arXiv subjects

Amos Uderzo

Publications and source records attributed to Amos Uderzo.

At least 19 recordsLinked to original sources

arXiv2026

First-order approximations of multifunctions defined implicitly by "split" feasibility problems with applications to optimization

In the present paper, a study is made of first-order differential properties of certain multifunctions, which are defined as a solution map associated with a family of parameterized ``split" feasibility problems. The latter are a class of convex feasibility problems, whose specific structure makes them suitable for well-established applications in several areas of engineering and systems biology. As a result, inner and outer approximations of the graphical (contingent) derivative of the solution map are provided, which are expressed in terms of derivatives of the problem data. As an application, a perspective is also discussed for employing some of these achievements in the formulation of optimality conditions for mathematical programs with ``split" feasibility constraints.

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arXiv2026

On Lipschitzian properties of multifunctions defined implicitly by "split" feasibility problems

In the present paper, a systematic study is made of quantitative semicontinuity (a.k.a. Lipschitzian) properties of certain multifunctions, which are defined as a solution map associated to a family of parameterized ``split" feasibility problems. The latter are a particular class of convex feasibility problems with well recognized applications to several areas of engineering and systems biology. As a part of a perturbation analysis of variational systems, this study falls within the framework of a line of research pursued by several authors. It is performed by means of techniques of variational analysis, which lead to establish sufficient conditions for the Lipschitz lower semicontinuity, calmness, isolated calmness, Lipschitz upper semicontinuity and Aubin property of the solution map. Along with each of these properties, a quantitative estimate of the related exact bound is also provided. The key elements emerging on the way to achieving the main results are dual regularity conditions qualifying the problem behaviour, which are expressed in terms of convex analysis constructions involving problem data. The approach here proposed tries to unify the study of the aforementioned properties.

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arXiv2025

On a robust approach to "split" feasibility problems: solvability and global error bound conditions

In the present paper, a robust approach to a special class of convex feasibility problems is considered. By techniques of convex and variational analysis, conditions for the existence of robust feasible solutions and related error bounds are investigated. This is done by reformulating the robust counterpart of a split feasibility problem as a set-valued inclusion, a problem for which one can take profit from the solvability and stability theory that has been recently developed. As a result, a sufficient condition for solution existence and error bounds is established in terms of problem data and discussed through several examples. A specific focus is devoted to error bound conditions in the case of the robust counterpart of polyhedral split feasibility problems.

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arXiv2024

Marginal Analysis of Convex Optimization Problems with Set-Valued Inclusion Constraints

In this paper, stability and sensitivity properties of a class of parametric constrained optimization problem, whose feasible region is defined by a set-valued inclusion, are investigated through the associated optimal value function. Set-valued inclusions are a kind of constraint system, which naturally emerges in contexts requiring the robust fulfilment of traditional cone constraints, where data are affected by uncertain elements having a non stochastic nature, or in (MPEC) as a vector equilibrium constraint, where feasible solutions are intended as equilibrium point in a strong sense. Under proper convexity assumptions on the objective function and the constraining set-valued term, combined with a global qualification condition, a class of parametric optimization problems is singled out, which displays a global Lipschitz behaviour. By employing recent results of variational analysis, elements for a sensitivity analysis of this class of problems are provided via exact subgradient formulae for the optimal value function. Further consequences of the stability behaviour are explored in terms of problem calmness and viability of penalization techniques.

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arXiv2024

On some global implicit function theorems for set-valued inclusions with applications to parametric vector optimization

The present paper deals with the perturbation analysis of set-valued inclusion problems, a problem format whose relevance has recently emerged in such contexts as robust and vector optimization as well as in vector equilibrium theory. The set-valued inclusions here considered are parameterized by variables belonging to a topological space, with and without constraints. By proper techniques of variational analysis, some qualitative global implicit function theorems are established, which ensure global solvability of these problems and continuous dependence on the parameter of the related solutions. Applications to parametric vector optimization are discussed, aimed at deriving sufficient conditions for the existence of ideal efficient solutions that depend continuously on the parameter perturbations.

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arXiv2023

On some generalizations of Hadamard's inversion theorem beyond differentiability

A recognized trend of research investigates generalizations of the Hadamard's inversion theorem to functions that may fail to be differentiable. In this vein, the present paper explores some consequences of a recent result about the existence of global Lipschitz continuous inverse by translating its metric assumptions in terms of nonsmooth analysis constructions. This exploration focuses on continuous, but possibly not locally Lipschitz mappings, acting in finite-dimensional Euclidean spaces. As a result, sufficient conditions for global invertibility are formulated by means of strict estimators, *-difference of convex compacta and regular/basic coderivatives. These conditions qualify the global inverse as a Lipschitz continuous mapping and provide quantitative estimates of its Lipschitz constant in terms of the above constructions.

