# Aris DaniilidisTU Wien | TU Wien · Institut für Stochastik und Wirtschaftsmathematik

Aris Daniilidis

PhD, Habilitation

## About

106

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Introduction

Aris Daniilidis is currently working at the TU Wien, heading the research group VADOR (Variational Analysis, Dynamics, Operational Research), former ORCOS, part of the Institute of Statistics and Mathematical Methods in Economics. His current research interest are Variational Analysis, Dynamical Systems and Tame and Stochastic Optimization.

Additional affiliations

September 2018 - December 2018

December 2017 - January 2018

November 2016 - November 2016

## Publications

Publications (106)

A classical result of variational analysis, known as Attouch theorem, establishes the equivalence between epigraphical convergence of a sequence of proper convex lower semicon-tinuous functions and graphical convergence of the corresponding subdifferential maps up to a normalization condition which fixes the integration constant. In this work, we s...

We establish existence of steepest descent curves emanating from almost every point of a regular locally Lipschitz quasiconvex functions, where regularity means that the sweeping process induced by the sublevel sets is reversible. We then use max-convolution to regularize general quasiconvex functions and obtain a result of the same nature in a mor...

It was established in [8] that Lipschitz inf-compact functions are uniquely determined by their local slope and critical values. Compactness played a paramount role in this result, ensuring in particular the existence of critical points. We hereby emancipate from this restriction and establish a determination result for merely bounded from below fu...

Daniilidis and Drusviatskiy, in 2017, extended the celebrated Kurdyka–Łojasiewicz inequality from definable functions to definable multivalued maps by establishing that the coderivative mapping admits a desingularization around every critical value. As was the case in the gradient dynamics, this desingularization yields a uniform control of the len...

We show that the deviation between the slopes of two convex functions controls the deviation between the functions themselves. This result reveals that the slope-a one dimensional construct-robustly determines convex functions, up to a constant of integration.

Rademacher theorem asserts that Lipschitz continuous functions between Euclidean spaces are differentiable almost everywhere. In this work we extend this result to set-valued maps using an adequate notion of set-valued differentiability relating to convex processes. Our approach uses Rademacher theorem but also recovers it as a special case.

The norm of the gradient ∇f (x) measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance dist(0, ∂f (x)) of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope |∇f |(x). In this work we propo...

The norm of the gradient $\nabla$f (x) measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance dist(0, $\partial$f (x)) of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope |$\nabla$f |(x)...

In [9], the celebrated K{\L}-inequality has been extended from definable functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R} $ to definable multivalued maps $S:\mathbb{R}\rightrightarrows\mathbb{R}^{n}$, by establishing that the co-derivative mapping $D^{\ast}S$ admits a desingularization around every critical value. As was the case in the gradient dy...

We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits of the function spiral around zero and at infinity. Besides, the function satisfies the Lojasiewicz gradient...

We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits of the function spiral around zero and at infinity. Besides, the function satisfies the Lojasiewicz gradient...

A construction analogous to that of Godefroy-Kalton for metric spaces allows to embed isometrically, in a canonical way, every quasi-metric space $(X,d)$ to an asymmetric normed space $\mathcal{F}_a(X,d)$ (its quasi-metric free space, also called asymmetric free space or semi-Lipschitz free space). The quasi-metric free space satisfies a universal...

The ordinary differential equation \(\dot{x}(t)=f(x(t)), \; t \ge 0 \), for f measurable, is not sufficiently regular to guarantee existence of solutions. To remedy this we may relax the problem by replacing the function f with its Filippov regularization \(F_{f}\) and consider the differential inclusion \(\dot{x}(t)\in F_{f}(x(t))\) which always h...

The convex cone SCSLip1(X) of real-valued smooth semi-Lipschitz functions on a Finsler manifold X is an order-algebraic structure that captures both the differentiable and the quasi-metric feature of X. In this work we show that the subset of smooth semi-Lipschitz functions of constant strictly less than 1, denoted SC1−1(X), can be used to classify...

We introduce the notion of trace convexity for functions and respectively, for subsets of a compact topological space. This notion generalizes both classical convexity of vector spaces, as well as Choquet convexity for compact metric spaces. We provide new notions of trace-convexification for sets and functions as well as a general version of Krein...

