicontinuous function y↦Ψ(t, y). In the framework of [14] (the case p ≥ 2), the strong solution found it there is the unique variational solution, via the uniqueness property proved in the present article.

A proximal point algorithm for finding minimizers and fixed points of quasi-pseudo-contractive mappings in CAT(0) spaces

The purpose of this article is twofold. One is to establish a proximal point algorithm for finding a minimizer of a proper convex and lower semi-continuous function and fixed points of quasi-pseudo-contractive mappings in CAT(0) spaces. The other is to point out and correct a basic and conceptual error in a paper of Ugwunnadi et al. [Theorem 3.1, J. Fixed Point Theory Appl. (2018) 20: 82].

Segre invariant and a stratification of the moduli space of coherent systems

The aim of this paper is to generalize the [Formula: see text]-Segre invariant for vector bundles to coherent systems. Let [Formula: see text] be a non-singular irreducible complex projective curve of genus [Formula: see text] and [Formula: see text] be the moduli space of [Formula: see text]-stable coherent systems of type [Formula: see text] on [Formula: see text]. For any pair of integers [Formula: see text] with [Formula: see text], [Formula: see text] we define the [Formula: see text]-Segre invariant, and prove that it defines a lower semicontinuous function on the families of coherent systems. Thus, the [Formula: see text]-Segre invariant induces a stratification of the moduli space [Formula: see text] into locally closed subvarieties [Formula: see text] according to the value [Formula: see text] of the function. We determine an above bound for the [Formula: see text]-Segre invariant and compute a bound for the dimension of the different strata [Formula: see text]. Moreover, we give some conditions under which the different strata are nonempty. To prove the above results, we introduce the notion of coherent systems of subtype [Formula: see text].

Ill-Posed Nonlinear Optimization Problems and Uniform Accuracy Estimates of Regularization Methods

We investigate the ill-posed problem of minimizing weakly lower semicontinuous functionals on a convex closed set in a Hilbert space. The functionals to be minimized are available with errors. We prove that a necessary condition for the existence of a regularization procedure with a uniform accuracy estimate on the class of weakly lower semicontinuous functionals is the well-posedness of related optimization problems. We also study regularization methods that provide uniform accuracy estimates, linear with respect to the error level in cost functionals. Under appropriate assumptions on the feasible set, we establish a necessary condition for the existence of such regularizing procedures. In the case of unconstrained problems, the obtained condition reduces to the local strong convexity of functionals under minimization.

Continuous Newton-like Inertial Dynamics for Monotone Inclusions

In a Hilbert framework ℌ, we study the convergence properties of a Newton-like inertial dynamical system governed by a general maximally monotone operator A ℌ: → 2ℌ. When A is equal to the subdifferential of a convex lower semicontinuous proper function, the dynamic corresponds to the introduction of the Hessian-driven damping in the continuous version of the accelerated gradient method of Nesterov. As a result, the oscillations are significantly attenuated. According to the technique introduced by Attouch-Peypouquet (Math. Prog. 2019), the maximally monotone operator is replaced by its Yosida approximation with an appropriate adjustment of the regularization parameter. The introduction into the dynamic of the Newton-like correction term (corresponding to the Hessian driven term in the case of convex minimization) provides a well-posed evolution system for which we will obtain the weak convergence of the generated trajectories towards the zeroes of A. We also obtain the fast convergence of the velocities towards zero. The results tolerate the presence of errors, perturbations. Then, we specialize our results to the case where the operator A is the subdifferential of a convex lower semicontinuous function, and obtain fast optimization results.

Finite-time stability of discrete autonomous systems

Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time. In this paper, we address finite time stability of discrete-time dynamical systems. Specifically, we show that finite time stability leads to uniqueness of solutions in forward time. Furthermore, we provide Lyapunov and converse Lyapunov theorems for finite-time stability of discrete autonomous systems involving scalar difference fractional inequalities and minimum operators. In addition, lower semicontinuity of the settling-time function capturing the finite settling time behavior of the dynamical system is studied and illustrated through several examples. In particular, it is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov theorems for finite time stability of discrete-time systems can only assure the existence of lower semicontinuous Lyapunov functions.

