Geodesic Convexity on R + n

Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities

Mathematics ◽

10.3390/math8122196 ◽

2020 ◽

Vol 8 (12) ◽

pp. 2196

Author(s):

Gabriel Ruiz-Garzón ◽

Rafaela Osuna-Gómez ◽

Antonio Rufián-Lizana ◽

Beatriz Hernández-Jiménez

Keyword(s):

Vector Optimization ◽

Optimization Problem ◽

Equilibrium Points ◽

Variational Problems ◽

Efficient Solutions ◽

Vector Optimization Problem ◽

Stackelberg Equilibrium ◽

Hadamard Manifolds ◽

Geodesic Convexity ◽

Weakly Efficient Solutions

This article has two objectives. Firstly, we use the vector variational-like inequalities problems to achieve local approximate (weakly) efficient solutions of the vector optimization problem within the novel field of the Hadamard manifolds. Previously, we introduced the concepts of generalized approximate geodesic convex functions and illustrated them with examples. We see the minimum requirements under which critical points, solutions of Stampacchia, and Minty weak variational-like inequalities and local approximate weakly efficient solutions can be identified, extending previous results from the literature for linear Euclidean spaces. Secondly, we show an economical application, again using solutions of the variational problems to identify Stackelberg equilibrium points on Hadamard manifolds and under geodesic convexity assumptions.

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Geodesic convexity on Rn1+

Optimization ◽

10.1080/02331939608844226 ◽

1996 ◽

Vol 37 (4) ◽

pp. 341-355 ◽

Cited By ~ 3

Author(s):

T. Rapcsak

Keyword(s):

Geodesic Convexity

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Generalized Geodesic Convexity on Riemannian Manifolds

Mathematics ◽

10.3390/math7060547 ◽

2019 ◽

Vol 7 (6) ◽

pp. 547 ◽

Cited By ~ 1

Author(s):

Izhar Ahmad ◽

Meraj Ali Khan ◽

Amira A. Ishan

Keyword(s):

Riemannian Manifold ◽

Riemannian Manifolds ◽

Global Minimum ◽

Mean Value ◽

Mathematical Programming Problem ◽

Hadamard Manifolds ◽

Geodesic Convexity ◽

Invex Functions ◽

Mean Value Inequality ◽

Global Minimum Point

We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and relationships of these functions are discussed. A characterization for the existence of a global minimum point of a mathematical programming problem is presented. Moreover, a mean value inequality under geodesic log-preinvexity is extended to Cartan-Hadamard manifolds.

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A Helly theorem for geodesic convexity in strongly dismantlable graphs

Discrete Mathematics ◽

10.1016/0012-365x(93)e0178-7 ◽

1995 ◽

Vol 140 (1-3) ◽

pp. 119-127 ◽

Cited By ~ 11

Author(s):

Norbert Polat

Keyword(s):

Geodesic Convexity ◽

Helly Theorem

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Helly and exchange numbers of geodesic and Steiner convexities in lexicographic product of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830915500494 ◽

2015 ◽

Vol 07 (04) ◽

pp. 1550049 ◽

Cited By ~ 3

Author(s):

Bijo S. Anand ◽

Manoj Changat ◽

Prasanth G. Narasimha-Shenoi

Keyword(s):

Convex Sets ◽

Lexicographic Product ◽

Geodesic Convexity ◽

Induced Path ◽

Arbitrary Graphs ◽

Geodesic Convex ◽

Product Of Graphs ◽

The Way

We discuss the convexity invariants, namely, the exchange and Helly numbers of the Steiner and geodesic convexity in lexicographic product of graphs. We use the structure of both the Steiner and geodesic convex sets in the lexicographic product for proving the results. Along the way the exchange number of the induced path convexity in arbitrary graphs is also determined.

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