Proper Lower Semicontinuous Convex Function Research Articles

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

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.

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).

Iterative Sequences for a Finite Number of Resolvent Operators on Complete Geodesic Spaces

We consider Halpern’s and Mann’s types of iterative schemes to find a common minimizer of a finite number of proper lower semicontinuous convex functions defined on a complete geodesic space with curvature bounded above.

The FM and BCQ Qualifications for Inequality Systems of Convex Functions in Normed Linear Spaces

For an inequality system defined by an infinite family of proper lower semicontinuous convex functions in normed linear space, we consider the Farkas--Minkowski (FM for short) type qualification an...

A parabolic–elliptic chemotaxis system with nonlinear diffusion approached from a Cahn–Hilliard-type system

This paper deals with a parabolic–elliptic chemotaxis system with nonlinear diffusion. It was proved that there exists a solution of a Cahn–Hilliard system as an approximation of a nonlinear diffusion equation by applying an abstract theory by Colli–Visintin (Commun Partial Differ Equ 15:737–756, 1990) for a doubly nonlinear evolution inclusion with some bounded monotone operator and subdifferential operator of a proper lower semicontinuous convex function (cf. Colli–Fukao in J Math Anal Appl 429, 2015:1190–1213). Moreover, Colli–Fukao (J Differ Equ 260:6930–6959, 2016) established existence of solutions to the nonlinear diffusion equation by passing to the limit in the Cahn–Hilliard equation. However, Cahn–Hilliard approaches to chemotaxis systems with nonlinear diffusions seem to be not studied yet. This paper will try to derive existence of solutions to a parabolic–elliptic chemotaxis system with nonlinear diffusion by passing to the limit in a Cahn–Hilliard-type chemotaxis system. Although in this paper, we employ a time discretization scheme to prove existence for the Cahn–Hilliard-type chemotaxis system in reference to Colli–Kurima (Nonlinear Anal 190:111613, 2020), please note that this reference does not deal with chemotaxis terms.

Stochastic Primal Dual Fixed Point Method for Composite Optimization

In this paper we propose a stochastic primal dual fixed point method for solving the sum of two proper lower semi-continuous convex function and one of which is composite. The method is based on the primal dual fixed point method proposed in Chen et al. (Inverse Probl 29:025011, 2013) that does not require subproblem solving. Under some mild condition, the convergence is established based on two sets of assumptions: bounded and unbounded gradients and the convergence rate of the expected error of iterate is of the order $${\mathcal {O}}(k^{-\alpha })$$ where k is iteration number and $$\alpha \in (0,1]$$ . Finally, numerical examples on graphic Lasso and logistic regressions are given to demonstrate the effectiveness of the proposed algorithm.

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.

A New Machine Learning Algorithm Based on Optimization Method for Regression and Classification Problems

A convex minimization problem in the form of the sum of two proper lower-semicontinuous convex functions has received much attention from the community of optimization due to its broad applications to many disciplines, such as machine learning, regression and classification problems, image and signal processing, compressed sensing and optimal control. Many methods have been proposed to solve such problems but most of them take advantage of Lipschitz continuous assumption on the derivative of one function from the sum of them. In this work, we introduce a new accelerated algorithm for solving the mentioned convex minimization problem by using a linesearch technique together with a viscosity inertial forward–backward algorithm (VIFBA). A strong convergence result of the proposed method is obtained under some control conditions. As applications, we apply our proposed method to solve regression and classification problems by using an extreme learning machine model. Moreover, we show that our proposed algorithm has more efficiency and better convergence behavior than some algorithms mentioned in the literature.

Lipschitz Continuity of Convex Functions

Fondecyt Postdoc Project 3180080 Basal Program from CONICYT-Chile CMM-AFB 170001 National Foundation for Science & Technology Development (NAFOSTED) 101.01-2017.325

On the global shape of convex functions on locally convex spaces

In the recent paper [1] D. Azagra studies the global shape of continuous convex functions defined on a Banach space X. More precisely, when X is separable, it is shown that for every continuous convex function f:X→R there exist a unique closed linear subspace Y of X, a convex function h:X/Y→R with the property that limt→∞⁡h(u+tv)=∞ for all u,v∈X/Y, v≠0, and x⁎∈X⁎ such that f=h∘π+x⁎, where π:X→X/Y is the natural projection. Our aim is to characterize those proper lower semicontinuous convex functions defined on a locally convex space which have the above representation. In particular, we show that the continuity of the function f and the completeness of X can be removed from the hypothesis of Azagra's theorem. For achieving our goal we study general sublinear functions as well as recession functions associated to convex ones.

