Starting from a general Hoffman-type estimate for inequalities defined via convex functions, we d... more Starting from a general Hoffman-type estimate for inequalities defined via convex functions, we derive estimates of the same type for inequality constraints expressed in terms of eigenvalue functions (as in eigenvalue optimization) or positive semidefiniteness (as in semidefinite programming).
The topic of this monograph ranges beyond its title. We are in presence of a volume which gives a... more The topic of this monograph ranges beyond its title. We are in presence of a volume which gives a large survey of calculus of variations in several variables with detours into partial differential equations of variational type, but also captures some parts of mathematical programming and convex analysis. It is destined for graduate and postgraduate students, but more advanced students in mathematics will also find useful information in these pages, along with other scientists interested in applied mathematics. A clear presentation of the historical Dirichlet problem in Chapter 2 kicks off the first part. The classical tools of variational analysis such as weak topologies and the introduction of Sobolev spaces are introduced in this chapter. The Lax–Milgram theorem and the direct method in the calculus of variations are developed in Chapter 3 along with an introduction to convex optimization. An original and intuitive proof is given for the Ekeland variational principle based on the convergence of a discrete dynamical system. Chapter 4 contains complements on measure theory: Hausdorff measures, duality and introduction to Young measures, which makes the link between classical and modern calculus of variations. Chapter 5 deals with a complete survey of Sobolev spaces with some useful complements on capacity and potential theory. The next chapter contains classical applications to boundary value problems of the Dirichlet and Neumann type, along with an introduction to the p-Laplacian.
In the lines of H. Attouch and R. Wets, two kinds of variational metrics are introduced between c... more In the lines of H. Attouch and R. Wets, two kinds of variational metrics are introduced between closed proper classes of convex-concave functions. The comparison between these two distances gives rise to a metric stability result for the associated saddle-points.
Proceedings of the American Mathematical Society, 2006
We generalize, in a metric space setting, the result due to Lim (2000), that a weakly inward mult... more We generalize, in a metric space setting, the result due to Lim (2000), that a weakly inward multivalued contraction, defined on a nonempty closed subset of a Banach space, has a fixed point. The simple proof of this generalization, avoiding the use of a transfinite induction as in Lim’s paper, is based on Ekeland’s variational principle (1974), along the lines of Hamel (1994) and Takahashi (1991). Moreover, we give a sharp estimate for the distance from any point to the fixed point set.
We survey ancient and recent results on global error bounds for the distance to a sublevel set of... more We survey ancient and recent results on global error bounds for the distance to a sublevel set of a lower semicontinuous function defined on a complete metric space. We emphasize the case of a recent characterization of this property which appeared in [5, 7]. We also review the convex case and show how the known result on sufficient condition for a global error bound can be derived from the quoted characterization. Résumé. Nous revenons sur d'anciens et nouveaux résultats de borne d'erreur pour la distanceà un ensemble de sous-niveau d'une fonction semi-continue inférieurement définie sur un espace métrique complet. Nous soulignons en particulier la caractérisation récente de cette propriété apparue dans [5,7]. On revient aussi sur le cas convexe, et on montre comment les résultats connus de conditions suffisantes pour une borne globale d'erreur découlent de la caractérisation citée.
