Lower Semicontinuous Function Research Articles

Local properties and augmented Lagrangians in fully nonconvex composite optimization

A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper investigates the versatile template of composite optimization without any convexity assumptions. First- and second-order optimality conditions are discussed. We highlight the difficulties that stem from the lack of convexity when dealing with necessary conditions in a Lagrangian framework and when considering error bounds. Building upon these characterizations, a local convergence analysis is delineated for a recently developed augmented Lagrangian method, deriving rates of convergence in the fully nonconvex setting.

Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces

This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on R n \mathbb {R}^n . While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on R n \mathbb {R}^n .

The Strong Ekeland Variational Principle in Quasi-Pseudometric Spaces

Roughly speaking, Ekeland’s Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324–353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. 131 (1988), no. 1, 1–21) and Tomonari Suzuki (J. Math. Anal. Appl. 320 (2006), no. 2, 787–794 and Nonlinear Anal. 72 (2010), no. 5, 2204–2209)) proved a Strong Ekeland Variational Principle, meaning the existence of strong minima for such perturbations. Please note that Suzuki also considered the case of functions defined on Banach spaces, emphasizing the key role played by reflexivity. In recent years, an increasing interest was manifested by many researchers to extend EkVP to the asymmetric case, i.e., to quasi-metric spaces (see references). Applications to optimization, behavioral sciences, and others were obtained. The aim of the present paper is to extend the strong Ekeland principle, both Georgiev’s and Suzuki’s versions, to the quasi-pseudometric case. At the end, we ask for the possibility of extending it to asymmetric normed spaces (i.e., the extension of Suzuki’s results).

Tracial oscillation zero and stable rank one

Abstract Let A be a separable (not necessarily unital) simple $C^*$ -algebra with strict comparison. We show that if A has tracial approximate oscillation zero, then A has stable rank one and the canonical map $\Gamma $ from the Cuntz semigroup of A to the corresponding lower-semicontinuous affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple $C^*$ -algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map $\Gamma $ is surjective.

New minimax theorems for lower semicontinuous functions and applications

In this paper we prove a version of the Fountain Theorem for a class of nonsmooth functionals that are sum of a $C^1$ functional and a convex lower semicontinuous functional, and also a version of a theorem due to Heinz for this class of functionals. The new abstract theorems will be used to prove the existence of infinitely many solutions for some elliptic problems whose the associated energy functional is of the above mentioned type. We study problems with logarithmic nonlinearity and a problem involving the 1-Laplacian operator.

Finite approximation properties of C*-modules III

Abstract We introduce and study a notion of module nuclear dimension for a C * \mathrm{C}^{*} -algebra A which is a C * \mathrm{C}^{*} -module over another C * \mathrm{C}^{*} -algebra 𝔄 {\mathfrak{A}} with compatible actions. We show that the module nuclear dimension of A is zero if A is 𝔄 {\mathfrak{A}} -NF. The converse is shown to hold when 𝔄 {\mathfrak{A}} is a C ⁢ ( X ) {C(X)} -algebra with simple fibers, with X compact and totally disconnected. We also introduce a notion of module decomposition rank, and show that when 𝔄 {\mathfrak{A}} is unital and simple, if the module decomposition rank of A is finite then A is 𝔄 {\mathfrak{A}} -QD. We study the set 𝒯 𝔄 ⁢ ( A ) {\mathcal{T}_{\mathfrak{A}}(A)} of 𝔄 {\mathfrak{A}} -valued module traces on A and relate the Cuntz semigroup of A with lower semicontinuous affine functions on the set 𝒯 𝔄 ⁢ ( A ) {\mathcal{T}_{\mathfrak{A}}(A)} . Along the way, we also prove a module Choi–Effros lifting theorem. We give estimates of the module nuclear dimension for a class of examples.

Forward-backward splitting algorithm with self-adaptive method for finite family of split minimization and fixed point problems in Hilbert spaces

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique

In this paper we would like to address the classical optimization problem of minimizing a proper, convex and lower semicontinuous function via the second order in time dynamics, combining viscous and Hessian-driven damping with a Tikhonov regularization term. In our analysis we heavily exploit the Moreau envelope of the objective function and its properties as well as Tikhonov regularization properties, which we extend to a nonsmooth case. We introduce the setting, which at the same time guarantees the fast convergence of the function (and Moreau envelope) values and strong convergence of the trajectories of the system to a minimal norm solution—the element of the minimal norm of all the minimizers of the objective. Moreover, we deduce the precise rates of convergence of the values for the particular choice of parameters. Various numerical examples are also included as an illustration of the theoretical results.

