Lower Semicontinuous Convex Function Research Articles

Education

The Education section in this issue presents two contributions. In `"Nesterov's Method for Convex Optimization," Noel J. Walkington proposes a teaching guide for a first course in optimization of this well-known algorithm for computing the minimum of a convex function. This algorithm, first proposed in 1983 by Yuri Nesterov, though well recognized in computational optimization in the presence of large data as a more efficient tool than the steepest descent method, is still absent in most modern textbooks on optimization. The author of the present article develops an elementary analysis of Nesterov's first order algorithm that parallels that of steepest descent but with an additional requirement proposed by Nesterov. Two cases are discussed. The first concerns an unconstrained minimization problem, while the second includes closed convex constraints represented using infinite penalization of the cost. More generally, the cost function becomes the sum of a smooth convex function and a lower semicontinuous convex function. Several student-level exercises are included in this paper. Results are nicely illustrated by an example of a signal recovery problem and a discussion of the Uzawa algorithm for optimization problems with constraints defined by inequalities involving convex functions. The second paper, "A Comprehensive Proof of Bertrand's Theorem," is presented by Patrick De Leenheer, John Musgrove, and Tyler Schimleck. It concerns the behavior of the solutions of the classical two-body problem and states that, among all possible gravitational laws, there are only two exhibiting the property that all bounded orbits are closed: Newtonian gravitation and Hookean gravitation. Historically, even if Newton was aware that there are to specific gravitational laws having the above property, it was only two centuries later, in 1873, that Bertrand realized that these are the only ones. Bertrand's theorem, due to its important consequences, has been integrated into the undergraduate curriculum in theoretical mechanics, but its proof, accessible to undergraduate mathematics or physics students, seems to be absent from modern textbooks. Although Bertrand's original paper did not contain a precise proof, V. Arnold proposed a sketch of it based on six subproblems. Among other contributions, this article provides a complete proof of the sixth subproblem under a specific assumption imposed on the magnitude of the force in the motion model. Under this assumption, a complete proof of Bertrand's theorem is then given, incorporating also earlier contributions by other authors. Still, comprehensive does not mean simple here, and this paper may be used to conceive several research projects for advanced-level undergraduate students in mathematics or physics.

On the sum rules and maximality of generalized subdifferentials

Some properties of σ-convex functions are studied. Then some relations between the notions of σ-subdifferentials and σ-convex functions are established. Also, we present some results regarding the sum formula for the σ-subdifferential. Moreover, we obtain some particular relationships between the σ-subdifferential and the -conjugate. Finally, by imposing some assumptions on f and σ, the maximal -monotonicity of σ-subdifferential is studied. Indeed, the maximality of the usual Fenchel subdifferentials of lower semicontinuous convex functions follows from our result.

Duality for Sets of Strong Slater Points

The strong Slater condition plays a significant role in the stability analysis of linear semi-infinite inequality systems. This piece of work studies the set of strong Slater points, whose non-emptiness guarantees the fullfilment of the strong Slater condition. Given a linear inequality system, we firstly establish some basic properties of the set of strong Slater points. Then, we derive dual characterizations for this set in terms of the data of the system, following similar characterizations provided also for the set of Slater points and the solution set of the given system, which are based on the polarity operators for evenly convex and closed convex sets. Finally, we present two geometric interpretations and apply our results to analyze the strict inequality systems defined by lower semicontinuous convex functions.

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.

Fast convex optimization via a third-order in time evolution equation: TOGES-V an improved version of TOGES*

In a Hilbert space setting H , for convex optimization, we analyse the fast convergence properties as t → + ∞ of the trajectories t ↦ u ( t ) ∈ H generated by a third-order in time evolution system. The function f : H → R to minimize is supposed to be convex, continuously differentiable, with argmin H f ≠ ∅ . It enters into the dynamic through its gradient. Based on this new dynamical system, we improve the results obtained by Attouch et al. [Fast convex optimization via a third-order in time evolution equation. Optimization. 2020;71(5):1275–1304]. As a main result, when the damping parameter α satisfies α > 3 , we show that f ( u ( t ) ) − inf H f = o ( 1 / t 3 ) as t → + ∞ , as well as the convergence of the trajectories. We complement these results by introducing into the dynamic an Hessian-driven damping term, which reduces the oscillations. In the case of a strongly convex function f, we show an autonomous evolution system of the third-order in time with an exponential rate of convergence. All these results have natural extensions to the case of a convex lower semicontinuous function f : H → R ∪ { + ∞ } . Just replace f with its Moreau envelope.

