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Aris Daniilidis

Publications and source records attributed to Aris Daniilidis.

At least 19 recordsLinked to original sources
arXiv2026

Feature weighting for data analysis via evolutionary simulation

We analyze an algorithm for assigning weights prior to scalarization in discrete multi-objective problems arising from data analysis. The algorithm evolves weights (interpreted as the relevance of features) by a replicator-type dynamic on the standard simplex, with update indices computed from a normalized data matrix. We prove that the resulting sequence converges globally to a unique interior equilibrium, yielding non-degenerate limiting weights.

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arXiv2026

A convexity criterion via the De Giorgi slope

Let $X$ be a Banach space and $f\in\mathcal{C}^1(X)$ be bounded from below. We show that if for some $m\geq 1$, the function $x\mapsto \|\nabla f(x)\|^m$ is convex, then $f$ is convex. We also establish a more general version of this result: if $f$ is continuous and bounded from below, then it is convex, provided $x\mapsto s_f(x)^m$ is convex for some $m\geq 1$, where $s_f$ denotes the (De Giorgi) metric slope of $f$.

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arXiv2026

Characterization of regularity via variational stability of alternating projections sequences

The notion of regular pair $(A,B)$ for two nonempty closed convex subsets $A$ and~$B$ of a Hilbert space $\H$ was introduced by Borwein and Bauschke in 1993 to ensure convergence (in norm) of the alternating projection method to some point of the best approximation set. In 2022, De Bernardi and Miglierina showed that regularity of the pair $(A,B)$ guarantees, additionally, the convergence for any variational perturbation of the alternating projection method, provided the corresponding best approximation sets are bounded. In this work, we show that the converse assertion is also true. Moreover, this converse assertion holds without requiring the best approximation sets to be bounded.

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arXiv2026

Solving Nonlinear Absolute Value Equations

In this work, we show that several problems naturally represented as Nonlinear Absolute Value Equations (NAVE) can be reformulated as Nonlinear Complementarity Problems (NCP) and efficiently solved using smoothing regularization techniques under mild assumptions. As far as we know, this is the first numerical approach that directly deals with NAVE. We also identify a technical assumption commonly utilized in smoothing techniques and prove its equivalence to a classical __ojasiewicz inequality at infinity, validating its non-restrictive nature. Furthermore, we extend established error estimates for NCP solvers to derive error bounds for NAVE problems under weaker assumptions. We illustrate the effectiveness of our approach through applications including asymmetric ridge optimization and nonlinear ordinary differential equations.

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arXiv2025

Determination of (unbounded) convex functions via Crandall-Pazy directions

It has been recently discovered that a convex function can be determined by its slopes and its infimum value, provided this latter is finite. The result was extended to nonconvex functions by replacing the infimum value by the set of all critical and asymptotically critical values. In all these results boundedness from below plays a crucial role and is generally admitted to be a paramount assumption. Nonetheless, this work develops a new technique that allows to also determine a large class of unbounded from below convex functions, by means of a Neumann-type condition related to the Crandall-Pazy direction.

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arXiv2025

Characterizing Maximal Monotone Operators with Unique Representation

We study maximal monotone operators $A : X \rightrightarrows X^*$ whose Fitzpatrick family reduces to a singleton; such operators will be called uniquely representable. We show that every such operator is cyclically monotone (hence, $A=\partial f$ for some convex function $f$) if and only if it is 3-monotone. In Radon-Nikodým spaces, under mild conditions (which become superfluous in finite dimensions), we prove that a subdifferential operator $A=\partial f$ is uniquely representable if and only if $f$ is the sum of a support and an indicator function of suitable convex sets.

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arXiv2024

All convex bodies are in the subdifferential of some everywhere differentiable locally Lipschitz function

We construct a differentiable locally Lipschitz function $f$ in $\mathbb{R}^{N}$ with the property that for every convex body $K\subset \mathbb{R}^N$ there exists $\bar x \in \mathbb{R}^N$ such that $K$ coincides with the set $\partial_L f(\bar x)$ of limits of derivatives $\{Df(x_n)\}_{n\geq 1}$ of sequences $\{x_n\}_{n\geq 1}$ converging to~$\bar x$. The technique can be further refined to recover all compact connected subsets with nonempty interior, disclosing an important difference between differentiable and continuously differentiable functions. It stems out from our approach that the class of these pathological functions contains an infinite dimensional vector space and is dense in the space of all locally Lipschitz functions for the uniform convergence.

