Author details

Constantin Udriste<sup>1</sup> \*, Henri Bonnel<sup>2</sup> , Ionel Tevy1 and Ali Sapeeh Rasheed1

\*Address all correspondence to: udriste@mathem.pub.ro

1 Department of Mathematics and Informatics, Faculty of Applied Sciences, University Politehnica of Bucharest, Bucharest, Romania

2 ERIM, Université de la Nouvelle-Calédonie, Nouma Cédex, New Caledonia, France

### References

[1] Aké LA, Sánchez M. Compact affine manifolds with precompact holonomy are geodesically complete. Journal of Mathematical Analysis and Applications. 2016;436(2):1369-1371. DOI: 10.1016/j.jmaa.2015.12.037

[2] Antonelli PL, Ingarden RS, Matsumoto M. The Theory of Sparays and Finsler Spaces with Applications in Physics and Biology. Netherlands: Kluwer Academic Publishers, Springer; 1993. pp. 97-132. DOI: 10.1007/978-94-015-8194-3-4

On the other hand

134 Optimization Algorithms - Examples

dyϕð Þ¼ y dyf xy ð Þ¼ ð Þ; y

¼ � <sup>∂</sup><sup>2</sup>

Theorem 6.4 Let f : <sup>M</sup><sup>1</sup> � <sup>M</sup><sup>2</sup> ! <sup>R</sup> be a C<sup>2</sup> function and

If the set A ¼ f g ð Þ x yð Þ; y : y∈ M<sup>2</sup> is affine convex and f j

f xy ð Þ¼ ð Þ; y

\*, Henri Bonnel<sup>2</sup>

\*Address all correspondence to: udriste@mathem.pub.ro

Politehnica of Bucharest, Bucharest, Romania

10.1016/j.jmaa.2015.12.037

0 ≤ d<sup>2</sup>

Author details

Constantin Udriste<sup>1</sup>

References

Proof. Suppose <sup>f</sup> is a <sup>C</sup><sup>2</sup> function. At points ð Þ x yð Þ; <sup>y</sup> , we have

∂2 f ∂xi ∂xj ∂xi ∂y<sup>α</sup>

<sup>¼</sup> <sup>∂</sup><sup>2</sup> f ∂xi ∂y<sup>α</sup>

∂2 f ∂y<sup>α</sup>∂xi

∂xi ∂y<sup>β</sup> þ

f ∂xj ∂xi ∂xi ∂y<sup>β</sup>

d2 <sup>y</sup>ϕð Þ¼ y

∂f ∂xi ∂xi ∂y<sup>α</sup> þ

∂2 f ∂y<sup>α</sup>∂y<sup>β</sup>

<sup>ϕ</sup>ð Þ¼ <sup>y</sup> min<sup>x</sup> f xð Þ¼ ; <sup>y</sup> f xy ð Þ ð Þ; <sup>y</sup> :

∂xj <sup>∂</sup>y<sup>β</sup> <sup>þ</sup> <sup>2</sup> <sup>∂</sup><sup>2</sup>

∂xi ∂y<sup>β</sup> þ

1 Department of Mathematics and Informatics, Faculty of Applied Sciences, University

2 ERIM, Université de la Nouvelle-Calédonie, Nouma Cédex, New Caledonia, France

[1] Aké LA, Sánchez M. Compact affine manifolds with precompact holonomy are geodesically complete. Journal of Mathematical Analysis and Applications. 2016;436(2):1369-1371. DOI:

∂xj ∂y<sup>α</sup> þ

∂f ∂y<sup>α</sup>

dy<sup>α</sup>dy<sup>β</sup>

f ∂xi ∂y<sup>α</sup>

, Ionel Tevy1 and Ali Sapeeh Rasheed1

∂2 f ∂y<sup>α</sup>∂y<sup>β</sup>

∂xi ∂y<sup>β</sup> þ

dy<sup>α</sup>dy<sup>β</sup> <sup>¼</sup> <sup>d</sup><sup>2</sup>

∂2 f ∂y<sup>α</sup>∂y<sup>β</sup>

ϕð Þy :

dy<sup>α</sup> <sup>¼</sup> <sup>∂</sup><sup>f</sup>

dy<sup>α</sup>dy<sup>β</sup> ≤ 0:

<sup>∂</sup>y<sup>α</sup> dy<sup>α</sup>

<sup>A</sup> is affine convex, then ϕð Þy is affine convex.

dy<sup>α</sup>dy<sup>β</sup>

□

∂2 f ∂y<sup>α</sup>∂y<sup>β</sup>


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136 Optimization Algorithms - Examples
