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306:. Nonconvex Optimization and its Applications. Vol. 19. Dordrecht: Kluwer Academic Publishers.
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40:. It is common to drop the prefix "geodesic" and refer simply to "convexity" of a set or function.
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is a (strictly) convex function in the usual sense for every unit speed geodesic arc
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with its usual metric is geodesically convex. However, the subset
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A geodesically convex (subset of a) Riemannian manifold is also a
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Convex functions and optimization methods on
Riemannian manifolds
279:) arc joining two distinct points on the southern boundary of
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it is convex in the usual sense, and similarly for functions.
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The "northern hemisphere" of the 2-dimensional sphere
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geodesically convex, since the minimizing geodesic (
248:with its usual flat metric is geodesically convex
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291:, the geodesic arc passes over the south pole).
188:{\displaystyle f\circ \gamma :\to \mathbf {R} }
287:(e.g. in the case of two points 180° apart in
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227:with respect to the geodesic distance.
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122:{\displaystyle f:C\to \mathbf {R} }
91:be a geodesically convex subset of
304:Smooth nonlinear optimization in R
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271:further north than 45° south is
267:consisting of those points with
205: : →
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76:, there is a unique minimizing
28:is a natural generalization of
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331:Udriste, Constantin (1994).
135:geodesically convex function
84:that joins those two points.
72:if, given any two points in
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20:— specifically, in
302:Rapcsák, Tamás (1997).
70:geodesically convex set
370:Geodesic (mathematics)
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365:Riemannian manifolds
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38:Riemannian manifolds
360:Convex optimization
225:convex metric space
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137:if the composition
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30:convexity for sets
26:geodesic convexity
209:contained within
129:is said to be a (
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237:A subset of
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44:Definitions
18:mathematics
354:Categories
296:References
218:Properties
289:longitude
178:→
157:γ
154:∘
112:→
60:A subset
34:functions
269:latitude
232:Examples
131:strictly
78:geodesic
24:—
322:1480415
283:leaves
52:,
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203:γ
48:Let (
337:ISBN
308:ISBN
87:Let
32:and
273:not
263:of
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36:to
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318:MR
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285:A
281:A
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261:A
257:S
246:E
239:n
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211:C
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182:R
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172:T
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166:0
163:[
160::
151:f
116:R
109:C
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103:f
93:M
89:C
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54:g
50:M
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