Journal of Optimization Theory and Applications, 1998
Recently, Moussaoui and Seeger (Ref. 1) studied the monotonicity of first-order and second-order ... more Recently, Moussaoui and Seeger (Ref. 1) studied the monotonicity of first-order and second-order difference quotients with primary goal the simplification of epilimits. It is well known that epilimits (lim inf and lim sup) can be written as pointwise limits in the case of a sequence of functions that is equi-lsc. In this paper, we introduce equicalmness as a condition that
Lecture Notes in Economics and Mathematical Systems, 1992
It is known that the inf-convolution (or epigraphical sum) $$ (f\mathop + \limits_e g)(x) = \inf ... more It is known that the inf-convolution (or epigraphical sum) $$ (f\mathop + \limits_e g)(x) = \inf \{ f(u) + g(x - u):u \in X\} $$ of convex proper lower semicontinuous functions defined on a normed linear space is continuous, under some assumptions, with respect to the usual variational convergences (Mosco convergence in the reflexive case, bounded Hausdorff topology in the general case). The deconvolution (or epigraphical difference) defined by $$ (k\mathop {{\text{ }} - }\limits_e g)(x) = \sup \{ k(x + u) - g(u):g(u) < + \infty \} $$ is the smallest solution, when it exists, of the equation \( \xi \mathop {{\text{ }} + }\limits_e g = k \). In this paper we present continuity results on the mappings \( k \mapsto k\mathop {{\text{ }} - }\limits_e g \) and \( g \mapsto k\mathop {{\text{ }} - }\limits_e g \) with respect to the quoted topologies and to some intermediate topologies lying between Mosco and bounded Hausdorff topologies. Applications are given to continuity of the parallel subtraction of operators in Hilbert spaces.
Control of Distributed Parameter Systems 1989, 1990
We present recent results of open mapping type for multifunctions between general Banach spaces. ... more We present recent results of open mapping type for multifunctions between general Banach spaces. Applications are sketched to the problem of local controllability of distributed parameter systems.
On the Sensitivity Analysis of Hoffman Constants for Systems of Linear Inequalities. [SIAM Journa... more On the Sensitivity Analysis of Hoffman Constants for Systems of Linear Inequalities. [SIAM Journal on Optimization 12, 913 (2002)]. D. Azé, JN Corvellec. Abstract. Relying on a general variational method developed by the authors ...
We give estimates of the modulus of continuity of the Legendre-Fenchel Transform with respect to ... more We give estimates of the modulus of continuity of the Legendre-Fenchel Transform with respect to the Attouch-Wets uniformity. We point out the local Lipschitz behaviour of the conjugacy operation when endowing the set of closed proper convex functions defined on a normed vector space with suitable families of semimetrics.
Nonlinear Analysis: Theory, Methods & Applications, 2002
... Authors, D. Azé, UMR CNRS MIP, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulous... more ... Authors, D. Azé, UMR CNRS MIP, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse cedex, France. ... 10. {10} D. Azé, CC Chou, J.-P. Penot, Subtraction theorems and approximate openness for multifunctions: topological and infinitesimal viewpoints, J. Math. Anal ...
We introduce a notion of strict differentiability for multifunctions by means of a notion of tang... more We introduce a notion of strict differentiability for multifunctions by means of a notion of tangency based on a uniform property of Clarke's tangent cone. Given a multifunction G and a point (a, b) ∈ G and assuming that the derivative DG(a, b) is surjective and has a bounded inverse, we build a sequence ((xn, yn)) ⊂ G, such that d(xn+1 − xn, DG(a, b)−1(−yn)) converges to 0. The sequence ((xn, yn)) is shown to converge to (x, 0) where x is a solution of 0 ∈ G(x) provided the norm of ‖y0‖ is small enough. As a consequence we obtain an open mapping theorem for multifunctions whose proof is constructive.
Combining the Clarke-Ekeland dual least action principle and the epi-convergence, we state an exi... more Combining the Clarke-Ekeland dual least action principle and the epi-convergence, we state an existence result and study the asymptotic behaviour for the periodic solution of a nonlinear Sturm-Liouville problem deriving from a convex subquadratic potential, when the data are perturbed in a suitable sense. The result appears like a stability result for the minimizers of a sequence of DC functions.
Journal of Optimization Theory and Applications, 1990
Combining a result on the lower semicontinuity of the intersection of two convex-valued multifunc... more Combining a result on the lower semicontinuity of the intersection of two convex-valued multifunctions and the level set approach of epi-convergence, we obtain results on the epi-upper semicontinuity of the supremum and the sum of two families of quasi-convex functions. As a consequence, we give some condition ensuring the stability of a quasi-convex program under a perturbation of the objective functions and the constraint sets.