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arXiv2023

First-order approximation of strong vector equilibria with application to nondifferentiable constrained optimization

Vector equilibrium problems are a natural generalization to the context of partially ordered spaces of the Ky Fan inequality, where scalar bifunctions are replaced with vector bifunctions. In the present paper, the local geometry of the strong solution set to these problems is investigated through its inner/outer conical approximations. Formulae for approximating the contingent cone to the set of strong vector equilibria are established, which are expressed via Bouligand derivatives of the bifunctions. These results are subsequently employed for deriving both necessary and sufficient optimality conditions for problems, whose feasible region is the strong solution set to a vector equilibrium problem, so they can be cast in mathematical programming with equilibrium constraints.

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arXiv2022

On some optimality conditions for a class of problems in mathematical programming with equilibrium constraints

This paper considers mathematical programs, whose constraints are expressed by a parameterized vector equilibrium problem. The latter is a well recognized framework, which is able to cover multicriteria optimization, vector variational inequalities and complementarity problems. As the solutions to vector equilibrium problems are here intended in a strong sense, the consequent MPEC problems result in a class still little explored by the existing literature. Some necessary optimality conditions for such programs are established following a penalization approach. To derive and express these conditions, concepts and tools of nonsmooth analysis are employed. In treating equilibrium constraints, by techniques of variational analysis some error bounds are obtained, which may be of independent interest.

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arXiv2022

Some enhanced existence results for strong vector equilibrium problems

This paper explores some sufficient conditions for the enhanced solvability of strong vector equilibrium problems, which can be established via a variational approach. Enhanced solvability here means existence of solutions, which are strong with respect to the partial ordering, complemented with inequalities estimating the distance from the solution set (namely, error bounds). This kind of estimates plays a crucial role in the tangential (first-order) approximation of the solution set as well as in formulating optimality conditions for mathematical programming with equilibrium constraints (MPEC). The approach here followed characterizes solutions as zeros (or global minimizers) of some merit functions associated to the original problem. Thus, to achieve the main results the traditional employment of the KKM theory is replaced by proper conditions on the slope of the merit functions. In turn, to make such conditions verifiable, some tools of nonsmooth analysis are exploited. As a result, several conditions for the enhanced solvability of strong equilibrium problems are derived, which are expressed in terms of generalized (Bouligand) derivatives, convex normals and various (Fenchel and Mordukhovich) subdifferentials.

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arXiv2021

A condition for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

In present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the ideal value mapping associated to the parametric class of vector optimization problems. Within such an analysis, a refinement of a recent existence result specific for ideal efficient solutions to unperturbed problem is also discussed.

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arXiv2021

On some efficiency conditions for vector optimization problems with uncertain cone constraints: a robust approach

In the present paper, several types of efficiency conditions are established for vector optimization problems with cone constraints affected by uncertainty, but with no information of stochastic nature about the uncertain data. Following a robust optimization approach, data uncertainty is faced by handling set-valued inclusion problems. The employment of recent results about error bounds and tangential approximations of the solution set to the latter enables one to achieve necessary conditions for weak efficiency via a penalization method as well as via the modern revisitation of the Euler-Lagrange method, with or without generalized convexity assumptions. The presented conditions are formulated in terms of various nonsmooth analysis constructions, expressing first-order approximations of mappings and sets, while the metric increase property plays the role of a constraint qualification.

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arXiv2021

On the quantitative solution stability of parameterized set-valued inclusions

The subject of the present paper are stability properties of the solution set to set-valued inclusions. The latter are problems emerging in robust optimization and mathematical economics, which can not be cast in traditional generalized equations. The analysis here reported focuses on several quantitative forms of semicontinuity for set-valued mappings, widely investigated in variational analysis, which include, among others, calmness. Sufficient conditions for the occurrence of these properties in the case of the solution mapping to a parameterized set-valued inclusion are established. Consequences on the calmness of the optimal value function, in the context of parametric optimization, are explored. Some specific tools for the analysis of the sufficient conditions, in the case of set-valued inclusion with concave data, are provided in a Banach space setting.