Approximating properties of metric and generalized metric projections in uniformly convex and uniformly smooth Banach spaces

This note conducts a comparative study of some approximating properties of the metric projection, generalized projection, and generalized metric projection in uniformly convex and uniformly smooth Banach spaces. We prove that the inverse images of the metric projections are closed and convex cones, but they are not necessarily convex. In contrast, inverse images of the generalized projection are closed and convex cones. Furthermore, the inverse images of the generalized metric projection are neither a convex set nor a cone. We also prove that the distance from a point to its projection on a convex set is a weakly lower semicontinuous function for all three notions of projections. We provide illustrating examples to highlight the different behavior of the three projections in Banach spaces.

Proximal gradient methods beyond monotony

We address composite optimization problems, which consist in minimizing the sum of a smooth and a merely lower semicontinuous function, without any convexity assumptions. Numerical solutions of these problems can be obtained by proximal gradient methods, which often rely on a line search procedure as globalization mechanism. We consider an adaptive nonmonotone proximal gradient scheme based on an averaged merit function and establish asymptotic convergence guarantees under weak assumptions, delivering results on par with the monotone strategy. Global worst-case rates for the iterates and a stationarity measure are also derived. Finally, a numerical example indicates the potential of nonmonotonicity and spectral approximations.

Orbital b-metric spaces and related fixed point results on advanced Nashine–Wardowski–Feng–Liu type contractions with applications

In this article, we prove some novel fixed-point results for a pair of multivalued dominated mappings obeying a new generalized Nashine–Wardowski–Feng–Liu-type contraction for orbitally lower semi-continuous functions in a complete orbital b-metric space. Furthermore, some new fixed-point theorems for dominated multivalued mappings are established in the scenario of ordered complete orbital b-metric spaces. Some examples are offered to demonstrate the validity of our new results’ premise. To demonstrate the applicability of our findings, applications for a system of nonlinear Volterra-type integral equations and fractional differential equations are shown. These results extend the theoretical results of Nashine et al. (Nonlinear Anal., Model. Control 26(3):522–533, 2021).

On Semi-Continuous and Clisquish Functions in Generalized Topological Spaces

In this paper, we will focus on three types of functions in a generalized topological space, namely; lower and upper semi-continuous functions, and cliquish functions. We give some results for nowhere dense sets and for second category sets. Further, we discuss the nature of cliquish functions in generalized metric spaces and provide the characterization theorem for cliquish functions in terms of nowhere dense sets.

An abstract convergence framework with application to inertial inexact forward–backward methods

In this paper we introduce a novel abstract descent scheme suited for the minimization of proper and lower semicontinuous functions. The proposed abstract scheme generalizes a set of properties that are crucial for the convergence of several first-order methods designed for nonsmooth nonconvex optimization problems. Such properties guarantee the convergence of the full sequence of iterates to a stationary point, if the objective function satisfies the Kurdyka-Lojasiewicz property. The abstract framework allows for the design of new algorithms. We propose two inertial-type algorithms with implementable inexactness criteria for the main iteration update step. The first algorithm, i$^2$Piano, exploits large steps by adjusting a local Lipschitz constant. The second algorithm, iPila, overcomes the main drawback of line-search based methods by enforcing a descent only on a merit function instead of the objective function. Both algorithms have the potential to escape local minimizers (or stationary points) by leveraging the inertial feature. Moreover, they are proved to enjoy the full convergence guarantees of the abstract descent scheme, which is the best we can expect in such a general nonsmooth nonconvex optimization setup using first-order methods. The efficiency of the proposed algorithms is demonstrated on two exemplary image deblurring problems, where we can appreciate the benefits of performing a linesearch along the descent direction inside an inertial scheme.

Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

A modified proximal point algorithm in geodesic metric space

<abstract><p>Proximal point algorithm is one of the most popular technique to find either zero of monotone operator or minimizer of a lower semi-continuous function. In this paper, we propose a new modified proximal point algorithm for solving minimization problems and common fixed point problems in CAT(0) spaces. We prove $ \Delta $ and strong convergence of the proposed algorithm. Our results extend and improve the corresponding recent results in the literature.</p></abstract>

Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions

Given a set of functions defined on a set X, á function is called abstract -convex if it is the upper envelope of its -minorants, i.e. such its minorants which belong to the set ; and f is called regularly abstract -convex if it is the upper envelope of its maximal (with respect to the pointwise ordering) -minorants. In the paper we first present the basic notions of (regular) -convexity for the case when is an abstract set of functions. For this abstract case a general sufficient condition based on Zorn's lemma for a -convex function to be regularly -convex is formulated. The goal of the paper is to study the particular class of regularly -convex functions, when is the set of real-valued Lipschitz continuous classically concave functions defined on a real normed space X. For an extended-real-valued function to be -convex it is necessary and sufficient that f be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each -convex function is regularly -convex as well. We focus on -subdifferentiability of functions at a given point. We prove that the set of points at which an -convex function is -subdifferentiable is dense in its effective domain. This result extends the well-known classical Brøndsted-Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using the subset of the set consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of -subgradient and -subdifferential of a function at a point which generalize the corresponding notions of the classical convex analysis. Some properties and simple calculus rules for -subdifferentials as well as -subdifferential conditions for global extremum points are established. Symmetric notions of abstract -concavity and -superdifferentiability of functions where is the set of Lipschitz continuous convex functions are also considered.

Some nonlinear characterizations of reflexive Banach spaces

It is well known that in the calculus of variations and in optimization there exist many formulations of the fundamental propositions on the attainment of the infima of sequentially weakly lower semicontinuous coercive functions on reflexive Banach spaces. By either some constructive skills or the regularization skill by inf–convolutions we show in this paper that all these formulations together with their important variants are equivalent to each other and equivalent to the reflexivity of the underlying space. Motivated by this research, we also give a characterization for a normed space to be finite dimensional: a normed space is finite dimensional iff every continuous real–valued function defined on each bounded closed subset of this space can attain its minimum, namely the converse of the classical Weierstrass theorem also holds true.

Convergence Properties of Monotone and Nonmonotone Proximal Gradient Methods Revisited

Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the objective function is of simple enough structure. The available convergence theory associated with these methods (mostly) requires the derivative of the smooth part of the objective function to be (globally) Lipschitz continuous, and this might be a restrictive assumption in some practically relevant scenarios. In this paper, we readdress this classical topic and provide convergence results for the classical (monotone) proximal gradient method and one of its nonmonotone extensions which are applicable in the absence of (strong) Lipschitz assumptions. This is possible since, for the price of forgoing convergence rates, we omit the use of descent-type lemmas in our analysis.

Tightness of supercritical Liouville first passage percolation

Liouville first passage percolation (LFPP) with parameter $\xi >0$ is the family of random distance functions ${D\_h^\epsilon}{\epsilon >0}$ on the plane obtained by integrating $e^{\xi h\epsilon}$ along paths, where $h\_\epsilon$ for $\epsilon >0$ is a smooth mollification of the planar Gaussian free field. Previous work by Ding–Dubédat–Dunlap–Falconet and Gwynne–Miller has shown that there is a critical value $\xi\_{\mathrm{crit}} > 0$ such that for $\xi < \xi\_{\mathrm{crit}}$, LFPP converges under appropriate re-scaling to a random metric on the plane which induces the same topology as the Euclidean metric (the so-called $\gamma$-Liouville quantum gravity metric for $\gamma = \gamma(\xi)\in (0,2)$). We show that for all $\xi > 0$, the LFPP metrics are tight with respect to the topology on lower semicontinuous functions. For $\xi > \xi\_{\mathrm{crit}}$, every possible subsequential limit $D\_h$ is a metric on the plane which does not induce the Euclidean topology: rather, there is an uncountable, dense, Lebesgue measure-zero set of points $z\in\mathbb C$ such that $D\_h(z,w) = \infty$ for every $w\in\mathbb C\setminus {z}$. We expect that these subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in $(1,25)$.

Subdifferential properties of a perturbed minimal time function in normed spaces

In a normed space, we study the perturbed minimal time function determined by a bounded closed convex set and a proper lower semicontinuous function. It covers the perturbed distance function as a special case. In particular, we show that the ε-subdifferential and the limiting subdifferential of the perturbed minimal time function are representable by virtue of corresponding subdifferential of the associated function and the support function of the constraint set.