Lower Semicontinuous Function Research Articles

Capra-Convexity, Convex Factorization and Variational Formulations for the ℓ0 Pseudonorm

The so-called ℓ0 pseudonorm, or cardinality function, counts the number of nonzero components of a vector. In this paper, we analyze the ℓ0 pseudonorm by means of so-called Capra (constant along primal rays) conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the ℓp-norms, but for the extreme ones). We obtain three main results. First, we show that the ℓ0 pseudonorm is equal to its Capra-biconjugate, that is, is a Capra-convex function. Second, we deduce an unexpected consequence, that we call convex factorization: the ℓ0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex lower semicontinuous function. Third, we establish a variational formulation for the ℓ0 pseudonorm by means of generalized top-k dual norms and k-support dual norms (that we formally introduce).

Variational Principles for Maximization Problems with Lower-semicontinuous Goal Functions

Let X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all bounded continuous real-valued functions in X endowed with the usual sup-norm. We show that the following two properties are equivalent: X is α-favourable (in the sense of the Banach-Mazur game);The set of functions h in C(X) for which f + h attains its supremum in X contains a dense and Gδ-subset of the space C(X).In particular, property (b) has place if X is a compact space or, more generally, if X is homeomorphic to a dense Gδ subset of a compact space.We show also the equivalence of the following stronger properties: X contains some dense completely metrizable subset;the set of functions h in C(X) for which f + h has strong maximum in X contains a dense and Gδ-subset of the space C(X).If X is a complete metric space and f is bounded, then the set of functions h from C(X) for which f + h has both strong maximum and strong minimum in X contains a dense Gδ-subset of C(X).

An extension of the proximal point algorithm beyond convexity

We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity.

Inexact and Stochastic Generalized Conditional Gradient with Augmented Lagrangian and Proximal Step

In this paper we propose and analyze inexact and stochastic versions of the CGALP algorithm developed in [25], which we denote ICGALP , that allow for errors in the computation of several important quantities. In particular this allows one to compute some gradients, proximal terms, and/or linear minimization oracles in an inexact fashion that facilitates the practical application of the algorithm to computationally intensive settings, e.g., in high (or possibly infinite) dimensional Hilbert spaces commonly found in machine learning problems. The algorithm is able to solve composite minimization problems involving the sum of three convex proper lower-semicontinuous functions subject to an affine constraint of the form Ax = b for some bounded linear operator A. Only one of the functions in the objective is assumed to be differentiable, the other two are assumed to have an accessible proximal operator and a linear minimization oracle. As main results, we show convergence of the Lagrangian values (so-called convergence in the Bregman sense) and asymptotic feasibility of the affine constraint as well as strong convergence of the sequence of dual variables to a solution of the dual problem, in an almost sure sense. Almost sure convergence rates are given for the Lagrangian values and the feasibility gap for the ergodic primal variables. Rates in expectation are given for the Lagrangian values and the feasibility gap subsequentially in the pointwise sense. Numerical experiments verifying the predicted rates of convergence are shown as well.

Differential stability properties in convex scalar and vector optimization

This paper focuses on formulas for the ε-subdifferential of the optimal value function of scalar and vector convex optimization problems. These formulas can be applied when the set of solutions of the problem is empty. In the scalar case, both unconstrained problems and problems with an inclusion constraint are considered. For the last ones, limiting results are derived, in such a way that no qualification conditions are required. The main mathematical tool is a limiting calculus rule for the ε-subdifferential of the sum of convex and lower semicontinuous functions defined on a (non necessarily reflexive) Banach space. In the vector case, unconstrained problems are studied and exact formulas are derived by linear scalarizations. These results are based on a concept of infimal set, the notion of cone proper set and an ε-subdifferential for convex vector functions due to Taa.

On the Inverse Source Identification Problem in L^{infty } for Fully Nonlinear Elliptic PDE

In this paper we generalise the results proved in N. Katzourakis (SIAM J. Math. Anal. 51, 1349–1370, 2019) by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order L2 “viscosity term” for the L^{infty } minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.

Iterative methods for optimization problems and image restoration

In this paper, we introduce a new accelerated iterative method for finding a common fixed point of a countable family of nonexpansive mappings in the Hilbert spaces framework. Using our main result, we obtain a new accelerated image restoration iterative method for solving a minimization problem in the form of the sum of two proper lower semi-continuous and convex functions. As applications, we apply our algorithm to solving image restoration problems.