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arXiv2020

On Differential Properties of Multifunctions Defined Implicitly by Set-Valued Inclusions

In the present paper, several properties concerning generalized derivatives of multifunctions implicitly defined by set-valued inclusions are studied by techniques of variational analysis. Set-valued inclusions are problems formalizing the robust fulfilment of cone constraint systems, whose data are affected by a "crude knowledge" of uncertain elements, so they can not be casted in traditional generalized equations. The focus of this study in on the first-order behaviour of the solution mapping associated with a parameterized set-valued inclusion, starting with Lipschitzian properties and then considering its graphical derivative. In particular, a condition for the Aubin continuity of the solution mapping is established in terms of outer prederivative of the set-valued mapping defining the inclusion. A large class of parameterized set-valued inculsions is singled out, whose solution mapping turns out to be convex. Some relevant consequences on the graphical derivative are explored. In the absence of that, formulae for the inner and outer approximation of the graphical derivative are provided by means of prederivatives of the problem data. A representation useful to calculate the coderivative of the solution mapping is also obtained via the subdifferential of a merit function.

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arXiv2019

An extension of the Polyak convexity principle with application to nonconvex optimization

The main problem considered in the present paper is to single out classes of convex sets, whose convexity property is preserved under nonlinear smooth transformations. Extending an approach due to B.T. Polyak, the present study focusses on the class of uniformly convex subsets of Banach spaces. As a main result, a quantitative condition linking the modulus of convexity of such kind of set, the regularity behaviour around a point of a nonlinear mapping and the Lipschitz continuity of its derivative is established, which ensures the images of uniformly convex sets to remain uniformly convex. Applications of the resulting convexity principle to the existence of solutions, their characterization and to the Lagrangian duality theory in constrained nonconvex optimization are then discussed.

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arXiv2018

On finite-dimensional set-inclusive constraint systems: local analysis and related optimality conditions

In the present paper, some aspects of the finite-dimensional theory of set-inclusive generalized equations are studied. Set-inclusive generalized equations are problems arising in several contexts of optimization and variational analysis, involving multi-valued mappings and cones. The aim of this paper is to propose an approach to the local analysis of the solution set to such kind of generalized equations. In particular, a study of the contingent cone to the solution set is carry out by means of first-order approximations of set-valued mappings, which are expressed by prederivatives. Such an approach emphasizes the role of the metric increase property for set-valued mappings, as a condition triggering crucial error bound estimates for the tangential description of solution sets. Some of the results obtained through this kind of analysis are then exploited for formulating necessary optimality conditions, which are valid for problems with constraints formalized by set-inclusive generalized equations.

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arXiv2018

Existence and continuity of solution trajectories of generalized equations with application in electronics

We consider a special form of parametric generalized equations arising from electronic circuits with AC sources and study the effect of perturbing the input signal on solution trajectories. Using methods of variational analysis and strong metric regularity property of an auxiliary map, we are able to prove the regularity properties of the solution trajectories inherited by the input signal. Furthermore, we establish the existence of continuous solution trajectories for the perturbed problem. This can be achieved via a result of uniform strong metric regularity for the auxiliary map. Key words and phrases: generalized equations, electronic circuits, strong metric regularity, uniform strong metric regularity, perturbations.

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arXiv2017

An implicit multifunction theorem for the hemiregularity of mappings with application to constrained optimization

The present paper contains some investigations about a uniform variant of the notion of metric hemiregularity, the latter being a less explored property obtained by weakening metric regularity. The introduction of such a quantitative stability property for set-valued mappings is motivated by applications to the penalization of constrained optimization problems, through the notion of problem calmness. As a main result, an implicit multifunction theorem for parameterized inclusion problems is established, which measures the uniform hemiregularity of the related solution mapping in terms of problem data. A consequence on the exactness of penalty functions is discussed.

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arXiv2015

On a class of convex sets with convex images and its application to nonconvex optimization

In the present paper, conditions under which the images of uniformly convex sets through $C^{1,1}$ regular mappings between Banach spaces remain convex are established. These conditions are expressed by a certain quantitative relation betweeen the modulus of convexity of a given set and the global regularity behaviour of the mapping on it. Such a result enables one to extend to a wide subclass of convex sets the Polyak's convexity principle, which was originally concerned with images of small balls around points of Hilbert spaces. In particular, the crucial phenomenon of the preservation of convexity under regular $C^{1,1}$ transformations is shown to include the class of $r$-convex sets, where the value of $r$ depends on the regularity behaviour of the involved transformation. Two consequences related to nonconvex optimization are discussed: the first one is a sufficient condition for the global solution existence for infinite-dimensional constrained extremum problems; the second one provides a zero-order Lagrangian type characterization of optimality in nonlinear mathematical programming.

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