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De Lara Michel - One of the best experts on this subject based on the ideXlab platform.
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Capra-Convexity, Convex Factorization and Variational Formulations for the l0 Pseudonorm
2021Co-Authors: Chancelier Jean-philippe, De Lara MichelAbstract:The so-called l0 pseudonorm on the Euclidean space Rd counts the number of nonzero components of a vector. In this paper, we analyze the l0 pseudonorm by means of so-called Capra conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the lp norms, but for the extreme ones). We obtain three main results. First, we show that the l0 pseudonorm is equal to its Capra-biconjugate, that is, is a Capra-convex Function. Second, we deduce an unexpected consequence, that we call convex factorization: the l0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex Lower Semicontinuous Function. Third, we establish variational formulations for the l0 pseudonorm by means of generalized top-k dual~norms and k-support dual~norms (that we formally introduce).Comment: arXiv admin note: text overlap with arXiv:2001.1044
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Capra-Convexity, Convex Factorization and Variational Formulations for the l0 Pseudonorm
HAL CCSD, 2021Co-Authors: Chancelier Jean-philippe, De Lara MichelAbstract:The so-called l0 pseudonorm on the Euclidean space Rd counts the number of nonzero components of a vector. In this paper, we analyze the l0 pseudonorm by means of so-called Capra conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the lp norms, but for the extreme ones). We obtain three main results. First, we show that the l0 pseudonorm is equal to its Capra-biconjugate, that is, is a Capra-convex Function. Second, we deduce an unexpected consequence, that we call convex factorization: the l0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex Lower Semicontinuous Function. Third, we establish variational formulations for the l0 pseudonorm by means of generalized top-k dual~norms and k-support dual~norms (that we formally introduce)
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Capra-Convexity, Convex Factorization and Variational Formulations for the l0 Pseudonorm
HAL CCSD, 2021Co-Authors: Chancelier Jean-philippe, De Lara MichelAbstract:The so-called l0 pseudonorm, or cardinality Function, counts the number of nonzero components of a vector. In this paper, we analyze the l0 pseudonorm by means of so-called Capra (constant along primal rays) conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the lp norms, but for the extreme ones). We obtain three main results. First, we show that the l0 pseudonorm is equal to its Capra-biconjugate, that is, is a Capra-convex Function. Second, we deduce an unexpected consequence, that we call convex factorization: the l0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex Lower Semicontinuous Function. Third, we establish a variational formulation for the l0 pseudonorm by means of generalized top-k dual~norms and k-support dual~norms (that we formally introduce)
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Hidden Convexity in the l0 Pseudonorm
HAL CCSD, 2020Co-Authors: Chancelier Jean-philippe, De Lara MichelAbstract:The so-called l0 pseudonorm counts the number of nonzero components of a vector of a Euclidian space. It is well-known that the l0 pseudonorm is not convex, as its Fenchel biconjugate is zero. In this paper, we introduce a suitable conjugacy, induced by a novel coupling, E-Capra, that has the property of being constant along primal rays like the l0 pseudonorm. The coupling E-Capra belongs to the class of one-sided linear couplings, that we introduce; we show that they induce conjugacies that share nice properties with the classic Fenchel conjugacy. For the E-Capra conjugacy, induced by the coupling E-Capra, we relate the E-Capra conjugate and biconjugate of the l0 pseudonorm, the characteristic Functions of its level sets and the sequence of so-called top-k norms. In particular, we prove that the l0 pseudonorm is equal to its biconjugate: hence, the l0 pseudonorm is E-Capra-convex in the sense of generalized convexity. As a corollary, we show that there exists a proper convex Lower Semicontinuous Function such that this Function and the l0 pseudonorm coincide on the Euclidian unit sphere. This hidden convexity property is somewhat surprising as the l0 pseudonorm is a highly nonconvex Function of combinatorial nature. We provide different expressions for this proper convex Lower Semicontinuous Function, and we give explicit formulas in the two-dimensional case
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Hidden Convexity in the l0 Pseudonorm
2019Co-Authors: Chancelier Jean-philippe, De Lara MichelAbstract:The so-called l0 pseudonorm on R d counts the number of nonzero components of a vector. It is well-known that the l0 pseudonorm is not convex, as its Fenchel biconjugate is zero. In this paper, we introduce a suitable conjugacy, induced by a novel coupling, Caprac, having the property of being constant along primal rays, like the l0 pseudonorm. The coupling Caprac belongs to the class of one-sided linear couplings, that we introduce. We show that they induce conjugacies that share nice properties with the classic Fenchel conjugacy. For the Caprac conjugacy, induced by the coupling Caprac, we prove that the l0 pseudonorm is equal to its biconjugate: hence, the l0 pseudonorm is Caprac-convex in the sense of generalized convexity. We also provide expressions for conjugates in terms of two families of dual norms, the 2-k-symmetric gauge norms and the k-support norms. As a corollary, we show that the l0 pseudonorm coincides, on the sphere, with a proper convex Lower Semicontinuous Function-that we characterize, and for which we give explicit formulas in the two dimensional case. This is somewhat surprising as the l0 pseudonorm is a highly nonconvex Function of combinatorial nature.Comment: arXiv admin note: substantial text overlap with arXiv:1902.0481
Chong Li - One of the best experts on this subject based on the ideXlab platform.
