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Xianfu Wang - One of the best experts on this subject based on the ideXlab platform.
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Restricted Normal Cones and Sparsity Optimization with Affine Constraints
Foundations of Computational Mathematics, 2014Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely behaved nonconvex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted Normal Cone which generalizes the classical Mordukhovich Normal Cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted Normal Cone together with the notion of superregularity, which is inherently satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint.
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Restricted Normal Cones and the Method of Alternating Projections: Applications
Set-Valued and Variational Analysis, 2013Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, Luke, and Malick derived local linear convergence results provided that a condition involving Normal Cones holds and at least one of the sets is superregular (a property less restrictive than convexity). However, their results failed to capture very simple classical convex instances such as two lines in a three-dimensional space. In this paper, we extend and develop the Lewis-Luke-Malick framework so that not only any two linear subspaces but also any two closed convex sets whose relative interiors meet are covered. We also allow for sets that are more structured such as unions of convex sets. The key tool required is the restricted Normal Cone, which is a generalization of the classical Mordukhovich Normal Cone. Numerous examples are provided to illustrate the theory.
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Restricted Normal Cones and sparsity optimization with affine constraints
arXiv: Optimization and Control, 2012Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved non convex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted Normal Cone which generalizes the classical Mordukhovich Normal Cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted Normal Cone together with the notion of superregularity, which is naturally satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint.
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Restricted Normal Cones and the method of alternating projections
arXiv: Functional Analysis, 2012Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, Luke, and Malick derived local linear convergence results provided that a condition involving Normal Cones holds and at least one of the sets is superregular (a property less restrictive than convexity). However, their results failed to capture very simple classical convex instances such as two lines in three-dimensional space. In this paper, we extend and develop the Lewis-Luke-Malick framework so that not only any two linear subspaces but also any two closed convex sets whose relative interiors meet are covered. We also allow for sets that are more structured such as unions of convex sets. The key tool required is the restricted Normal Cone, which is a generalization of the classical Mordukhovich Normal Cone. We thoroughly study restricted Normal Cones from the viewpoint of constraint qualifications and regularity. Numerous examples are provided to illustrate the theory.
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an answer to s simons question on the maximal monotonicity of the sum of a maximal monotone linear operator and a Normal Cone operator
arXiv: Functional Analysis, 2009Co-Authors: Heinz H Bauschke, Xianfu WangAbstract:The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar's constraint qualification - that is, whether or not "the sum theorem" is true - is the most famous open problem in Monotone Operator Theory. In his 2008 monograph "From Hahn-Banach to Monotonicity", Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and a Normal Cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the domain of the linear operator. In this note, we provide an affirmative answer to Simons' question. In fact, we show that the sum theorem is true for a maximal monotone linear relation and a Normal Cone operator. The proof relies on Rockafellar's formula for the Fenchel conjugate of the sum as well as some results featuring the Fitzpatrick function.
D. Russell Luke - One of the best experts on this subject based on the ideXlab platform.
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Tangent and Normal Cones for Low-Rank Matrices
Nonsmooth Optimization and Its Applications, 2019Co-Authors: Seyedehsomayeh Hosseini, D. Russell Luke, André UschmajewAbstract:In (D. R. Luke, J. Math. Imaging Vision, 47 (2013), 231–238) the structure of the Mordukhovich Normal Cone to varieties of low-rank matrices at rank-deficient points has been determined. A simplified proof of that result is presented here. As a corollary we obtain the corresponding Clarke Normal Cone. The results are put into the context of first-order optimality conditions for low-rank matrix optimization problems.
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Restricted Normal Cones and Sparsity Optimization with Affine Constraints
Foundations of Computational Mathematics, 2014Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely behaved nonconvex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted Normal Cone which generalizes the classical Mordukhovich Normal Cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted Normal Cone together with the notion of superregularity, which is inherently satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint.