Determining functions by slopes

It is shown that by knowing the slope of a bounded from below lower semicontinuous convex function at each point of a Banach space, we know the function up to an additive constant.

The subdifferentiability of functions convex with respect to the set of Lipschitz concave functions

A function defined on normed vector spaces X is called convex with respect to the set LĈ := LĈ (X,R ) ofLipschitz continuous classically concave functions (further, for brevity, LĈ -convex), if it is the upper envelope of some subset of functions from LĈ. A function f is LĈ -convex if and only if it is lower semicontinuous and bounded from below by a Lipschitz function. We introduce the notion of LĈ -subdifferentiability of a function at a point, i. e., subdifferentiability with respect to Lipschitz concave functions, which generalizes the notion of subdifferentiability of classically convex functions, and prove that for each LĈ -convex function the set of points at which it is LĈ -subdifferentiable is dense in its effective domain. The last result extends the well-known Brondsted – Rockafellar theorem on the existence of the subdifferential for classically convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using elements of the subset LĈ θ ⊂ LĈ, which consists of Lipschitz continuous functions vanishing at the origin of X we introduce the notions of LĈ θ -subgradient and LĈ θ -subdifferential for a function at a point.The properties of LĈ -subdifferentials and their relations with the classical Fenchel – Rockafellar subdifferential are studied. Considering the set LČ := LČ (X,R ) of Lipschitz continuous classically convex functions as elementary ones we define the notions of LČ -concavity and LČ -superdifferentiability that are symmetric to the LĈ -convexity and LĈ -subdifferentiability of functions. We also derive criteria for global minimum and maximum points of nonsmooth functions formulated in terms of LĈ θ -subdifferentials and LČ θ -superdifferentials.

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.

Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

<p style='text-indent:20px;'>In this paper, for solving the variational inequality problem over the set of common fixed points of a finite family of demiclosed quasi-nonexpansive mappings in Hilbert spaces, we propose two new strongly convergent methods, constructed by specific combinations between the steepest-descent method and the block-iterative ones. The strong convergence is proved without the boundedly regular assumptions on the family of fixed point sets as well as the approximately shrinking property for each mapping of the family, that are usually assumed in recent literature for similar problems. Applications to the multiple-operator split common fixed point problem (MOSCFPP) and the problem of common minimum points of a finite family of lower semi-continuous convex functions with numerical experiments are given.</p>

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).

On convergence and complexity analysis of an accelerated forward\u2013backward algorithm with linesearch technique for convex minimization problems and applications to data prediction and classification

In this work, we introduce a new accelerated algorithm using a linesearch technique for solving convex minimization problems in the form of a summation of two lower semicontinuous convex functions. A weak convergence of the proposed algorithm is given without assuming the Lipschitz continuity on the gradient of the objective function. Moreover, the convexity of this algorithm is also analyzed. Some numerical experiments in machine learning are also discussed, namely regression and classification problems. Furthermore, in our experiments, we evaluate the convergent behavior of this new algorithm, then compare it with various algorithms mentioned in the literature. It is found that our algorithm performs better than the others.

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.

Solving Optimization Problems over the Weakly Efficient Set

In this paper, we study the optimization problem (PWE) of minimizing a convex function over the set of weakly efficient solutions of a convex multiobjective problem. This is done by using the fact that each lower semicontinuous convex function is an upper envelope of its affine minorants together with a generalized cutting plane method. We give necessary conditions for optimal solutions of the problem Moreover, a novel algorithm for solving the problem (PWE) together with numerical results are presented. We also prove that the proposed algorithm terminates after a finite numbers of iterations, and the algorithm is coded in MATLAB language and evaluated by numerical examples.

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.

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.

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.

Closed convex sets with an open or closed Gauss range

We characterize the closed convex subsets of $${\mathbb {R}}^{n}$$ which have open or closed Gauss ranges. Some special attention is paid to epigraphs of lower semicontinuous convex functions.

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.

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.