Resolvents of convex functions in complete geodesic metric spaces with negative curvature

In this paper, we define a resolvent of a proper lower semicontinuous convex function in a complete geodesic space with negative curvature and we show its well-definedness and fundamental properties. We further show a fixed point theorem for hyperbolically nonspreading mappings, which can be applied to the resolvent defined in this paper. We also apply these results to convex optimization problems in complete geodesic spaces with negative curvature.

Inertial Forward–Backward Algorithms with Perturbations: Application to Tikhonov Regularization

In a Hilbert space, we analyze the convergence properties of a general class of inertial forward–backward algorithms in the presence of perturbations, approximations, errors. These splitting algorithms aim to solve, by rapid methods, structured convex minimization problems. The function to be minimized is the sum of a continuously differentiable convex function whose gradient is Lipschitz continuous and a proper lower semicontinuous convex function. The algorithms involve a general sequence of positive extrapolation coefficients that reflect the inertial effect and a sequence in the Hilbert space that takes into account the presence of perturbations. We obtain convergence rates for values and convergence of the iterates under conditions involving the extrapolation and perturbation sequences jointly. This extends the recent work of Attouch–Cabot which was devoted to the unperturbed case. Next, we consider the introduction into the algorithms of a Tikhonov regularization term with vanishing coefficient. In this case, when the regularization coefficient does not tend too rapidly to zero, we obtain strong ergodic convergence of the iterates to the minimum norm solution. Taking a general sequence of extrapolation coefficients makes it possible to cover a wide range of accelerated methods. In this way, we show in a unifying way the robustness of these algorithms.

Two modified proximal point algorithms in geodesic spaces with curvature bounded above

We obtain existence and convergence theorems for two variants of the proximal point algorithm involving proper lower semicontinuous convex functions in complete geodesic spaces with curvature bounded above.

A New Primal–Dual Algorithm for Minimizing the Sum of Three Functions with a Linear Operator

In this paper, we propose a new primal–dual algorithm for minimizing $$f({\mathbf {x}})+g({\mathbf {x}})+h({\mathbf {A}}{\mathbf {x}})$$ , where f, g, and h are proper lower semi-continuous convex functions, f is differentiable with a Lipschitz continuous gradient, and $${\mathbf {A}}$$ is a bounded linear operator. The proposed algorithm has some famous primal–dual algorithms for minimizing the sum of two functions as special cases. E.g., it reduces to the Chambolle–Pock algorithm when $$f=0$$ and the proximal alternating predictor–corrector when $$g=0$$ . For the general convex case, we prove the convergence of this new algorithm in terms of the distance to a fixed point by showing that the iteration is a nonexpansive operator. In addition, we prove the O(1 / k) ergodic convergence rate in the primal–dual gap. With additional assumptions, we derive the linear convergence rate in terms of the distance to the fixed point. Comparing to other primal–dual algorithms for solving the same problem, this algorithm extends the range of acceptable parameters to ensure its convergence and has a smaller per-iteration cost. The numerical experiments show the efficiency of this algorithm.

A convergence rate of the proximal point algorithm in Banach spaces

We consider the convergence rate of the proximal point algorithm (PPA) for finding a minimizer of proper lower semicontinuous convex functions. In the Hilbert space setting, Güler showed that the big-O rate of the PPA can be improved to little-o when the sequence generated by the algorithm converges strongly to a minimizer. In this paper, we establish little-o rate of the PPA in Banach spaces without requiring this assumption. Then we apply the result to give new results on the convergence rate for sequences of alternating and averaged projections.