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Prox-Regularity of Rank Constraint Sets and Implications for Algorithms
Journal of Mathematical Imaging and Vision, 2013Co-Authors: D. Russell LukeAbstract:We present an analysis of sets of matrices with rank less than or equal to a specified number s . We provide a simple formula for the Normal Cone to such sets, and use this to show that these sets are prox-regular at all points with rank exactly equal to s . The Normal Cone formula appears to be new. This allows for easy application of prior results guaranteeing local linear convergence of the fundamental alternating projection algorithm between sets, one of which is a rank constraint set. We apply this to show local linear convergence of another fundamental algorithm, approximate steepest descent. Our results apply not only to linear systems with rank constraints, as has been treated extensively in the literature, but also nonconvex systems with rank constraints.
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Restricted Normal Cones and the Method of Alternating Projections: Applications
Set-Valued and Variational Analysis, 2013Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, Luke, and Malick derived local linear convergence results provided that a condition involving Normal Cones holds and at least one of the sets is superregular (a property less restrictive than convexity). However, their results failed to capture very simple classical convex instances such as two lines in a three-dimensional space. In this paper, we extend and develop the Lewis-Luke-Malick framework so that not only any two linear subspaces but also any two closed convex sets whose relative interiors meet are covered. We also allow for sets that are more structured such as unions of convex sets. The key tool required is the restricted Normal Cone, which is a generalization of the classical Mordukhovich Normal Cone. Numerous examples are provided to illustrate the theory.
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Restricted Normal Cones and sparsity optimization with affine constraints
arXiv: Optimization and Control, 2012Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved non convex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted Normal Cone which generalizes the classical Mordukhovich Normal Cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted Normal Cone together with the notion of superregularity, which is naturally satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint.
Heinz H Bauschke - One of the best experts on this subject based on the ideXlab platform.
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Restricted Normal Cones and Sparsity Optimization with Affine Constraints
Foundations of Computational Mathematics, 2014Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely behaved nonconvex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted Normal Cone which generalizes the classical Mordukhovich Normal Cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted Normal Cone together with the notion of superregularity, which is inherently satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint.
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Restricted Normal Cones and the Method of Alternating Projections: Applications
Set-Valued and Variational Analysis, 2013Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, Luke, and Malick derived local linear convergence results provided that a condition involving Normal Cones holds and at least one of the sets is superregular (a property less restrictive than convexity). However, their results failed to capture very simple classical convex instances such as two lines in a three-dimensional space. In this paper, we extend and develop the Lewis-Luke-Malick framework so that not only any two linear subspaces but also any two closed convex sets whose relative interiors meet are covered. We also allow for sets that are more structured such as unions of convex sets. The key tool required is the restricted Normal Cone, which is a generalization of the classical Mordukhovich Normal Cone. Numerous examples are provided to illustrate the theory.
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Restricted Normal Cones and sparsity optimization with affine constraints
arXiv: Optimization and Control, 2012Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved non convex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted Normal Cone which generalizes the classical Mordukhovich Normal Cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted Normal Cone together with the notion of superregularity, which is naturally satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint.
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Restricted Normal Cones and the method of alternating projections
arXiv: Functional Analysis, 2012Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, Luke, and Malick derived local linear convergence results provided that a condition involving Normal Cones holds and at least one of the sets is superregular (a property less restrictive than convexity). However, their results failed to capture very simple classical convex instances such as two lines in three-dimensional space. In this paper, we extend and develop the Lewis-Luke-Malick framework so that not only any two linear subspaces but also any two closed convex sets whose relative interiors meet are covered. We also allow for sets that are more structured such as unions of convex sets. The key tool required is the restricted Normal Cone, which is a generalization of the classical Mordukhovich Normal Cone. We thoroughly study restricted Normal Cones from the viewpoint of constraint qualifications and regularity. Numerous examples are provided to illustrate the theory.
