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Lionel Thibault - One of the best experts on this subject based on the ideXlab platform.
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Well-posedness and Subdifferentials of Optimal Value and Infimal Convolution
Set-Valued and Variational Analysis, 2019Co-Authors: Grigorii E Ivanov, Lionel ThibaultAbstract:We show that well-posedness (namely approximative well-posedness) properties of optimization problems are very efficient tools in Subdifferential calculus of optimal value (marginal) function and in particular of infimal convolution. Under well-posedness conditions we establish an inclusion for the Mordukhovich limiting Subdifferential of the marginal function and obtain new properties and descriptions of the Fréchet, proximal and Mordukhovich limiting Subdifferentials of the infimal convolution. We also formulate sufficient conditions for well-posedness properties under consideration.
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well posedness and Subdifferentials of optimal value and infimal convolution
Set-valued and Variational Analysis, 2019Co-Authors: Grigorii E Ivanov, Lionel ThibaultAbstract:We show that well-posedness (namely approximative well-posedness) properties of optimization problems are very efficient tools in Subdifferential calculus of optimal value (marginal) function and in particular of infimal convolution. Under well-posedness conditions we establish an inclusion for the Mordukhovich limiting Subdifferential of the marginal function and obtain new properties and descriptions of the Frechet, proximal and Mordukhovich limiting Subdifferentials of the infimal convolution. We also formulate sufficient conditions for well-posedness properties under consideration.
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Chain Rules for a Proper ε -Subdifferential of Vector Mappings
Journal of Optimization Theory and Applications, 2015Co-Authors: César Gutiérrez, Lidia Huerga, Vicente Novo, Lionel ThibaultAbstract:In this paper, we derive exact chain rules for a proper epsilon-Subdifferential in the sense of Benson of extended vector mappings, recently introduced by ourselves. For this aim, we use a new regularity condition and a new strong epsilon-Subdifferential for vector mappings. In particular, we determine chain rules when one of the mappings is linear, obtaining formulations easier to handle in the finite-dimensional case by considering the componentwise order. This Benson proper epsilon-Subdifferential generalizes and improves several of the most important proper epsilon-Subdifferentials of vector mappings given in the literature and, consequently, the results presented in this work extend known chain rules stated for the last ones. As an application, we derive a characterization of approximate Benson proper solutions of implicitly constrained convex Pareto problems. Moreover, we estimate the distance between the objective values of these approximate proper solutions and the set of nondominated attained values.
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Chain Rules for a Proper $$\varepsilon $$ ε -Subdifferential of Vector Mappings
Journal of Optimization Theory and Applications, 2015Co-Authors: César Gutiérrez, Lidia Huerga, Vicente Novo, Lionel ThibaultAbstract:In this paper, we derive exact chain rules for a proper epsilon-Subdifferential in the sense of Benson of extended vector mappings, recently introduced by ourselves. For this aim, we use a new regularity condition and a new strong epsilon-Subdifferential for vector mappings. In particular, we determine chain rules when one of the mappings is linear, obtaining formulations easier to handle in the finite-dimensional case by considering the componentwise order. This Benson proper epsilon-Subdifferential generalizes and improves several of the most important proper epsilon-Subdifferentials of vector mappings given in the literature and, consequently, the results presented in this work extend known chain rules stated for the last ones. As an application, we derive a characterization of approximate Benson proper solutions of implicitly constrained convex Pareto problems. Moreover, we estimate the distance between the objective values of these approximate proper solutions and the set of nondominated attained values.
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Noncoincidence of Approximate and Limiting Subdifferentials of Integral Functionals
SIAM Journal on Control and Optimization, 2011Co-Authors: Abderrahim Jourani, Lionel ThibaultAbstract:For a locally Lipschitz integral functional $I_f$ on $L^1(T,\mathbf{R}^n)$ associated with a measurable integrand f, the limiting Subdifferential and the approximate Subdifferential never coincide at a point $x_0$ where $f(t,\cdot)$ is not Subdifferentially regular at $x_0(t)$ for a.e. $t\in T$. The coincidence of both Subdifferentials occurs on a dense set of $L^1(T,\mathbf{R}^n)$ if and only if $f(t,\cdot)$ is convex for a.e. $t\in T$. Our results allow us to characterize Aubin's Lipschitz-like property as well as the convexity of multivalued mappings between $L^1$-spaces. New necessary optimality conditions for some Bolza problems are also obtained.
