Субдифференцируемость функций, выпуклых относительно множества липшицевых вогнутых функций

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.Функция, определенная на нормированном пространстве X, называется выпуклой относительно множества LĈ := LĈ (X,R ) липшицевых классически вогнутых функций (далее для краткости – LĈ -выпуклой), если она является верхней огибающей некоторого подмножества функций из LĈ. Функция является LĈ –выпуклой в том и только том случае, когда она полунепрерывна снизу и, кроме того, ограничена снизу некоторой липшицевой функцией. В статье вводится понятие LĈ -субдифференцируемости функции в точке, т. е. субдифференцируемости относительно липшицевых вогнутых функций, обобщающее понятие субдифференцируемости классически выпуклых функций, и доказывается, что для любой LĈ -выпуклой функции множество точек, в которых она является LĈ -субдифференцируемой, является плотным в ее эффективной области. Данное утверждение распространяет на более широкий класс полунепрерывных снизу функций известную теорему Брондстеда – Рокафеллара о существовании субдифференциала для классически выпуклых полунепрерывных снизу функций. Используя элементы подмножества LĈ θ ⊂ LĈ , состоящего из таких липшицевых вогнутых функций, которые принимают нулевое значение в нулевой точке пространства X, определяются понятия LĈ θ LĈ -субградиента и LĈ θ  -субдифференциала функции в точке. Исследуются свойства LĈ θ -субдифференциалов и их связь с классическим субдифференциалом Фенхеля – Рокафеллара. Рассматривая в качестве элементарных функций множество LČ := LČ (X,R ) липшицевых выпуклых (в классическом смысле) функций, вводятся симметричные LĈ -выпуклости и LĈ -субдифференцируемости понятия LČ -вогнутости и LČ -супердифференцируемости функций. В терминах LĈθ –субдифференциалов и LČθ -супердифференциалов устанавливаются критерии для точек глобального минимума и максимума функций

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

The goal of the paper is to study the particular class of regularly ${\mathcal{H}}$-convex functions, when ${\mathcal{H}}$ is the set ${\mathcal{L}\widehat{C}}(X,{\mathbb{R}})$ of real-valued Lipschitz continuous classically concave functions defined on a real normed space $X$. For an extended-real-valued function $f:X \mapsto \overline{\mathbb{R}}$ to be ${\mathcal{L}\widehat{C}}$-convex it is necessary and sufficient that $f$ be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each ${\mathcal{L}\widehat{C}}$-convex function is regularly ${\mathcal{L}\widehat{C}}$-convex as well. We focus on ${\mathcal{L}\widehat{C}}$-subdifferentiability of functions at a given point. We prove that the set of points at which an ${\mathcal{L}\widehat{C}}$-convex function is ${\mathcal{L}\widehat{C}}$-subdifferentiable is dense in its effective domain. Using the subset ${\mathcal{L}\widehat{C}}_\theta$ of the set ${\mathcal{L}\widehat{C}}$ consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of ${\mathcal{L}\widehat{C}}_\theta$-subgradient and ${\mathcal{L}\widehat{C}}_\theta$-subdifferential of a function at a point which generalize the corresponding notions of the classical convex analysis. Symmetric notions of abstract ${\mathcal{L}\widecheck{C}}$-concavity and ${\mathcal{L}\widecheck{C}}$-superdifferentiability of functions where ${\mathcal{L}\widecheck{C}}:= {\mathcal{L}\widecheck{C}}(X,{\mathbb{R}})$ is the set of Lipschitz continuous convex functions are also considered. Some properties and simple calculus rules for ${\mathcal{L}\widehat{C}}_\theta$-subdifferentials as well as ${\mathcal{L}\widehat{C}}_\theta$-subdifferential conditions for global extremum points are established.Comment: 18 page

Abstract convex optimal antiderivatives

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A primal--dual algorithm as applied to optimal control problems

We propose a primal--dual technique that applies to infinite dimensional equality constrained problems, in particular those arising from optimal control. As an application of our general framework, we solve a control-constrained double integrator optimal control problem and the challenging control-constrained free flying robot optimal control problem by means of our primal--dual scheme. The algorithm we use is an epsilon-subgradient method that can also be interpreted as a penalty function method. We provide extensive comparisons of our approach with a traditional numerical approach

Proximal Algorithms for a class of abstract convex functions

In this paper we analyze a class of nonconvex optimization problem from the viewpoint of abstract convexity. Using the respective generalizations of the subgradient we propose an abstract notion proximal operator and derive a number of algorithms, namely an abstract proximal point method, an abstract forward-backward method and an abstract projected subgradient method. Global convergence results for all algorithms are discussed and numerical examples are give

Zero duality gap conditions via abstract convexity

vital:16776Using tools provided by the theory of abstract convexity, we extend conditions for zero duality gap to the context of non-convex and nonsmooth optimization. Mimicking the classical setting, an abstract convex function is the upper envelope of a family of abstract affine functions (being conventional vertical translations of the abstract linear functions). We establish new conditions for zero duality gap under no topological assumptions on the space of abstract linear functions. In particular, we prove that the zero duality gap property can be fully characterized in terms of an inclusion involving (abstract) (Formula presented.) -subdifferentials. This result is new even for the classical convex setting. Endowing the space of abstract linear functions with the topology of pointwise convergence, we extend several fundamental facts of functional/convex analysis. This includes (i) the classical Banach–Alaoglu–Bourbaki theorem (ii) the subdifferential sum rule, and (iii) a constraint qualification for zero duality gap which extends a fact established by Borwein, Burachik and Yao (2014) for the conventional convex case. As an application, we show with a specific example how our results can be exploited to show zero duality for a family of non-convex, non-differentiable problems. © 2021 Informa UK Limited, trading as Taylor & Francis Group

G-coupling functions and properties of strongly star-shaped cones

The main part of this thesis presents a new approach to the topic of conjugation, with applications to various optimization problems. It does so by introducing (what we call) G-coupling functions.Doctor of Philosoph

Extremality and stationarity of collections of sets : metric, slope and normal cone characterisations

Variational analysis, a relatively new area of research in mathematics, has become one of the most powerful tools in nonsmooth optimisation and neighbouring areas. The extremal principle, a tool to substitute the conventional separation theorem in the general nonconvex environment, is a fundamental result in variational analysis. There have seen many attempts to generalise the conventional extremal principle in order to tackle certain optimisation models. Models involving collections of sets, initiated by the extremal principle, have proved their usefulness in analysis and optimisation, with non-intersection properties (or their absence) being at the core of many applications: recall the ubiquitous convex separation theorem, extremal principle, Dubovitskii Milyutin formalism and various transversality/regularity properties. We study elementary nonintersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish nonlinear primal (slope) and linear/nonlinear dual (generalised separation) characterisations of these non-intersection properties. We establish a series of consequences of our main results covering all known formulations of extremality/ stationarity and generalised separability properties. This research develops a universal theory, unifying all the current extensions of the extremal principle, providing new results and better understanding for the exquisite theory of variational analysis. This new study also results in direct solutions for many open questions and new future research directions in the fields of variational analysis and optimisation. Some new nonlinear characterisations of the conventional extremality/stationarity properties are obtained. For the first time, the intrinsic transversality property is characterised in primal space without involving normal cones. This characterisation brings a new perspective on intrinsic transversality. In the process, we thoroughly expose and classify all quantitative geometric and metric characterisations of transversality properties of collections of sets and regularity properties of set-valued mappings.Doctor of Philosoph