Nonsmooth Riemannian Optimization with Applications to Sphere Packing and Grasping

Abstract

Summary. This paper presents a survey on Riemannian geometry methods for smooth and nonsmooth constrained optimization. Gradient and subgradient descent algorithms on a Riemannian manifold are discussed. We illustrate the methods by applications from robotics and multi antenna communication. Gradient descent algorithms for dextrous hand grasping and for sphere packing problems on Grassmann manifolds are presented respectively.

ε-subgradient algorithms for locally lipschitz functions on Riemannian manifolds

“…The main result is the global convergence property of our minimization algorithm which is stated in Theorem 3.8. Moreover, comparing with subgradient algorithm [16], the ε-subgradient algorithm is much more general, because in this algorithm we do not need to have an explicit formula for the subdifferential and it can be computed approximately. An implementation of our proposed minimization algorithm is given in Matlab environment and tested on some problems.…”

“…In Table 1, we illustrate the results of the nonsmooth subgradient (SB) method and our method for the sphere packing in Gr (16,2) with m = 10 after 80 iterations with the same arbitrary starting points for both methods and the same step lengths. The computation time for both methods is the same.…”

“…The problem (4.4) is equivalent to the minimizing the following nonsmooth function; on Gr(n, k) × ... × Gr(n, k); see [16].…”

“…Then we would like to find m points P 1 , ..., P m in Gr(n, k) such that max r|∀i = j : B(P i , r) ∩ B(P j , r) = ∅ , (4.4) is maximized. This problem has been solved in [16] using a subgradient method. Indeed, Gr(n, k) can be identified with the set {P ∈ S(n)|P 2 = P , tr(P ) = k}.…”

A Riemannian subgradient algorithm for economic dispatch with valve-point effect

“…[DHL07], a contribution of our development is to incorporate bound constraints in order to handle the generator capacity constraints (2) present in the ELDP. Another contribution of this work is that, whereas many heuristic algorithms for the ELDP consist of (a combination of) existing black-box optimization techniques, the proposed method strives to exploit as much as possible the very particular structure of the ELDP.…”