Starting from a general Hoffman-type estimate for inequalities defined via convex functions, we d... more Starting from a general Hoffman-type estimate for inequalities defined via convex functions, we derive estimates of the same type for inequality constraints expressed in terms of eigenvalue functions (as in eigenvalue optimization) or positive semidefiniteness (as in semidefinite programming).
The topic of this monograph ranges beyond its title. We are in presence of a volume which gives a... more The topic of this monograph ranges beyond its title. We are in presence of a volume which gives a large survey of calculus of variations in several variables with detours into partial differential equations of variational type, but also captures some parts of mathematical programming and convex analysis. It is destined for graduate and postgraduate students, but more advanced students in mathematics will also find useful information in these pages, along with other scientists interested in applied mathematics. A clear presentation of the historical Dirichlet problem in Chapter 2 kicks off the first part. The classical tools of variational analysis such as weak topologies and the introduction of Sobolev spaces are introduced in this chapter. The Lax–Milgram theorem and the direct method in the calculus of variations are developed in Chapter 3 along with an introduction to convex optimization. An original and intuitive proof is given for the Ekeland variational principle based on the convergence of a discrete dynamical system. Chapter 4 contains complements on measure theory: Hausdorff measures, duality and introduction to Young measures, which makes the link between classical and modern calculus of variations. Chapter 5 deals with a complete survey of Sobolev spaces with some useful complements on capacity and potential theory. The next chapter contains classical applications to boundary value problems of the Dirichlet and Neumann type, along with an introduction to the p-Laplacian.
In the lines of H. Attouch and R. Wets, two kinds of variational metrics are introduced between c... more In the lines of H. Attouch and R. Wets, two kinds of variational metrics are introduced between closed proper classes of convex-concave functions. The comparison between these two distances gives rise to a metric stability result for the associated saddle-points.
Proceedings of the American Mathematical Society, 2006
We generalize, in a metric space setting, the result due to Lim (2000), that a weakly inward mult... more We generalize, in a metric space setting, the result due to Lim (2000), that a weakly inward multivalued contraction, defined on a nonempty closed subset of a Banach space, has a fixed point. The simple proof of this generalization, avoiding the use of a transfinite induction as in Lim’s paper, is based on Ekeland’s variational principle (1974), along the lines of Hamel (1994) and Takahashi (1991). Moreover, we give a sharp estimate for the distance from any point to the fixed point set.
We survey ancient and recent results on global error bounds for the distance to a sublevel set of... more We survey ancient and recent results on global error bounds for the distance to a sublevel set of a lower semicontinuous function defined on a complete metric space. We emphasize the case of a recent characterization of this property which appeared in [5, 7]. We also review the convex case and show how the known result on sufficient condition for a global error bound can be derived from the quoted characterization. Résumé. Nous revenons sur d'anciens et nouveaux résultats de borne d'erreur pour la distanceà un ensemble de sous-niveau d'une fonction semi-continue inférieurement définie sur un espace métrique complet. Nous soulignons en particulier la caractérisation récente de cette propriété apparue dans [5,7]. On revient aussi sur le cas convexe, et on montre comment les résultats connus de conditions suffisantes pour une borne globale d'erreur découlent de la caractérisation citée.