In this work we show that various algorithms, ubiquitous in convex optimization (e.g. proximal-gradient, alternating projections and averaged projections) generate self-contracted sequences $\{x_{k}\}_{k\in\mathbb{N}}$. As a consequence, a novel universal bound for the \emph{length} ($\sum_{k\ge 0}\Vert x_{k+1}-x_k\Vert$) can be deduced. In additio...

The ordinary differential equation $\dot{x}(t)=f(x(t)), \; t \geq 0 $, for $f$ measurable, is not sufficiently regular to guarantee existence of solutions. To remedy this we may relax the problem by replacing the function $f$ with its Filippov regularization $F_{f}$ and consider the differential inclusion $\dot{x}(t)\in F_{f}(x(t))$ which always ha...

A construction analogous to that of Godefroy-Kalton for metric spaces allows to embed iso-metrically, in a canonical way, every quasi-metric space (X, d) to an asymmetric normed space Fa(X, d) (its quasi-metric free space, also called asymmetric free space or semi-Lipschitz free space). The quasi-metric free space satisfies a universal property (li...

The convex cone $SC_{\mathrm{SLip}}^1(\mathcal{X})$ of real-valued smooth semi-Lipschitz functions on a Finsler manifold $\mathcal{X}$ is an order-algebraic structure that captures both the differentiable and the quasi-metric feature of $\mathcal{X}$. In this work we show that the subset of smooth semi-Lipschitz functions of constant strictly less...

Convergence of projection-based methods for nonconvex set feasibility problems has been established for sets with ever weaker regularity assumptions. What has not kept pace with these developments is analogous results for convergence of optimization problems with correspondingly weak assumptions on the value functions. Indeed, one of the earliest c...

We construct examples of Lipschitz continuous functions, with pathological subgradient dynamics both in continuous and discrete time. In both settings, the iterates generate bounded trajectories, and yet fail to detect any (generalized) critical points of the function.

Self-contractedness (or self-expandedness, depending on the orientation) is hereby extended in two natural ways giving rise, for any $\lambda\in\lbrack-1,1)$, to the metric notion of $\lambda $-curve and the (weaker) geometric notion of $\lambda$-cone property ($\lambda$-eel). In the Euclidean space $\mathbb{R}^{d}$ it is established that for $\lam...

In this paper we establish that the set of Lipschitz functions f : U → R (U a nonempty open subset of ` ¹d ) with maximal Clarke subdifferential contains a linear subspace of uncountable dimension (in particular, an isometric copy of ` ∞ (N)). This result follows along a similar line to that of a previous result of Borwein and Wang (see [Proc. Amer...

It is hereby established that the set of Lipschitz functions $f:\mathcal{U}\rightarrow \mathbb{R}$ ($\mathcal{U}$ nonempty open subset of $\ell_{d}^{1}$) with maximal Clarke subdifferential contains a linear subspace of uncountable dimension (in particular, an isometric copy of $\ell^{\infty}(\mathbb{N})$). This result goes in the line of a previou...

Convergence of projection-based methods for nonconvex set feasibility problems has been established for sets with ever weaker regularity assumptions. What has not kept pace with these developments is analogous results for convergence of optimization problems with correspondingly weak assumptions on the value functions. Indeed, one of the earliest c...

In this work we are interested in the Demyanov--Ryabova conjecture for a finite family of polytopes. The conjecture asserts that after a finite number of iterations (successive dualizations), either a 1-cycle or a 2-cycle eventually comes up. In this work we establish a strong version of this conjecture under the assumption that the initial family...

We disclose an interesting connection between the gradient flow of a $\mathcal{C}^2$-smooth function $\psi$ and evanescent orbits of the second order gradient system defined by the square-norm of $\nabla\psi$, under adequate convexity assumption. As a consequence, we obtain the following surprising result for two $\mathcal{C}^2$, convex and bounded...

We give a simple alternative proof for the $C^{1,1}$--convex extension problem which has been introduced and studied by D. Azagra and C. Mudarra [2]. As an application, we obtain an easy constructive proof for the Glaeser-Whitney problem of $C^{1,1}$ extensions on a Hilbert space. In both cases we provide explicit formulae for the extensions. For t...