Upper Semismooth Functions and the Subdifferential Determination Property

An upper semismooth function is a lower semicontinuous function whose radial subderivative satisfies a mild directional upper semicontinuity property. Examples of upper semismooth functions are the proper lower semicontinuous convex functions, the lower-C1 functions, the Clarke regular functions, the Mifflin semismooth functions, the Thibault-Zagrodny directionally stable functions. It is shown that the radial subderivative of such functions can be recovered from any subdifferential of the function. It is also shown that these functions are subdifferentially determined, in the sense that if two functions have the same subdifferential and one of the functions is upper semismooth, then the two functions are equal up to an additive constant.

Convergence of functions and their Moreau envelopes on Hadamard spaces

A well known result of H. Attouch states that the Mosco convergence of a sequence of proper convex lower semicontinuous functions defined on a Hilbert space is equivalent to the pointwise convergence of the associated Moreau envelopes. In the present paper we generalize this result to Hadamard spaces. More precisely, while it has already been known that the Mosco convergence of a sequence of convex lower semicontinuous functions on a Hadamard space implies the pointwise convergence of the corresponding Moreau envelopes, the converse implication was an open question. We now fill this gap.Our result has several consequences. It implies, for instance, the equivalence of the Mosco and Frolík–Wijsman convergences of convex sets. As another application, we show that there exists a complete metric on the cone of proper convex lower semicontinuous functions on a separable Hadamard space such that a sequence of functions converges in this metric if and only if it converges in the sense of Mosco.

Local Convergence Properties of Douglas–Rachford and Alternating Direction Method of Multipliers

The Douglas---Rachford and alternating direction method of multipliers are two proximal splitting algorithms designed to minimize the sum of two proper lower semi-continuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local linear convergence behaviour of Douglas---Rachford (resp. alternating direction method of multipliers) when the involved functions (resp. their Legendre---Fenchel conjugates) are moreover partly smooth. More precisely, when the two functions (resp. their conjugates) are partly smooth relative to their respective smooth submanifolds, we show that Douglas---Rachford (resp. alternating direction method of multipliers) (i) identifies these manifolds in finite time; (ii) enters a local linear convergence regime. When both functions are locally polyhedral, we show that the optimal convergence radius is given in terms of the cosine of the Friedrichs angle between the tangent spaces of the identified submanifolds. Under polyhedrality of both functions, we also provide conditions sufficient for finite convergence. The obtained results are illustrated by several concrete examples and supported by numerical experiments.

A Unified Approach to Robust Farkas-Type Results with Applications to Robust Optimization Problems

In this paper we consider the inequality of the form $ \sup_{u \in \mathcal{U}} F_u(\cdot, 0_Y) \geq h,$ where $X, Y$ are locally convex Hausdorff topological vector spaces, $\mathcal{U} \not= \emptyset$ is an uncertainty set, $F_u : X \times Y \rightarrow \overline{\mathbb{R}}$ for each $u \in \mathcal{U}$, and $h : X \rightarrow \overline{\mathbb{R}}$ is a lower semicontinuous proper convex function. Characterizations of such an inequality in terms of robust abstract perturbational duality are established and applied to diverse robust composite functional inequalities to obtain variants of robust Farkas-type results, such as robust Farkas lemmas for general nonconvex conical systems, robust Farkas lemmas for convex-DC systems, robust/stable robust Farkas lemmas for convex systems, robust Farkas lemmas for general linear systems in infinite dimensional spaces, and robust semi-infinite Farkas lemmas. The results are then applied to classes of robust DC and robust convex optimization problems, and strong F...

Existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation

An evolution inclusion is considered in a separable Hilbert space. The right-hand side of the inclusion contains the subdifferential of a time-dependent proper convex lower semicontinuous function and a multivalued perturbation with nonempty closed, not necessarily, bounded values. Along with this inclusion we consider the inclusion with the perturbation term being convexified. We prove the existence of solutions and a density of the solution set of the original inclusion in a closure of the solution set of the inclusion with the convexified perturbation. In contrast to the known results of this kind we do not suppose that the convex function has the compactness property and that the values of the perturbation are bounded sets. An example of a perturbation with closed unbounded values satisfying the conditions of the main theorems is given.