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limiting subdifferentials of perturbed distance Functions in banach spaces
Nonlinear Analysis-theory Methods & Applications, 2012Co-Authors: Chong Li, Li MengAbstract:Abstract We explore in arbitrary Banach spaces the Frechet type e -subdifferentials and the limiting subdifferentials for the perturbed distance Function d S J ( ⋅ ) determined by a closed subset S and a Lower Semicontinuous Function J defined on S . In particular, upper and Lower estimates for the Frechet type e -subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated Functions J and δ S .
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subdifferentials of perturbed distance Functions in banach spaces
Journal of Global Optimization, 2010Co-Authors: Jinhua Wang, Chong Li, Hongkun XuAbstract:In general Banach space setting, we study the perturbed distance Function $${d_S^J(\cdot)}$$ determined by a closed subset S and a Lower Semicontinuous Function J (·). In particular, we show that the Frechet subdifferential and the proximal subdifferential of a perturbed distance Function are representable by virtue of corresponding normal cones of S and subdifferentials of J (·).
Teffera M Asfaw - One of the best experts on this subject based on the ideXlab platform.
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maximality theorems on the sum of two maximal monotone operators and application to variational inequality problems
Abstract and Applied Analysis, 2016Co-Authors: Teffera M AsfawAbstract:Let be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space . Let and be maximal monotone operators. The maximality of the sum of two maximal monotone operators has been an open problem for many years. In this paper, new maximality theorems are proved for under weaker sufficient conditions. These theorems improved the well-known maximality results of Rockafellar who used condition and Browder and Hess who used the quasiboundedness of and condition . In particular, the maximality of is proved provided that , where is a proper, convex, and Lower Semicontinuous Function. Consequently, an existence theorem is proved addressing solvability of evolution type variational inequality problem for pseudomonotone perturbation of maximal monotone operator.
Chancelier Jean-philippe - One of the best experts on this subject based on the ideXlab platform.