A Differential Perspective on Gradient Flows on textsf {CAT} (kappa )-Spaces and Applications

We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on textsf {CAT} (kappa )-spaces and prove that they can be characterized by the same differential inclusion y_t'in -partial ^-textsf {E} (y_t) one uses in the smooth setting and more precisely that y_t' selects the element of minimal norm in -partial ^-textsf {E} (y_t). This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar–Schoen energy functional on the space of L^2 and CAT(0) valued maps: we define the Laplacian of such L^2 map as the element of minimal norm in -partial ^-textsf {E} (u), provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is L^2-dense. Basic properties of this Laplacian are then studied.

A New Forward–Backward Algorithm with Line Searchand Inertial Techniques for Convex Minimization Problems with Applications

For the past few decades, various algorithms have been proposed to solve convex minimization problems in the form of the sum of two lower semicontinuous and convex functions. The convergence of these algorithms was guaranteed under the L-Lipschitz condition on the gradient of the objective function. In recent years, an inertial technique has been widely used to accelerate the convergence behavior of an algorithm. In this work, we introduce a new forward–backward splitting algorithm using a new line search and inertial technique to solve convex minimization problems in the form of the sum of two lower semicontinuous and convex functions. A weak convergence of our proposed method is established without assuming the L-Lipschitz continuity of the gradient of the objective function. Moreover, a complexity theorem is also given. As applications, we employed our algorithm to solve data classification and image restoration by conducting some experiments on these problems. The performance of our algorithm was evaluated using various evaluation tools. Furthermore, we compared its performance with other algorithms. Based on the experiments, we found that the proposed algorithm performed better than other algorithms mentioned in the literature.

Existence, uniqueness and regularity of the projection onto differentiable manifolds

We investigate the maximal open domain {mathscr {E}}(M) on which the orthogonal projection map p onto a subset Msubseteq {{mathbb {R}}}^d can be defined and study essential properties of p. We prove that if M is a C^1 submanifold of {{mathbb {R}}}^d satisfying a Lipschitz condition on the tangent spaces, then {mathscr {E}}(M) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is C^2 or if the topological skeleton of M^c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a C^k-submanifold M with kge 2, the projection map is C^{k-1} on {mathscr {E}}(M), and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion Msubseteq {mathscr {E}}(M) is that M is a C^1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with Msubseteq {mathscr {E}}(M), then M must be C^1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between {mathscr {E}}(M) and the topological skeleton of M^c.

Monotone Vector Fields and Generation of Nonexpansive Semigroups in Complete CAT(0) Spaces

In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in spaces stands in opposed to the monotonicity defined earlier in spaces by Khatibzadeh and Ranjbar [J. Aust. Math. Soc. 103(1), 70–90 (2017).] and Chaipunya and Kumam [Optimization 66(10), 1647–1665 (2017).]. In particular, this new concept extends the theory from both Hilbert spaces and Hadamard manifolds, while the known concept barely has any obvious relationship to the theory in Hadamard manifolds. We also study the corresponding resolvents and Yosida approximations of a given monotone vector field and derive many of their important properties. Finally, we prove a generation theorem by showing convergence of an exponential formula applied to resolvents of a monotone vector field. Our findings improve several known results in the literature including generation theorems of Jost [AMS/IP Stud. Adv. Math., vol. 8, pp. 1–47. Amer. Math. Soc., Providence, RI (1998)], Mayer [Comm. Anal. Geom. 6(2), 199–253 (1998).], Stojkovic [Adv. Calc. Var. 5(1), 77–126 (2012).], and Bačák [Convex analysis and optimization in Hadamard spaces, De Gruyter Series in Nonlinear Analysis and Applications, vol. 22. De Gruyter, Berlin (2014).] for proper, convex, lower semicontinuous functions in the context of complete spaces, and also by Iwamiya and Okochi [Nonlinear Anal. 54(2), 205–214 (2003).] for monotone vector fields in the context of Hadamard manifolds.

Iterative Algorithms for a System of Variational Inclusions Involving Set-Valued Quasi-Contractive Mappings in Banach Spaces

In this paper, a new system of variational inclusions involving set-valued quasi-contractive, lower semi-continuous mappings with nonempty closed and convex values is introduced and studied in real Banach spaces. First, both the existence of fixed points of the set-valued quasi-contractive mappings and the solution of the system of variational inclusions are proved. Then, using the existence result, a new iterative algorithm is constructed. And then the strong convergence of the iterative sequences is proved, and the proof is novel. Finally, a numerical example is given to demonstrate the algorithm. Our results improve and extend some known results.