We show that any semi-algebraic sweeping process admits piecewise absolutely continuous solutions (trajectories), and any such bounded trajectory must have finite length. Analogous results hold more generally for sweeping processes definable in o-minimal structures. This extends previous work on (sub)gradient dynamical systems beyond monotone sweep...

We establish a “preparatory Sard theorem” for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lips...

This paper studies stability properties of linear optimization problems with finitely many variables and an arbitrary number of constraints, when only left hand side coefficients can be perturbed. The coefficients of the constraints are assumed to be continuous functions with respect to an index which ranges on certain compact Hausdorff topological...

We consider the separation problem for sets X that are pre-images of a given set S by a linear mapping. Classical examples occur in integer programming, as well as in other optimization problems such as complementarity. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. To...

We show that bounded trajectories of the semi-algebraic, or more generally o-minimally definable, sweeping process have finite length. This result generalizes previous work on gradient/subgradient dynamical systems and paves the way for further extensions in control systems and mathematical mechanics.

It is established that every self-contracted curve in a Riemannian manifold has finite length, provided its image is contained in a compact set.

This paper studies structural properties of locally symmetric submanifolds. One of the main result states that a locally symmetric submanifold M of R-n admits a locally symmetric tangential parametrization in an appropriately reduced ambient space. This property has its own interest and is the key element to establish, in a follow-up paper [7], tha...

A set of n × n symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in ℝn
is called spectral or isotropic. In this paper, we establish that every locally symmetric C
k
submanifoldMof ℝn
gives rise to a C
k
spectral manifold for k ∈ {2, 3, …,∞,ω}. An explicit formula for the dimension of the spectral manifold in terms of the...

We prove that quasiconvex functions always admit descent trajectories
bypassing all non-minimizing critical points.

The Morse-Sard theorem states that the set of critical values of a C k smooth function defined on a Euclidean space R d has Lebesgue measure zero, provided k ≥ d. This result is hereby extended for (generalized) critical values of continuous selections over a compactly indexed countable family of C k functions: it is shown that these functions are...

Orthogonally invariant functions of symmetric matrices often inherit
properties from their diagonal restrictions: von Neumann's theorem on
matrix norms is an early example. We discuss the example of
"identifiability", a common property of nonsmooth functions associated
with the existence of a smooth manifold of approximate critical points.
Identifi...

It is hereby established that, in Euclidean spaces of finite dimension, bounded self-contracted curves have finite length. This extends the main result of Daniilidis et al. (J. Math. Pures Appl. 94:183–199, 2010) concerning continuous planar self-contracted curves to any dimension, and dispenses entirely with the continuity requirement. The proof b...

In optimization problems such as integer programs or their relaxations, one encounters feasible regions of the form \(\{x\in\mathbb{R}_+^n:\: Rx\in S\}\) where R is a general real matrix and S ⊂ ℝq
is a specific closed set with 0 ∉ S. For example, in a relaxation of integer programs introduced in [ALWW2007], S is of the form ℤq
− b where \(b \not\i...

This paper deals with stability properties of the feasible set of linear inequality systems having a finite number of variables and an arbitrary number of constraints. Several types of perturbations preserving consistency are considered, affecting respectively, all of the data, the left-hand side data, or the right-hand side coefficients.

It is hereby established that, in Euclidean spaces of finite dimension,
bounded self-contracted curves have finite length. This extends the main result
of Daniilidis, Ley, and Sabourau (J. Math. Pures Appl. 2010) concerning
continuous planar self-contracted curves to any dimension, and dispenses
entirely with the continuity requirement. The proof b...

We consider linear optimization over a nonempty convex semialgebraic feasible region F. Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique "active" manifold, around which F is "partly smooth," and the second-order sufficient conditions hold. Perturbin...

Given a C 1;1 {function f : U ! R (where U ‰ R n open) we deal with the question of whether or not at a given point x0 2 U there exists a local minorant ' of f of class C 2 that satisfles '(x0) = f(x0), D'(x0) = Df(x0) and D 2 '(x0) 2 Hf(x0) (the generalized Hessian of f at x0). This question is motivated by the second-order viscosity theory of the...

