Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds

Azagra, Daniel; Ferrera, Juan; López-Mesas, Fernando

doi:10.1016/j.jfa.2004.10.008

Cited by 133 publications

(135 citation statements)

References 36 publications

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“…In Section 2 we recall a Fan-type variational inequality proved by McClendon [10] concerning compact acyclic ANR sets, which is the main tool in the proof of Theorems 1.1 and 1.3. In order to prove Theorem 1.3, we also recall some basic elements from non-smooth analysis on Riemannian manifolds developed by Azagra, Ferrera and Lópes-Mesas [2]. In Section 3 we prove our results, while in Section 4 we give two applications which illustrate the applicability of the developed method.…”

Section: Proposition 12 Any Nash Equilibrium Point Formentioning

confidence: 94%

Location of Nash equilibria: A Riemannian geometrical approach

Kristály

2009

Proc. Amer. Math. Soc.

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Abstract. Existence and location of Nash equilibrium points are studied for a large class of a finite family of payoff functions whose domains are not necessarily convex in the usual sense. The geometric idea is to embed these non-convex domains into suitable Riemannian manifolds regaining certain geodesic convexity properties of them. By using recent non-smooth analysis on Riemannian manifolds and a variational inequality for acyclic sets, an efficient location result of Nash equilibrium points is given. Some examples show the applicability of our results.

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Section: Proposition 12 Any Nash Equilibrium Point Formentioning

confidence: 94%

Location of Nash equilibria: A Riemannian geometrical approach

Kristály

2009

Proc. Amer. Math. Soc.

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“…On the other hand, the maximal monotonicity notion in Banach spaces extended to Riemannian manifolds makes it possible to develop a proximal-type method to find singular points for multivalued vector fields on Riemannian manifolds with nonpositive sectional curvatures, i.e., on Hadamard manifolds; see, e.g., [44] with other references. Furthermore, various derivative-like and subdifferential constructions for nondifferentiable functions on spaces with no linear structure are developed in [3,5,28,39,46,47,51] and applied therein to the study of constrained optimization problems, nonclassical problems of the calculus of variations and optimal control, and generalized solutions to the first-order partial differential equations on Riemannian manifolds and other important classes of spaces with no linearity. This paper is devoted to the study of weak sharp minimizers for constrained optimization problems on Riemannian and Hadamard manifolds.…”

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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“…For instance, he proves that if noncompact initial data Γ 0 are allowed then one loses uniqueness of the generalized geometric evolution. In recent years, an interest has grown in the use of viscosity solutions of (first order) Hamilton-Jacobi equations defined on Riemannian manifolds (in relation to dynamical systems, to geometric problems, or from a theoretical point of view), see [32,11,12,4,31,10,23], but no second order theory, apart from Ilmanen's paper, has apparently been developed for parabolic equations (in the case of stationary, degenerate elliptic equations, such a study was recently started in [5]). …”

Section: Introductionmentioning

confidence: 99%

Generalized motion of level sets by functions of their curvatures on Riemannian manifolds

2008

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Abstract. We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M . The level sets of a function u : M → R evolve in such a way whenever u solves an equation ut + F (Du, D 2 u) = 0, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation under the assumptions that M has nonnegative curvature, F is continuous off {Du = 0}, (degenerate) elliptic, and locally invariant by parallel translation. We then prove that this approach is geometrically consistent, hence it allows to define a generalized evolution of level sets by very general, singular functions of their curvatures. For instance, these assumptions on F are satisfied when F is given by the evolutions of level sets by their mean curvature (even in arbitrary codimension) or by their positive Gaussian curvature. We also prove that the generalized evolution is consistent with the classical motion by the corresponding function of the curvature, whenever the latter exists. When M is not of nonnegative curvature, the same results hold if one additionally requires that F is uniformly continuous with respect to D 2 u. Finally we give some counterexamples showing that several well known properties of the evolutions in R n are no longer true when M has negative sectional curvature.

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Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds

Cited by 133 publications

References 36 publications

Location of Nash equilibria: A Riemannian geometrical approach

Location of Nash equilibria: A Riemannian geometrical approach

Weak Sharp Minima on Riemannian Manifolds

Generalized motion of level sets by functions of their curvatures on Riemannian manifolds

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