Geodesic convexity in nonlinear optimization

Rapcsák, Tamás

doi:10.1007/bf00940467

Cited by 105 publications

(49 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the literature, many papers deal with the problem of generalizing usual convexity from different points of view: c-spaces [7], simplicial convexity [2], geodesic convexity [20], L-convexity [1] or convexity induced by an order [9] are some of the generalizations dealt with.…”

Section: Introductionmentioning

confidence: 99%

Abstract Convexity, Some Relations and Applications

Llinares

2002

Optimization

View full text Add to dashboard Cite

The aim of this article is to analyze the relationship between various notions of abstract convexity structures that we find in the literature, in connection with the problem of the existence of continuous selections and fixed points of correspondences. We focus mainly on the notion of mc-spaces, which was introduced in [J. V. LLinares (1998). Unified treatment of the problem of the existence of maximal elements in binary relations: a characterization. Journal of Mathematical Economics, 29, 285-302], and its relationship with c-spaces [Ch.D. Horvath (1991). Contractibility and generalized convexity. Journal of Mathematical Analysis and Applications, 156, 341-357], simplicial convexity [R. Bielawski (1987). Simplicial convexity and its applications. Journal of Mathematical Analysis and Applications, 127, 155-171], order convexity (used in [Ch.D. Horvath and J.V. LLinares (1996). Maximal elements and fixed points for binary relations on topological ordered spaces. Journal of Mathematical Economics, 25, 291-306]), B 0 -simplicial convexity and L-spaces [H. Ben-El-Mechaiekh, S. Chebbi, M. Florenzano and J.V. LLinares (1998). Abstract convexity and fixed points. Journal of Mathematical Analysis and Applications, 222, 138-150]. Moreover, in the context of mc-spaces, a characterization result of nonempty finite intersection, in the line with the KnasterKuratowski-Mazurkiewicz Lemma, some consequences of it and some generalizations of Browder's existence of continuous selection and fixed point theorem are presented.

show abstract

Section: Introductionmentioning

confidence: 99%

Abstract Convexity, Some Relations and Applications

Llinares

2002

Optimization

View full text Add to dashboard Cite

show abstract

“…Definition 2.1 ( [12]). A subset K of M is said to be geodesic convex if and only if for any two points x, y ∈ K, the geodesic joining x to y is contained in K, that is, if γ : [0, 1] → M is a geodesic with x = γ(0) and y = γ(1), then γ(t) ∈ K, f or 0 ≤ t ≤ 1.…”

Section: Convexitymentioning

confidence: 99%

“…There are some advantages for a generalization of optimization methods from Euclidean spaces to Riemannian manifolds, because nonconvex and nonsmooth constrained optimization problems can be seen as convex and smooth unconstrained optimization problems from the Riemannian geometry point of view; see, for example, [15], [11], [12]. Nemeth [10] and Wang et al [16] studied monotone and accretive vector fields on Riemannian manifolds.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium and mixed equilibrium problems under weak monotonicity on Hadamard manifolds

2018

Commun. Optim. Theory

View full text Add to dashboard Cite

show abstract

“…For a metric-compatible connection, convexity is in fact determined by the choice of metric tensor (Rapcsák 1991): convexity is determined by the geodesics, which are determined by the connection, which is in turn determined by the metric tensor. Convexity is not, however, determined by the coordinate system used (Udrişte 1996a).…”

Section: Theoretical Background For Riemannian Optimizationmentioning

confidence: 99%

Riemannian optimization and multidisciplinary design optimization

Bakker

Parks

2016

Optim Eng

View full text Add to dashboard Cite

Riemannian Optimization (RO) generalizes standard optimization methods from Euclidean spaces to Riemannian manifolds. Multidisciplinary Design Optimization (MDO) problems exist on Riemannian manifolds, and with the differential geometry framework which we have previously developed, we can now apply RO techniques to MDO. Here, we provide background theory and a literature review for RO and give the necessary formulae to implement the Steepest Descent Method (SDM), Newton's Method (NM), and the Conjugate Gradient Method (CGM), in Riemannian form, on MDO problems. We then compare the performance of the Riemannian and Euclidean SDM, NM, and CGM algorithms on several test problems (including a satellite design problem from the MDO literature); we use a calculated step size, line search, and geodesic search in our comparisons. With the framework's induced metric, the RO algorithms are generally not as effective as their Euclidean counterparts, and line search is consistently better than geodesic search. In our post-experimental analysis, we also show how the optimization trajectories for the Riemannian SDM and CGM relate to design coupling and thereby provide some explanation for the observed optimization behaviour. This work is only a first step in applying RO to MDO, however, and the use of quasi-Newton methods and different metrics should be explored in future research.

show abstract

Geodesic convexity in nonlinear optimization

Cited by 105 publications

References 14 publications

Abstract Convexity, Some Relations and Applications

Abstract Convexity, Some Relations and Applications

Equilibrium and mixed equilibrium problems under weak monotonicity on Hadamard manifolds

Riemannian optimization and multidisciplinary design optimization

Contact Info

Product

Resources

About