Determination of convex functions via subgradients of minimal norm

We show, in Hilbert space setting, that any two convex proper lower semicontinuous functions bounded from below, for which the norm of their minimal subgradients coincide, they coincide up to a constant. Moreover, under classic boundary conditions, we provide the same results when the functions are continuous and defined over an open convex domain. These results show that for convex functions bounded from below, the slopes provide sufficient first-order information to determine the function up to a constant, giving a positive answer to the conjecture posed in Boulmezaoud et al. (SIAM J Optim 28(3):2049–2066, 2018) .

On the continuous points of semi-continuous functions

Let (X, d) be a complete metric space and E be a non-empty closed set in X. Suppose that f is a lower semi-continuous functions which maps E into $${\mathbb {R}}$$ . In this paper, we show that f has continuous points on E. Furthermore, we get the set of all such continuous points denote by $$C_f(E)$$ is dense on E. It should be pointed out that the same problem was also considered by professor Zaslavski (PanAmer Math J 17:1–10, 2007) by using a totally different argument.

Abstract Convexity of Functions with Respectto the Set of Lipschitz (Concave) Functions

The paper is devoted to the abstract $${\mathcal{H}}$$ -convexity of functions (where $${\mathcal{H}}$$ is a given set of elementary functions) and its realization in the cases when $${\mathcal{H}}$$ is the space of Lipschitz functions or the set of Lipschitz concave functions. The notion of regular $${\mathcal{H}}$$ -convex functions is introduced. These are functions representable as the upper envelopes of the set of their maximal (with respect to the pointwise order) $${\mathcal{H}}$$ -minorants. As a generalization of the global subdifferential of a convex function, we introduce the set of maximal support $${\mathcal{H}}$$ -minorants at a point and the set of lower $${\mathcal{H}}$$ -support points. Using these tools, we formulate both a necessary condition and a sufficient one for global minima of nonsmooth functions. In the second part of the paper, the abstract notions of $${\mathcal{H}}$$ -convexity are realized in the specific cases when functions are defined on a metric or normed space $$X$$ and the set of elementary functions is the space $${\mathcal{L}}(X,{\mathbb{R}})$$ of Lipschitz functions or the set $${\mathcal{L}}\widehat{C}(X,{\mathbb{R}})$$ of Lipschitz concave functions, respectively. An important result of this part of the paper is the proof of the fact that, for a lower semicontinuous function lower bounded by a Lipschitz function, the set of lower $${\mathcal{L}}$$ -support points and the set of lower $${\mathcal{L}}\widehat{C}$$ -support points coincide and are dense in the effective domain of the function. These results extend the known Brondsted–Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the wider class of lower semicontinuous functions and go back to the Bishop–Phelps theorem on the density of support points in the boundary of a closed convex set, which is one of the most important results of classical convex analysis.

Reprint of: On countably selective spaces

A space X is strongly Y-selective (resp., Y-selective) if every lower semicontinuous mapping from Y to the nonempty subsets (resp., nonempty closed subsets) of X has a continuous selection. We also call X (strongly) C-selective if it is (strongly) Y-selective for any countable and regular space Y. E. Michael showed that every first countable space is strongly C-selective. We extend this by showing that every W-space in the sense of the second author is strongly C-selective. We also show that every GO-space is C-selective, and that every (ω+1)-selective space has Arhangel'skii's property α1. We obtain an example under p=c of a strongly (ω+1)-selective space that is not C-selective, and we show that it is consistent with and independent of ZFC that a space is strongly (ω+1)-selective iff it is (ω+1)-selective and Fréchet. Finally, we answer a question of the third author and Junnila by showing that the ordinal space ω1+1 is not self-selective.

