Invariant monotone vector fields on Riemannian manifolds

Barani, A.; Pouryayevali, Mohamad R.

doi:10.1016/j.na.2008.02.085

Cited by 31 publications

(15 citation statements)

References 16 publications

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“…In the past few years, a number of results have been obtained on numerous aspects of nonsmooth analysis and their applications on Riemannian manifolds; see, e.g. [3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds

Hosseini

Pouryayevali

2011

Nonlinear Analysis: Theory, Methods & Applications

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“…In the past few years, a number of results have been obtained on numerous aspects of nonsmooth analysis and their applications on Riemannian manifolds; see, e.g. [3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds

Hosseini

Pouryayevali

2011

Nonlinear Analysis: Theory, Methods & Applications

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“…That is, the function of time γ (t), X(γ(t)) G must be increasing faster than linearly. We refer the reader to [43,44,10,45] for further details. Since we are more interested in stable systems, we will in fact reverse this definition and say that X is strongly monotone if there exists a λ > 0 such that ϕ(t) γ (t), X(γ(t)) G + λt γ (0), γ (0) G is monotone decreasing.…”

Section: Connections To Contraction Theory In Mathematics and Physicsmentioning

confidence: 99%

Contraction theory on Riemannian manifolds

Simpson-Porco

Bullo

2014

Systems & Control Letters

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Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. In this work, we present the fundamental results of contraction theory in an intrinsic, coordinate-free setting. The presentation highlights both the underlying geometric foundations of contraction theory, and the coordinate invariance of the resulting stability properties. We provide coordinate-free proofs of the main results for autonomous vector fields, and clarify the assumptions under which the results hold. In addition, we state and prove several interesting corollaries, study cascade and feedback interconnections, and highlight how contraction theory has arisen independently in other scientific disciplines. We conclude by illustrating the developed theory for the case of gradient dynamics.

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“…Among various problems of this type we mention geometric models for human spine [2], eigenvalue optimization problems [15,45,58], nonconvex and nonsmooth problems of constrained optimization in JRn that can be reduced to convex and smooth unconstrained optimization problems on Riemannian manifolds as in [19,27,48,54,59], etc. We refer the reader to [2,6,24,33,45,49,58,59] and the bibliographies therein for more examples and discussions.…”

Section: Introductionmentioning

confidence: 99%

“…The seminal Ekeland's paper [25] contains applications of his variational principle to the existence of minimal geodesics on Riemannian manifolds; see also [26] for further developments. More recently, a number of important results have been obtained on various aspects of optimization theory and applications for problems formulated on Riemannian and Hadamard manifolds as well as on other spaces with nonlinear structures; see, e.g., [1,2,9,6,20,24,29,33,43,44,45,58,59] and the references therein. Let us particularly mention Newton's method, the conjugate gradient method, the trust-region method, and their modifications extended from optimization problems on linear spaces to their Riemannian counterparts.…”

Section: Introductionmentioning

confidence: 99%

Weak Sharp Minima on Riemannian Manifolds

Li¹,

Mordukhovich²,

Wang³

et al. 2011

SIAM J. Optim.

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Abstract. This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and obtain their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and their Hadamard counterparts. A number of the results obtained in this paper are also new for the case of conventional problems in linear spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper.Key words. Variational analysis and optimization, Weak sharp minima, Riemannian manifolds, Hadamard manifolds, Convexity, Generalized differentiability AMS subject classifications. Primary 49J52; Secondary 90C31 1. Introduction. A vast majority of problems considered in optimization theory are formulated in finite-dimensional or infinite-dimensional Banach spaces, where the linear structure plays a crucial role to employ conventional tools of variational analysis and (classical or generalized) differentiation to deriving optimality conditions and then develop numerical algorithms. At the same time many optimization problems arising in various applications cannot be posted in linear spaces and require a Riemannian manifold (in particular, a Hadamard manifold) structure for their formalization and study. Among various problems of this type we mention geometric models for human spine [2], eigenvalue optimization problems [15,45,58],

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Invariant monotone vector fields on Riemannian manifolds

Cited by 31 publications

References 16 publications

Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds

Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds

Contraction theory on Riemannian manifolds

Weak Sharp Minima on Riemannian Manifolds

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