Feng–Liu-type fixed point result in orbital b-metric spaces and application to fractal integral equation

In this manuscript, we establish two Wardowski–Feng–Liu-type fixed point theorems for orbitally lower semicontinuous functions defined in orbitally complete b-metric spaces. The obtained results generalize and improve several existing theorems in the literature. Moreover, the findings are justified by suitable nontrivial examples. Further, we also discuss ordered version of the obtained results. Finally, an application is presented by using the concept of fractal involving a certain kind of fractal integral equations. An illustrative example is presented to substantiate the applicability of the obtained result in reducing the energy of an antenna.

Supercaloric functions for the parabolic p-Laplace equation in the fast diffusion case

We study a generalized class of supersolutions, so-called p-supercaloric functions, to the parabolic p-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for pge 2, but little is known in the fast diffusion case 1<p<2. Every bounded p-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic p-Laplace equation for the entire range 1<p<infty . Our main result shows that unbounded p-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case frac{2n}{n+1}<p<2. The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case 1<ple frac{2n}{n+1} and the theory is not yet well understood.

A Projected Forward-Backward Algorithm for Constrained Minimization with Applications to Image Inpainting

In this research, we study the convex minimization problem in the form of the sum of two proper, lower-semicontinuous, and convex functions. We introduce a new projected forward-backward algorithm using linesearch and inertial techniques. We then establish a weak convergence theorem under mild conditions. It is known that image processing such as inpainting problems can be modeled as the constrained minimization problem of the sum of convex functions. In this connection, we aim to apply the suggested method for solving image inpainting. We also give some comparisons to other methods in the literature. It is shown that the proposed algorithm outperforms others in terms of iterations. Finally, we give an analysis on parameters that are assumed in our hypothesis.

On applications of the calmness moduli for multifunctions to error bounds

In this paper, we mainly study applications of the calmness moduli for multifunctions to error bounds of several non-convex systems. Based on the work given in Shen et al. [Calmness and the Abadie CQ for multifunctions and linear regularity for a collection of closed sets. SIAM J Optim. 2019;29(3):2291–2319], we use results on the calmness modulus of the multifunction therein to study error bounds of differentiable inclusions, weak sharp minima of a lower semicontinuous function and linear regularity of finitely many closed subsets. Several primal equivalent conditions for these regularity properties of the corresponding non-convex systems are provided with some mild assumptions.

Periodic Solutions for Systems with p-Relativistic Operator and Unbounded Discontinuous Nonlinearities

We are concerned with the existence of periodic solutions for potential differential inclusions involving the p-relativistic operator $$\begin{aligned} {\mathcal {R}}_pu:= \left( \frac{|u'|^{p-2}u'}{(1-|u'|^p)^{1-1/p}} \right) ' \end{aligned}$$ and an (possible) unbounded discontinuous gradient. The approach relies on critical point theory for locally Lipschitz perturbations of convex, lower semicontinuous functions. The solutions we obtain appear as either minimizers or saddle points of the corresponding energy functional. Some examples of applications illustrating the general results are provided.

Moreau Envelope of Supremum Functions with Applications to Infinite and Stochastic Programming

In this paper, we investigate the Moreau envelope of the supremum of a family of convex, proper, and lower semicontinuous functions. Under mild assumptions, we prove that the Moreau envelope of a supremum is the supremum of Moreau envelopes, which allows us to approximate possibly nonsmooth supremum functions by smooth functions that are also the suprema of functions. Consequently, we propose and study approximated optimization problems from infinite and stochastic programming for which we obtain zero-duality gap results and optimality conditions without the verification of constraint qualification conditions.

PAIRS OF HAHN AND SEPARATELY CONTINUOUS FUNCTION

In this paper we continue the study of interconnections between separately continuous function which was started by V. K. Maslyuchenko. A pair (g, h) of functions on a topological space is called a pair of Hahn if g ≤ h, g is an upper semicontinuous function and h is a lower semicontinuous function. We say that a pair of Hahn (g, h) is generated by a function f, which depends on two variables, if the infimum of f and the supremum of f with respect to the second variable equals g and h respectively. We prove that for any perfectly normal space X and non-pseudocompact space Y every pair of Hahn on X is generated by a continuous function on X x Y . We also obtain that for any perfectly normal space X and for any space Y having non-scattered compactification any pair of Hahn on X is generated by a separately continuous function on X x Y .

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.