Marc Lassonde - One of the best experts on this subject based on the ideXlab platform.
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Characterization of the monotone polar of Subdifferentials
Optimization Letters, 2013Co-Authors: Marc LassondeAbstract:We show that a point is solution of the Minty variational inequality of Subdifferential type for a given lower semicontinuous function if and only if the function is increasing along rays starting from that point. This provides a characterization of the monotone polar of Subdifferentials of lower semicontinuous functions: it is a common subset of their graphs which depends only on the function.
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Characterization of the monotone polar of Subdifferentials
Optimization Letters, 2013Co-Authors: Marc LassondeAbstract:We show that a point is solution of the Minty variational inequality of Subdifferential type for a given function if and only if the function is increasing along rays starting from that point. This provides a characterization of the monotone polar of Subdifferentials of lower semicontinuous functions: it is a common subset of their graphs which depends only on the function.
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Subdifferential estimate of the directional derivative optimality criterion and separation principles
Optimization, 2013Co-Authors: Florence Jules, Marc LassondeAbstract:We provide an inequality relating the radial directional derivative and the Subdifferential of proper lower semicontinuous functions, which extends the known formula for convex functions. We show that this property is equivalent to other Subdifferential properties of Banach spaces, such as controlled dense subdifferentiability, optimality criterion, mean value inequality and separation principles. As an application, we obtain a first-order sufficient condition for optimality, which extends the known condition for differentiable functions in finite-dimensional spaces and which amounts to the maximal monotonicity of the Subdifferential for convex lower semicontinuous functions. Finally, we establish a formula describing the Subdifferential of the sum of a convex lower semicontinuous function with a convex inf-compact function in terms of the sum of their approximate ϵ-Subdifferentials. Such a formula directly leads to the known formula relating the directional derivative of a convex lower semicontinuous fun...
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Brøndsted-Rockafellar property of Subdifferentials of prox-bounded functions
arXiv: Optimization and Control, 2013Co-Authors: Marc LassondeAbstract:We provide a new proof that the Subdifferential of a proper lower semicontinuous convex function on a Banach space is maximal monotone by adapting the pattern commonly used in the Hilbert setting. We then extend the arguments to show more precisely that Subdifferentials of proper lower semicontinuous prox-bounded functions possess the Br{\o}ndsted-Rockafellar property.
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Dense subdifferentiability and trustworthiness for arbitrary Subdifferentials
Serdica. Mathematical Journal, 2010Co-Authors: Florence Jules, Marc LassondeAbstract:We show that the properties of dense subdifferentiability and of trustworthiness are equivalent for any Subdifferential satisfying a small set of natural axioms. The proof relies on a remarkable property of the Subdifferential of the inf-convolution of two (non necessarily convex) functions. We also show the equivalence of the dense subdifferentiability property with other basic properties of Subdifferentials such as a weak∗ Lipschitz Separation property, a strong Compact Separation property and a Minimal property for the analytic approximate Subdifferential of Ioffe.
Dariusz Zagrodny - One of the best experts on this subject based on the ideXlab platform.
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enlarged inclusion of Subdifferentials
Canadian Mathematical Bulletin, 2005Co-Authors: Lionel Thibault, Dariusz ZagrodnyAbstract:This paper studies the integration of inclusion of Subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions f and g have the Subdifferential of f included in the gamma-enlargement of the Subdifferential of g, then the difference of those functions is gamma-Lipschitz over their effective domain.