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Capra-Convexity, Convex Factorization and Variational Formulations for the l0 Pseudonorm
2021Co-Authors: Chancelier Jean-philippe, De Lara MichelAbstract:The so-called l0 pseudonorm on the Euclidean space Rd counts the number of nonzero components of a vector. In this paper, we analyze the l0 pseudonorm by means of so-called Capra conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the lp norms, but for the extreme ones). We obtain three main results. First, we show that the l0 pseudonorm is equal to its Capra-biconjugate, that is, is a Capra-convex Function. Second, we deduce an unexpected consequence, that we call convex factorization: the l0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex Lower Semicontinuous Function. Third, we establish variational formulations for the l0 pseudonorm by means of generalized top-k dual~norms and k-support dual~norms (that we formally introduce).Comment: arXiv admin note: text overlap with arXiv:2001.1044
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Capra-Convexity, Convex Factorization and Variational Formulations for the l0 Pseudonorm
HAL CCSD, 2021Co-Authors: Chancelier Jean-philippe, De Lara MichelAbstract:The so-called l0 pseudonorm on the Euclidean space Rd counts the number of nonzero components of a vector. In this paper, we analyze the l0 pseudonorm by means of so-called Capra conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the lp norms, but for the extreme ones). We obtain three main results. First, we show that the l0 pseudonorm is equal to its Capra-biconjugate, that is, is a Capra-convex Function. Second, we deduce an unexpected consequence, that we call convex factorization: the l0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex Lower Semicontinuous Function. Third, we establish variational formulations for the l0 pseudonorm by means of generalized top-k dual~norms and k-support dual~norms (that we formally introduce)
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Capra-Convexity, Convex Factorization and Variational Formulations for the l0 Pseudonorm
HAL CCSD, 2021Co-Authors: Chancelier Jean-philippe, De Lara MichelAbstract:The so-called l0 pseudonorm, or cardinality Function, counts the number of nonzero components of a vector. In this paper, we analyze the l0 pseudonorm by means of so-called Capra (constant along primal rays) conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the lp norms, but for the extreme ones). We obtain three main results. First, we show that the l0 pseudonorm is equal to its Capra-biconjugate, that is, is a Capra-convex Function. Second, we deduce an unexpected consequence, that we call convex factorization: the l0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex Lower Semicontinuous Function. Third, we establish a variational formulation for the l0 pseudonorm by means of generalized top-k dual~norms and k-support dual~norms (that we formally introduce)
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Hidden Convexity in the l0 Pseudonorm
HAL CCSD, 2020Co-Authors: Chancelier Jean-philippe, De Lara MichelAbstract:The so-called l0 pseudonorm counts the number of nonzero components of a vector of a Euclidian space. It is well-known that the l0 pseudonorm is not convex, as its Fenchel biconjugate is zero. In this paper, we introduce a suitable conjugacy, induced by a novel coupling, E-Capra, that has the property of being constant along primal rays like the l0 pseudonorm. The coupling E-Capra belongs to the class of one-sided linear couplings, that we introduce; we show that they induce conjugacies that share nice properties with the classic Fenchel conjugacy. For the E-Capra conjugacy, induced by the coupling E-Capra, we relate the E-Capra conjugate and biconjugate of the l0 pseudonorm, the characteristic Functions of its level sets and the sequence of so-called top-k norms. In particular, we prove that the l0 pseudonorm is equal to its biconjugate: hence, the l0 pseudonorm is E-Capra-convex in the sense of generalized convexity. As a corollary, we show that there exists a proper convex Lower Semicontinuous Function such that this Function and the l0 pseudonorm coincide on the Euclidian unit sphere. This hidden convexity property is somewhat surprising as the l0 pseudonorm is a highly nonconvex Function of combinatorial nature. We provide different expressions for this proper convex Lower Semicontinuous Function, and we give explicit formulas in the two-dimensional case
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Hidden Convexity in the l0 Pseudonorm
2019Co-Authors: Chancelier Jean-philippe, De Lara MichelAbstract:The so-called l0 pseudonorm on R d counts the number of nonzero components of a vector. It is well-known that the l0 pseudonorm is not convex, as its Fenchel biconjugate is zero. In this paper, we introduce a suitable conjugacy, induced by a novel coupling, Caprac, having the property of being constant along primal rays, like the l0 pseudonorm. The coupling Caprac belongs to the class of one-sided linear couplings, that we introduce. We show that they induce conjugacies that share nice properties with the classic Fenchel conjugacy. For the Caprac conjugacy, induced by the coupling Caprac, we prove that the l0 pseudonorm is equal to its biconjugate: hence, the l0 pseudonorm is Caprac-convex in the sense of generalized convexity. We also provide expressions for conjugates in terms of two families of dual norms, the 2-k-symmetric gauge norms and the k-support norms. As a corollary, we show that the l0 pseudonorm coincides, on the sphere, with a proper convex Lower Semicontinuous Function-that we characterize, and for which we give explicit formulas in the two dimensional case. This is somewhat surprising as the l0 pseudonorm is a highly nonconvex Function of combinatorial nature.Comment: arXiv admin note: substantial text overlap with arXiv:1902.0481
Hedy Attouch - One of the best experts on this subject based on the ideXlab platform.