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an answer to s simons question on the maximal monotonicity of the sum of a maximal monotone linear operator and a Normal Cone operator
arXiv: Functional Analysis, 2009Co-Authors: Heinz H Bauschke, Xianfu WangAbstract:The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar's constraint qualification - that is, whether or not "the sum theorem" is true - is the most famous open problem in Monotone Operator Theory. In his 2008 monograph "From Hahn-Banach to Monotonicity", Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and a Normal Cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the domain of the linear operator. In this note, we provide an affirmative answer to Simons' question. In fact, we show that the sum theorem is true for a maximal monotone linear relation and a Normal Cone operator. The proof relies on Rockafellar's formula for the Fenchel conjugate of the sum as well as some results featuring the Fitzpatrick function.
M. Ali Khan - One of the best experts on this subject based on the ideXlab platform.
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The Mordukhovich Normal Cone and the Foundations of Welfare Economics
Journal of Public Economic Theory, 1999Co-Authors: M. Ali KhanAbstract:The statement that Pareto optimal allocations require the equalization of marginal rates of substitution, or in an economy with public goods, require the equalization of the aggregate of the marginal rates in consumption to those in production, is formalized through the use of the Mordukhovich Normal Cone. Since this Cone is strictly contained, in general, in the Clarke Normal Cone, the results generalize earlier work of Khan and Vohra, Quinzii, Yun, and Cornet. The results are an application of Mordukhovich's 1980 theorem on necessary conditions for optimality in constrained optimization problems involving functions that are not necessarily differentiable or quasi-concave. As such, the results suggest a distinction between the mathematical programming approach to the “second welfare theorem,” as in the work of Hicks, Lange, and Samuelson, and that based on the separation of sets, as pioneered by Arrow and Debreu.
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Ioffe's Normal Cone and the foundations of welfare economics: The infinite dimensional theory
Journal of Mathematical Analysis and Applications, 1991Co-Authors: M. Ali KhanAbstract:Abstract We establish the relevance of Ioffe's Normal Cone for basic theorems of welfare economics in the context of a commodity space formalized as a vactor lattice and endowed with a locally convex linear topology in which the positive Cone is closed.
Hung M. Phan - One of the best experts on this subject based on the ideXlab platform.
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Restricted Normal Cones and Sparsity Optimization with Affine Constraints
Foundations of Computational Mathematics, 2014Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely behaved nonconvex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted Normal Cone which generalizes the classical Mordukhovich Normal Cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted Normal Cone together with the notion of superregularity, which is inherently satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint.
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Restricted Normal Cones and the Method of Alternating Projections: Applications
Set-Valued and Variational Analysis, 2013Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, Luke, and Malick derived local linear convergence results provided that a condition involving Normal Cones holds and at least one of the sets is superregular (a property less restrictive than convexity). However, their results failed to capture very simple classical convex instances such as two lines in a three-dimensional space. In this paper, we extend and develop the Lewis-Luke-Malick framework so that not only any two linear subspaces but also any two closed convex sets whose relative interiors meet are covered. We also allow for sets that are more structured such as unions of convex sets. The key tool required is the restricted Normal Cone, which is a generalization of the classical Mordukhovich Normal Cone. Numerous examples are provided to illustrate the theory.
-
Restricted Normal Cones and sparsity optimization with affine constraints
arXiv: Optimization and Control, 2012Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved non convex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted Normal Cone which generalizes the classical Mordukhovich Normal Cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted Normal Cone together with the notion of superregularity, which is naturally satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint.
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Restricted Normal Cones and the method of alternating projections
arXiv: Functional Analysis, 2012Co-Authors: Heinz H Bauschke, D. Russell Luke, Hung M. Phan, Xianfu WangAbstract:The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, Luke, and Malick derived local linear convergence results provided that a condition involving Normal Cones holds and at least one of the sets is superregular (a property less restrictive than convexity). However, their results failed to capture very simple classical convex instances such as two lines in three-dimensional space. In this paper, we extend and develop the Lewis-Luke-Malick framework so that not only any two linear subspaces but also any two closed convex sets whose relative interiors meet are covered. We also allow for sets that are more structured such as unions of convex sets. The key tool required is the restricted Normal Cone, which is a generalization of the classical Mordukhovich Normal Cone. We thoroughly study restricted Normal Cones from the viewpoint of constraint qualifications and regularity. Numerous examples are provided to illustrate the theory.