The classicalLojasiewicz inequality and its extensions for partial difierential equa- tion problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial difierential equations, tame g...

Continuity of set-valued maps is hereby revisited: after recalling some basic
concepts of variational analysis and a short description of the
State-of-the-Art, we obtain as by-product two Sard type results concerning
local minima of scalar and vector valued functions. Our main result though, is
inscribed in the framework of tame geometry, stating t...

Throughout, X stands for a real Banach space, SX for its unit sphere, X for its to- pological dual, andh ;i for the duality pairing. All the functions f : X!R(f+1g are lower semicontinuous. The Clarke subdier ential, the Hadamard subdier ential and the Fr echet subdier ential of f are respectively denoted by @Cf, @Hf and @Ff. The Zagrodny two point...

Superlinear convergence of the Newton method for nonsmooth equations requires a "semismoothness" assumption. In this work we prove that locally Lipschitz functions definable in an o-minimal structure (in particular semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings presen...

We consider linear optimization over a fixed compact convex feasible region that is semi-algebraic (or, more generally, "tame"). Generically, we prove that the optimal solution is unique and lies on a unique manifold, around which the feasible region is "partly smooth", ensuring finite identification of the manifold by many optimization algorithms....

We consider the problem of minimizing nonsmooth convex functions, dened piecewise by a nite number of functions each of which is either convex quadratic or twice continuously dieren tiable with positive denite Hessian on the set of interest. This is a particular case of functions with primal-dual gradient structure, a notion closely related to the...

In this work we prove that every locally symmetric smooth submanifold M of R n gives rise to a naturally defined smooth submanifold of the space of n × n symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to M. We also present an explicit formula for the dimension of the spectral man...

We hereby introduce and study the notion of self-contracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded self-contracted planar curves have a finite length. We also give an example of a convex function defined in the plane whose gradient orbits spiral infinitely many ti...

Important properties such as differentiability and convexity of symmetric functions in R n can be transferred to the corresponding spectral functions and vice-versa. Continuing to built on this line of research, we hereby prove that a spectral function F : S n → R ∪ {+∞} is prox-regular if and only if the underlying symmetric function f : R n → R ∪...

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame ge...

More than half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals. Though somehow unprecise for what concerns the interpretation of the word “generically”, this statement is ostensibly true and is nowada...

Existence and asymptotic stability of the periodic solutions of the Lipschitz system x' (t) = εF(t, x, ε) is hereby studied via the averaging method. The traditional C1 dependence of F(s,·,ε) on z is relaxed to the mere strict differentiability of F(s,·,0) at z = z0 for ε = 0, giving room to potential applications for structured nonsmooth systems.

We introduce the notion of variational (semi-) strict quasimonotonicity for a multivalued operator T
:
X⇉X
*
relative to a nonempty subset A of X which is not necessarily included in the domain of T. We use this notion to characterize the subdifferentials of continuous (semi-) strictly quasiconvex functions. The proposed
definition is a relaxatio...

Existence and asymptotic stability of the periodic solutions of the
Lipschitz system $x'(t) = εF (t, x, ε)$ is hereby studied via the averaging method.
The traditional C1 dependence of $F(s, ·, ε)$ on z is relaxed to the mere strict
differentiability of $F(s, ·, 0)$ at $z = z0$ for $ε = 0$, giving room to potential
applications for structured nonsm...

We establish the following result: if the graph of a lower semicontinuous real-extended- valued function f : Rn ! R ( {+1} admits a Whitney stratification (so in particular if f is a semialgebraic function), then the norm of the gradient of f at x 2 domf relative to the stratum containingx bounds from below all norms of Clarke subgradients off atx....

Given a real-analytic function f : R n → R and a critical point a ∈ R n , the Lojasiewicz inequality asserts that there exists θ ∈ [ 1 2 , 1) such that the function |f − f (a)| θ ∇f −1 remains bounded around a. In this paper, we extend the above result to a wide class of nonsmooth functions (that possibly admit the value +∞), by establishing an ana...