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arXiv2024

Steepest geometric descent for regularized quasiconvex functions

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 flow 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 more general setting.

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arXiv2024

ABB theorems: Results and limitations in infinite dimensions

We construct a weakly compact convex subset of $\ell^2$ with nonempty interior that has an isolated maximal element, with respect to the lattice order $\ell _+^2$. Moreover, the maximal point cannot be supported by any strictly positive functional, showing that the Arrow-Barankin-Blackwell theorem fails. This example discloses the pertinence of the assumption that the cone has a bounded base for the validity of the result in infinite dimensions. Under this latter assumption, the equivalence of the notions of strict maximality and maximality is established

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arXiv2023

A slope generalization of Attouch theorem

A classical result of variational analysis, known as Attouch theorem, establishes the equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding subdifferential maps up to a normalization condition which fixes the integration constant. In this work, we show that in finite dimensions and under a mild boundedness assumption, we can replace subdifferentials (sets of vectors) by slopes (scalars, corresponding to the distance of the subdifferentials to zero) and still obtain the same characterization: namely, the epigraphical convergence of functions is equivalent to the epigraphical convergence of their slopes. This surprising result goes in line with recent developments on slope determination (Boulmezaoud, Cieutat, Daniilidis, 2018), (Pérez-Aros, Salas, Vilches, 2021) and slope sensitivity (Daniilidis, Drusvyatskiy, 2023) for convex functions.

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arXiv2023

Metric compatibility and determination in complete metric spaces

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 functions, by adding an assumption controlling the asymptotic behavior. This assumption is trivially fulfilled if $f$ is inf-compact. In addition, our result is not only valid for the (De Giorgi) local slope, but also for the main paradigms of average descent operators as well as for the global slope, case in which the asymptotic assumption becomes superfluous. Therefore, the present work extends simultaneously the metric determination results of [8] and [18].

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arXiv2023

The slope robustly determines convex functions

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.

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arXiv2022

Extending Rademacher Theorem to Set-Valued Maps

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.

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arXiv2022

Descent modulus and applications

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 this work we propose an axiomatic definition of descent modulus T [f ](x) of a real-valued function f at every point x, defined on a general (not necessarily metric) space. The definition encompasses all above instances as well as average descents for functions defined on probability spaces. We show that a large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope. The particular case of functions defined on finite spaces is studied in the last section. In this case, we obtain an explicit classification of descent operators that are, in some sense, typical.

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arXiv2021

Desingularization of the Sweeping Process Mapping

In [9], the celebrated KŁ-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 dynamics, this desingularization yields a uniform control of the lengths of all bounded orbits of the corresponding sweeping process $-\dotγ(t)\in N_{S(t)}(γ(t))$. In this paper, working outside the framework of o-minimal geometry, we characterize the existence of a desingularization for the coderivative in terms of the behaviour of the sweeping process orbits and the integrability of the talweg function. These results are close in spirit with the ones in [3], where characterizations for the desingularization of the (sub)gradient of functions had been obtained.

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arXiv2021

A convex function satisfying the Lojasiewicz inequality but failing the gradient conjecture both at zero and infinity

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 inequality at zero.

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arXiv2021

Asymmetric free spaces and canonical asymmetrizations

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 property (linearization of semi-Lipschitz functions). The (conic) dual of $\mathcal{F}_a(X,d)$ coincides with the nonlinear asymmetric dual of $(X,d)$, that is, the space $\mathrm{SLip}_0(X,d)$ of semi-Lipschitz functions on $(X,d)$, vanishing at a base point. In particular, for the case of a metric space $(X,D)$, the above construction yields its usual free space. On the other hand, every metric space $(X,D)$ inherits naturally a canonical asymmetrization coming from its free space $\mathcal{F}(X)$. This gives rise to a quasi-metric space $(X,D_+)$ and an asymmetric free space $\mathcal{F}_a(X,D_+)$. The symmetrization of the latter is isomorphic to the original free space $\mathcal{F}(X)$. The results of this work are illustrated with explicit examples.

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