Journal of Optimization Theory and Applications, 1998
Recently, Moussaoui and Seeger (Ref. 1) studied the monotonicity of first-order and second-order ... more Recently, Moussaoui and Seeger (Ref. 1) studied the monotonicity of first-order and second-order difference quotients with primary goal the simplification of epilimits. It is well known that epilimits (lim inf and lim sup) can be written as pointwise limits in the case of a sequence of functions that is equi-lsc. In this paper, we introduce equicalmness as a condition that
Lecture Notes in Economics and Mathematical Systems, 1992
It is known that the inf-convolution (or epigraphical sum) $$ (f\mathop + \limits_e g)(x) = \inf ... more It is known that the inf-convolution (or epigraphical sum) $$ (f\mathop + \limits_e g)(x) = \inf \{ f(u) + g(x - u):u \in X\} $$ of convex proper lower semicontinuous functions defined on a normed linear space is continuous, under some assumptions, with respect to the usual variational convergences (Mosco convergence in the reflexive case, bounded Hausdorff topology in the general case). The deconvolution (or epigraphical difference) defined by $$ (k\mathop {{\text{ }} - }\limits_e g)(x) = \sup \{ k(x + u) - g(u):g(u) < + \infty \} $$ is the smallest solution, when it exists, of the equation \( \xi \mathop {{\text{ }} + }\limits_e g = k \). In this paper we present continuity results on the mappings \( k \mapsto k\mathop {{\text{ }} - }\limits_e g \) and \( g \mapsto k\mathop {{\text{ }} - }\limits_e g \) with respect to the quoted topologies and to some intermediate topologies lying between Mosco and bounded Hausdorff topologies. Applications are given to continuity of the parallel subtraction of operators in Hilbert spaces.
Control of Distributed Parameter Systems 1989, 1990
We present recent results of open mapping type for multifunctions between general Banach spaces. ... more We present recent results of open mapping type for multifunctions between general Banach spaces. Applications are sketched to the problem of local controllability of distributed parameter systems.
On the Sensitivity Analysis of Hoffman Constants for Systems of Linear Inequalities. [SIAM Journa... more On the Sensitivity Analysis of Hoffman Constants for Systems of Linear Inequalities. [SIAM Journal on Optimization 12, 913 (2002)]. D. Azé, JN Corvellec. Abstract. Relying on a general variational method developed by the authors ...
We give estimates of the modulus of continuity of the Legendre-Fenchel Transform with respect to ... more We give estimates of the modulus of continuity of the Legendre-Fenchel Transform with respect to the Attouch-Wets uniformity. We point out the local Lipschitz behaviour of the conjugacy operation when endowing the set of closed proper convex functions defined on a normed vector space with suitable families of semimetrics.
Nonlinear Analysis: Theory, Methods & Applications, 2002
... Authors, D. Azé, UMR CNRS MIP, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulous... more ... Authors, D. Azé, UMR CNRS MIP, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse cedex, France. ... 10. {10} D. Azé, CC Chou, J.-P. Penot, Subtraction theorems and approximate openness for multifunctions: topological and infinitesimal viewpoints, J. Math. Anal ...
We introduce a notion of strict differentiability for multifunctions by means of a notion of tang... more We introduce a notion of strict differentiability for multifunctions by means of a notion of tangency based on a uniform property of Clarke's tangent cone. Given a multifunction G and a point (a, b) ∈ G and assuming that the derivative DG(a, b) is surjective and has a bounded inverse, we build a sequence ((xn, yn)) ⊂ G, such that d(xn+1 − xn, DG(a, b)−1(−yn)) converges to 0. The sequence ((xn, yn)) is shown to converge to (x, 0) where x is a solution of 0 ∈ G(x) provided the norm of ‖y0‖ is small enough. As a consequence we obtain an open mapping theorem for multifunctions whose proof is constructive.
Combining the Clarke-Ekeland dual least action principle and the epi-convergence, we state an exi... more Combining the Clarke-Ekeland dual least action principle and the epi-convergence, we state an existence result and study the asymptotic behaviour for the periodic solution of a nonlinear Sturm-Liouville problem deriving from a convex subquadratic potential, when the data are perturbed in a suitable sense. The result appears like a stability result for the minimizers of a sequence of DC functions.
Journal of Optimization Theory and Applications, 1990
Combining a result on the lower semicontinuity of the intersection of two convex-valued multifunc... more Combining a result on the lower semicontinuity of the intersection of two convex-valued multifunctions and the level set approach of epi-convergence, we obtain results on the epi-upper semicontinuity of the supremum and the sum of two families of quasi-convex functions. As a consequence, we give some condition ensuring the stability of a quasi-convex program under a perturbation of the objective functions and the constraint sets.
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Papers by Dominique Azé