A Relaxation Result in the Vectorial Setting and Power Law Approximation for Supremal Functionals

We provide relaxation for not lower semicontinuous supremal functionals defined on vectorial Lipschitz functions, where the Borel level convex density depends only on the gradient. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally, we discuss the power law approximation of supremal functionals, with nonnegative, coercive densities having explicit dependence also on the spatial variable, and satisfying minimal measurability assumptions.

Dual characterizations of set containments involving uncertain polyhedral sets in Banach spaces with applications

In this paper, we first give a dual characterization for set containments defined by lower semi-continuous and sublinear functions on Banach spaces. Next, we provide dual characterizations for robust polyhedral containments where a robust counterpart of an uncertain polyhedral set is contained in another polyhedral set or a polyhedral set is contained in a robust counterpart of an uncertain polyhedral set. Finally, as an application, we derive Lagrange multiplier characterizations for robust solutions of the robust uncertain linear programming problems.

Iterative algorithm for computing fixed points of demicontractive and zeros points of multivalued accretive operators in certain Banach spaces with application

In this paper, an iterative algorithm for finding a common point of the set of common zero of an infinite family of multivalued accretive operators and the set of fixed points of a demicontractive operator is constructed and studied in certain Banach spaces having a weakly continuous duality map. Under suitable control conditions, strong convergence of the sequence generated by proposed algorithm to a common point of the two sets is established. Moreover, application to convex minimization problems involving an infinite family of lower semi-continuous and convex functions are included.The main theorems develop and complement the recent results announced by researchers in this area.

A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces

The purpose of this paper is to propose and analyze a multi-step iterative sequence to solve a convex optimization problem and a fixed point problem in an Hadamard space. We aim to establish strong and $$ \triangle $$ -convergence results of the proposed iterative sequence by employing suitable conditions on the control parameters and the structural properties of the underlying space. As a consequence, we compute an optimal solution for a minimizer of proper convex lower semicontinuous function and a common fixed point of a finite family of total asymptotically quasi-nonexpansive mappings in Hadamard spaces. Our results can be viewed as an extension and generalization of various corresponding results in the existing literature.

Pseudo metric subregularity and its stability in Asplund spaces

As a variant of metric subregularity, pseudo metric subregularity is studied via general limit critical sets using the techniques of variational analysis. In terms of limit critical sets, we provide some sufficient conditions for the validity of pseudo/Holder metric subregularity. Usually, the property of pseudo metric subregularity is not stable under small smooth perturbation. We provide a characterization for pseudo metric subregularity to be stable under small $$C^{1,p}$$ smooth perturbation. In particular, some existing results on metric subregularity are extended to pseudo metric subregularity. Finally, we consider the pseudo weak sharp minimizer of a proper lower semicontinuous function and its relation with pseudo metric subregularity of the corresponding subdifferential mapping.

Application of new strongly convergent iterative methods to split equality problems

In this paper, we study the generalized problem of split equality variational inclusion problem. For this purpose, we introduced the problem of finding the zero of a nonnegative lower semicontinuous function over the common solution set of fixed point problem and monotone inclusion problem. We proposed and studied the convergence behaviour of different iterative techniques to solve the generalized problem. Furthermore, we study an inertial form of the proposed algorithm and compare the convergence speed. Numerical experiments have been conducted to compare the convergence speed of the proposed algorithm, its inertial form and already existing algorithms to solve the generalized problem.

An extreme value theorem on closed convex subsets of Lp spaces and two applications

In this paper we prove that each (R∪{∞})-valued bounded below lower semicontinuous function defined on E reaches its infimum, where E is a closed convex subset of Lp(X,A,μ) and each element of E is dominated by some h∈Lp(X,A,μ), p≥1. As a consequence of this extreme value theorem, a version of the minimax theorem is proved in the context of dominated p-integrable functions, as above. One more consequence is that certain Euler-Lagrange equation has a weak solution in a dominated Sobolev subspace of W01,1.