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Continuous Newton-like Inertial Dynamics for Monotone Inclusions
Set-Valued and Variational Analysis, 2020Co-Authors: Hedy Attouch, Szilárd Csaba LászlóAbstract:In a Hilbert framework ℌ, we study the convergence properties of a Newton-like inertial dynamical system governed by a general maximally monotone operator A ℌ: → 2^ℌ. When A is equal to the subdifferential of a convex Lower Semicontinuous proper Function, the dynamic corresponds to the introduction of the Hessian-driven damping in the continuous version of the accelerated gradient method of Nesterov. As a result, the oscillations are significantly attenuated. According to the technique introduced by Attouch-Peypouquet (Math. Prog. 2019), the maximally monotone operator is replaced by its Yosida approximation with an appropriate adjustment of the regularization parameter. The introduction into the dynamic of the Newton-like correction term (corresponding to the Hessian driven term in the case of convex minimization) provides a well-posed evolution system for which we will obtain the weak convergence of the generated trajectories towards the zeroes of A . We also obtain the fast convergence of the velocities towards zero. The results tolerate the presence of errors, perturbations. Then, we specialize our results to the case where the operator A is the subdifferential of a convex Lower Semicontinuous Function, and obtain fast optimization results.
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Fast convex optimization via a third-order in time evolution equation: TOGES-V an improved version of TOGES
2020Co-Authors: Hedy Attouch, Zaki Chbani, Hassan RiahiAbstract:In a Hilbert space setting H, for convex optimization, we analyze the fast convergence properties as t → +∞ of the trajectories t → u(t) ∈ H generated by a third-order in time evolution system. The Function f : H → R to minimize is supposed to be convex, continuously differentiable, with argmin H f = ∅. It enters into the dynamic through its gradient. Based on this new dynamical system, we improve the results obtained by [Attouch, Chbani, Riahi: Fast convex optimization via a third-order in time evolution equation, Optimization 2020]. As a main result, when the damping parameter α satisfies α > 3, we show that f (u(t)) − inf H f = o 1/t 3 as t → +∞, as well as the convergence of the trajectories. We complement these results by introducing into the dynamic an Hessian driven damping term, which reduces the oscillations. In the case of a strongly convex Function f , we show an autonomous evolution system of the third order in time with an exponential rate of convergence. All these results have natural extensions to the case of a convex Lower Semicontinuous Function f : H → R ∪ {+∞}. Just replace f with its Moreau envelope.
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dynamical systems and forward backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator
Optimization, 2015Co-Authors: Boushra Abbas, Hedy AttouchAbstract:In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators , where is the subdifferential of a convex Lower Semicontinuous Function , and is a monotone cocoercive operator. We first consider the extension to this setting of the regularized Newton dynamic with two potentials which was considered in Abbas, Attouch, Svaiter JOTA, 2014. Then, we revisit some related dynamical systems, namely the semigroup of contractions generated by , and the continuous gradient projection dynamic. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems, thereby extending the known results. The time discretization of these dynamics gives various forward–backward splitting methods (some new) for solving structured monotone inclusions involving non-potential terms. The convergence of these algorithms is obtained under classical step size limitation. Perspectives are given in the field of numerical spl...
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dynamical systems and forward backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator
arXiv: Optimization and Control, 2014Co-Authors: Boushra Abbas, Hedy AttouchAbstract:In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators $A=\partial\Phi+B$, where $\partial\Phi$ is the subdifferential of a convex Lower Semicontinuous Function $\Phi$, and $B$ is a monotone cocoercive operator. We first consider the extension to this setting of the regularized Newton dynamic with two potentials. Then, we revisit some related dynamical systems, namely the semigroup of contractions generated by $A$, and the continuous gradient projection dynamic. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems. The time discretization of these dynamics gives various forward-backward splitting methods (some new) for solving structured monotone inclusions involving non-potential terms. The convergence of these algorithms is obtained under classical step size limitation. Perspectives are given in the field of numerical splitting methods for optimization, and multi-criteria decision processes.