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integration of Subdifferentials of lower semicontinuous functions on banach spaces
Journal of Mathematical Analysis and Applications, 1995Co-Authors: Lionel Thibault, Dariusz ZagrodnyAbstract:Abstract This paper studies the lower semicontinuous functions ƒ on Banach spaces whose Subdifferentials are not disassociated from ƒ. Several conditions are given to ensure that two lower semicontinuous functions with the same Subdifferential are equal up to an additive constant.
Boris S. Mordukhovich - One of the best experts on this subject based on the ideXlab platform.
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Subdifferentials of nonconvex integral functionals in banach spaces with applications to stochastic dynamic programming
arXiv: Optimization and Control, 2015Co-Authors: Boris S. Mordukhovich, Nobusumi SagaraAbstract:The paper concerns the investigation of nonconvex and nondifferentiable integral functionals on general Banach spaces, which may not be reflexive and/or separable. Considering two major Subdifferentials of variational analysis, we derive nonsmooth versions of the Leibniz rule on subdifferentiation under the integral sign, where the integral of the Subdifferential set-valued mappings generated by Lipschitzian integrands is understood in the Gelfand sense. Besides examining integration over complete measure spaces and also over those with nonatomic measures, our special attention is drawn to a stronger version of measure nonatomicity, known as saturation, to invoke the recent results of the Lyapunov convexity theorem type for the Gelfand integral of the Subdifferential mappings. The main results are applied to the Subdifferential study of the optimal value functions and deriving the corresponding necessary optimality conditions in nonconvex problems of stochastic dynamic programming with discrete time on the infinite horizon.
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Subdifferentials of nonconvex supremum functions and their applications to semi infinite and infinite programs with lipschitzian data
Siam Journal on Optimization, 2013Co-Authors: Boris S. Mordukhovich, T T A NghiaAbstract:The paper is devoted to the Subdifferential study and applications of the supremum of uniformly Lipschitzian functions over arbitrary index sets with no topology. Based on advanced techniques of variational analysis, we evaluate major Subdifferentials of the supremum functions in the general framework of Asplund (in particular, reflexive) spaces with no convexity or relaxation assumptions. The results obtained are applied to deriving new necessary optimality conditions for nonsmooth and nonconvex problems of semi-infinite and infinite programming.
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second order Subdifferential calculus with applications to tilt stability in optimization
Siam Journal on Optimization, 2012Co-Authors: Boris S. Mordukhovich, R T RockafellarAbstract:This paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finite- dimensional spaces. The main focus is the so-called (full and partial) second-order Subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of first-order Subdifferential mappings. We develop an extended second-order Subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal second-order chain rule for strongly and fully amenable compositions. We also calculate the second- order Subdifferentials for some major classes of piecewise linear-quadratic functions. These results are applied to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms.
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second order Subdifferential calculus with applications to tilt stability in optimization
arXiv: Optimization and Control, 2011Co-Authors: Boris S. Mordukhovich, R T RockafellarAbstract:The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the so-called (full and partial) second-order Subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of frst-order Subdifferential mappings. We develop an extended second-order Subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal secondorder chain rule for strongly and fully amenable compositions. The calculus results obtained in this way and computing the second-order Subdifferentials for piecewise linear-quadratic functions and their major specifications are applied then to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms.
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infimal convolutions and lipschitzian properties of Subdifferentials for prox regular functions in hilbert spaces
Journal of Convex Analysis, 2010Co-Authors: Miroslav Bacak, Andrew Eberhard, Jonathan M. Borwein, Boris S. MordukhovichAbstract:We study infimal convolutions of extended-real-valued functions in Hilbert spaces paying a special attention to the rather broad and remarkable class of prox-regular functions. Such functions have been well recognized as highly important in many aspects of variational analysis and its applications in both finite-dimensional and infinite-dimensional settings. Based on advanced variational techniques, we discover some new Subdifferential properties of infimal convolutions and apply them to the study of Lipschitzian behavior of Subdifferentials for prox-regular functions in Hilbert spaces. It is shown, in particular, that the fulfillment of a natural Lipschitz-like property for (set-valued) Subdifferentials of prox-regular functions forces such functions, under weak assumptions, actually to be locally smooth with single-valued Subdifferentials reduced to Lipschitz continuous gradient mappings.