According to the Morse–Sard theorem, any sufficiently smooth function on a Euclidean space remains constant along any arc of critical points. We prove here a theorem of Morse–Sard type suitable as a tool in variational analysis: we broaden the definition of a critical point to the standard notion in nonsmooth optimization, while we restrict the fun...

It has been observed that in many optimization problems, nonsmooth objective functions often appear smooth on naturally arising manifolds. This has led to the development of optimization algorithms which attempt to exploit this smoothness. Many of these algorithms follow the same two-step pattern: first to predict a direction of decrease, and secon...

In this note, we prove the equivalence, under appropriate conditions, between several dynamical formalisms: projected dynamical systems, two types of differential inclusions, and a class of complementarity dynamical systems. Each of these dynamical systems can also be considered as a hybrid dynamical system. This work both generalizes previous resu...

Given a real-analytic function f : IR n → IR and a critical point a ∈ IR n , the Lojasiewicz inequality asserts that there exists θ ∈ [ 1 2 , 1) such that the function |f −f (a)| θ −1 remains bounded around a. In this paper, we extend the above result to a wide class of non-smooth functions (that admit possibly the value +∞), by establishing an ana...

Lagrangian relaxation is useful to bound the optimal value of a given optimization problem, and also to obtain relaxed solutions. To obtain primal solutions, it is conceivable to use a convexification procedure suggested by D.P. Bertsekas in 1979, based on the proximal algorithm in the primal space.
The present paper studies the theory assessing th...

Prox-regularity of a set (Poliquin-Rockafellar-Thibault [25]), or its global version, proximal smoothness (Clarke-Stern-Wolenski [5]) plays an important role in variational analysis, not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist (C; ·), or the local uniqueness of th...

The classes of lower-C 1,α functions (0 < α 1), that is, functions locally repre-sentable as a maximum of a compactly parametrized family of continuously differentiable functions with α-Hölder derivative, are hereby introduced. These classes form a strictly decreasing sequence from the larger class of lower-C 1 towards the smaller class of lower-C...

Every lower semicontinuous convex function can be represented through its subdifferential by means of an "integration" formula introduced in [10] by Rockafellar. We show that in Banach spaces with the Radon-Nikodym property this formula can be significantly refined under a standard coerciv-ity assumption. This yields an interesting application to t...

We prove that any subanalytic locally Lipschitz function has the Sard property. Such functions are typically nonsmooth and their lack of regularity necessitates the choice of some generalized notion of gradient and of critical point. In our framework these notions are defined in terms of the Clarke and of the convex-stable subdifferentials. The mai...

It is shown that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. Consequently, in finite dimensions, the class of locally Lipschitz approximately convex functions coincides with the class of lower-C1 functions. Directional approximate convexity is introduced and shown to be...

Rockafellar has shown that the subdifferentials of convex functions are always cyclically monotone operators. Moreover, maximal cyclically monotone operators are necessarily operators of this type, since one can construct explicitly a convex function, which turns out to be unique up to a constant, whose subdifferential gives back the operator. This...

In this note we prove the equivalence, under appropriate conditions, between several dynamical formalisms: projected dynamical systems, two types of unilateral differential inclusions, and a class of complementarity dynamical systems. Each of these dynamical systems can also be considered as a hybrid dynamical system. This work is of interest since...

We introduce a notion of cyclic submonotonicity for multivalued operators from a Banach space X to its dual. We show that if the Clarke subdifferential of a locally Lipschitz function is strictly submonotone on an open subset U of X, then it is also maximal cyclically submonotone on U, and, conversely, that every maximal cyclically submonotone oper...

The aim of this paper is to present a geometric characterization of even convexity in separable Banach spaces, which is not expressed in terms of dual functionals or separation theorems. As an application, an analytic equivalent definition for the class of evenly quasiconvex functions is derived.

It is shown that in finite dimensions Rockafellar's technique of integrating cyclically monotone operators, applied to the Fenchel subdifferential of an epi-pointed function, yields the closed convex hull of that function.

We consider the question of integration of a multivalued operator T, that is the question of finding a function f such that Tf. If is the Fenchel–Moreau subdifferential, the above problem has been completely solved by Rockafellar, who introduced cyclic monotonicity as a necessary and sufficient condition. In this article we consider the case where...

