Convex Functions and Optimization Methods on Riemannian by Constantin Udrişte (auth.)

By Constantin Udrişte (auth.)

The item of this publication is to provide the elemental evidence of convex capabilities, typical dynamical structures, descent numerical algorithms and a few machine courses on Riemannian manifolds in a sort appropriate for utilized mathematicians, scientists and engineers. It includes mathematical info on those matters and purposes disbursed in seven chapters whose subject matters are with regards to my very own components of analysis: Metric houses of Riemannian manifolds, First and moment diversifications of the p-energy of a curve; Convex services on Riemannian manifolds; Geometric examples of convex features; Flows, convexity and energies; Semidefinite Hessians and purposes; Minimization of services on Riemannian manifolds. all of the numerical algorithms, machine courses and the appendices (Riemannian convexity of features f:R ~ R, Descent tools at the Poincare aircraft, Descent tools at the sphere, Completeness and convexity on Finsler manifolds) represent an try to make accesible to all clients of this publication a few simple computational options and implementation of geometric constructions. To additional relief the readers,this e-book additionally incorporates a a part of the folklore approximately Riemannian geometry, convex services and dynamical structures since it is sadly "nowhere" to be present in an analogous context; current textbooks on convex capabilities on Euclidean areas or on dynamical structures don't point out what occurs in Riemannian geometry, whereas the papers facing Riemannian manifolds often stay away from discussing trouble-free proof. frequently a convex functionality on a Riemannian manifold is a true­ valued functionality whose restrict to each geodesic arc is convex.

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1) Let J y = r(l). Obviously be a Jacobi field JeT r Q and which vanishes at x = r(O), the second variation formula gives E .. e, J eN. p 2 2 p Conversely, suppose that W e N . Choose a division 0 1 =t < t p o l < ... First and second variations of the p-energy of a curve are COO We take W of the form 2 = f(t){IIVII 2 ( W (t) 2 where f(t) > 0 for u2 W U2 W __1 - R(V,W )v) + (P-2)g( V, _1 - R(V,W )v)v} , dt2 1 dt2 1 t ~ t 1 and =0 f(t) for t =ti. W,1 " ["V" - P-. W, IVlvll' • (P-21&[ o. dt It follows that 2 2 IIVII 2 [ U W1 _ R(V,W )Vl + (P-2)g[ V, U W1 - R(V,W )vlv 1 dt 2 dt2 1 =0 which (for peR - {0,1}) is equivalent to U2 W __1 _ R(V,W )V = dt2 1 o.

Then (M,g) is complete if (M,g) is complete and the tensor field g - g is positive semidefinite. 3. The Riemannian manifold (H,g) is complete if and only if there exists a proper COO function f: H R such that --7 g - df ® df is positive definite. Equivalently, a Riemannian manifold is complete if and only if it supports a proper COO function whose gradient has bounded length. Proof. Let (g - df ® df) > f: M --7 R be a proper o. 2 and of the relation g = (g - df ® df) + df ® df. Let Vf be the gradient of f.

3. 4. Corollary. If (M,g) is a Riemannian manifold and a COO proper function, then the Riemannian manifold (M, f:M g ~ ~ e is IIVfll2 ) g is complete. Proof. Let be - the gradient operators and the norms with respect to g and g respectively. Then IIVg fll~g = IIVg fll2 exp (- IIV f1l2) g g g ~ -1 e Remarks. 1) The existence of a proper function on M is obtained by the embedding theorem of Whitney which states that a smooth manifold M can be embedded as a closed submanifold of ~k (k sufficiently large).

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