Jonathan M. Borwein - One of the best experts on this subject based on the ideXlab platform.
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infimal convolutions and lipschitzian properties of Subdifferentials for prox regular functions in hilbert spaces
Journal of Convex Analysis, 2010Co-Authors: Miroslav Bacak, Andrew Eberhard, Jonathan M. Borwein, Boris S. MordukhovichAbstract:We study infimal convolutions of extended-real-valued functions in Hilbert spaces paying a special attention to the rather broad and remarkable class of prox-regular functions. Such functions have been well recognized as highly important in many aspects of variational analysis and its applications in both finite-dimensional and infinite-dimensional settings. Based on advanced variational techniques, we discover some new Subdifferential properties of infimal convolutions and apply them to the study of Lipschitzian behavior of Subdifferentials for prox-regular functions in Hilbert spaces. It is shown, in particular, that the fulfillment of a natural Lipschitz-like property for (set-valued) Subdifferentials of prox-regular functions forces such functions, under weak assumptions, actually to be locally smooth with single-valued Subdifferentials reduced to Lipschitz continuous gradient mappings.
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Subdifferentials whose graphs are not norm x weak closed
Canadian Mathematical Bulletin, 2003Co-Authors: Jonathan M. Borwein, Simon Fitzpatrick, Roland GirgensohnAbstract:In this note we give examples of convex functions whose Subdifferentials have unpleasant properties. Particularly, we exhibit a \pcf on separable Hilbert space such that the graph of its Subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed.
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Local Lipschitz-constant functions and maximal Subdifferentials
Set-Valued Analysis, 2003Co-Authors: Jonathan M. Borwein, Jon D. Vanderwerff, Xianfu WangAbstract:It is shown that if k(x) is an upper semicontinuous and quasi lower semicontinuous function on a Banach space X, then k(x)BX* is the Clarke Subdifferential of some locally Lipschitz function on X. Related results for approximate Subdifferentials are also given. Moreover, on smooth Banach spaces, for every locally Lipschitz function with minimal Clarke Subdifferential, one can obtain a maximal Clarke Subdifferential map via its ‘local Lipschitz-constant’ function. Finally, some results concerning the characterization and calculus of local Lipschitz-constant functions are developed.
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Local Lipschitz Constants and Maximal Subdifferentials
2003Co-Authors: Jonathan M. Borwein, Jon D. Vanderwerff, Shawn Xianfu WangAbstract:It is shown that if $k(x)$ is upper semicontinuous and quasi lower semicontinuou s on a Banach space $X$, then $k(x) B_{X^*}$ is the Clarke Subdifferential of some locally Lipschitz function on $X$. Related results for approximate Subdifferentials are also given. Moreover, on smooth Banach spaces, for every locally Lipschitz function with minimal Clarke Subdifferential, one can obtain a maximal Clarke Subdifferential map via its `local Lipschitz constant' function. Finally, some results concerning the calculus of local Lipschitz constants are developed.
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essential smoothness essential strict convexity and legendre functions in banach spaces
Communications in Contemporary Mathematics, 2001Co-Authors: Heinz H Bauschke, Jonathan M. Borwein, Patrick L CombettesAbstract:The classical notions of essential smoothness, essential strict convexity, and Legendreness for convex functions are extended from Euclidean to Banach spaces. A pertinent duality theory is developed and several useful characterizations are given. The proofs rely on new results on the more subtle behavior of Subdifferentials and directional derivatives at boundary points of the domain. In weak Asplund spaces, a new formula allows the recovery of the Subdifferential from nearby gradients. Finally, it is shown that every Legendre function on a reflexive Banach space is zone consistent, a fundamental property in the analysis of optimization algorithms based on Bregman distances. Numerous illustrating examples are provided.