In this paper we introduce and study a subdifferential that is related to the quasiconvex
functions, much as the Fenchel–Moreau subdifferential is related to the convex ones. It is
defined for any lower semicontinuous function, through an appropriate combination of an abstract subdifferential and the normal cone to sublevel sets. We show that this...

The paper aims at creating a new insight into our perception of convexity by focusing on two fundamental problems: the coincidence of two functions (at least one being convex) upon an information on a dense set and the clarification of the relation between convexity and Fenchel subdifferential. Various results are established into these directions....

Lagrangian relaxation is useful to bound the optimal value of a given optimization problem, and also to obtain relaxed solutions. To obtain primal solutions, it is conceivable to use a convexification procedure suggested by D.P. Bertsekas in 1979, based on the proximal algorithm in the primal space. The present paper studies the theory assessing th...

Various properties of continuity for the class of lower semicontinuous convex functions are considered and dual characterizations are established. In particular, it is shown that the restriction of a lower semicontinuous convex function to its domain (respectively, domain of subdifferentiability) is continuous if and only if its subdifferential is...

An axiomatic approach of normal operators to sublevel sets is given. Considering the Clarke-Rockafellar subdifferential (resp.
quasiconvex functions), the definition given in [4] (resp. [5]) is recovered. Moreover, the results obtained in [4] are extended in this more general setting. Under mild assumptions, quasiconvex continuous functions are cla...

We prove that a Banach space X has the Radon-Nikodym property if, and only if, every weak*-lower semicontinuous convex continuous function f of X* is Gâteaux differentiable at some point of its domain with derivative in the predual space x.

We prove that a Banach space X has the Radon-Nikodym property if, and only if, every weak*-lower semicontinuous convex continuous function f of X* is Gateaux differentiable at some point of its domain with derivative in the predual space X.

In this article we explore the concept of the normal cone to the sublevel sets (or strict sublevel sets) of a function. By slightly modifying the original definition of Borde and Crouzeix, we obtain here a new (but strongly related to the already existent) notion of a normal operator. This technique turns out to be appropriate in Quasiconvex Analys...

The property of σcyclic monotonicity is proposed here to describe subdifferentials of lsc convex functions that are continuous in their domains. It is shown that all monotone operators in R and all densely defined cyclically monotome operators in Rn share this property. Examples of a densely defined maximal cyclically monotone operator in a Hilbert...

In this paper we consider an abstract subdifferential that fulfills a prioria weak type of a mean value property. We survey and extend some recent results connecting the gener-alized convexity of nonsmooth functions with the generalized cyclic monotonidty of their subdifferentials. It is shown that, for a large class of subdifferentials, a Isc func...

We attempt a brief survey on the cone duality and on the density theorems of Arrow-Barankin and Blackwell’s type [K. Arrow, E. Barankin and D. Blackwell, Contrib. Theory of Games II, Ann. Math. Studies 28, 87-91 (1953; Zbl 0050.14203)]. Concerning the latter aspect we show the equivalence of two recent and ostensibly different results. We follow a...

In this paper we consider an abstract subdifferential that fulfills a priori a weak type of a mean value property. We survey and extend some recent results connecting the generalized convexity of nonsmooth functions with the generalized cyclic monotonicity of their subdifferentials. It is shown that, for a large class of subdifferentials, a Isc fun...

New concepts of semistrict quasimonotonicity and strict quasimonotonicity for multivalued maps are introduced. It is shown that a locally Lipschitz map is (semi)strictly quasiconvex if and only if its Clarke subdifferential is (semi)strictly quasimonotone. Finally, an existence result for the corresponding variational inequality problem is obtained...

The notions of cyclic quasimonotonicity and cyclic pseudomonotonicity are introduced. A classical result of convex analysis concerning the cyclic monotonicity of the (Fenchel–Moreau) subdifferential of a convex function is extended to corresponding results for the Clarke–Rockafellar subdifferential of quasiconvex and pseudoconvex functions.The noti...

Various coercivity conditions appear in the literature in order to guarantee solutions for the variational inequality problem. We show that these conditions are equivalent to each other and that they are not only sufficient, but also necessary for the set of solutions to be non-empty and bounded.