Using Riemannian geometry to obtain new results on Dikin and Karmarkar methods
We are motivated by a 1990 Karmarkar paper on Riemannian geometry and Interior Point Methods. In this talk we show 3 results. (1) Karmarkar direction can be derived from the Dikin one. This is obtained by constructing a certain Z(x) representation of the null space of the unitary simplex (e, x) = 1; then the projective direction is the image under Z(x) of the affine-scaling one, when it is restricted to that simplex. (2) Second order information on Dikin and Karmarkar methods. We establish computable Hessians for each of the metrics corresponding to both directions, thus permitting the generation of {open_quotes}second order{close_quotes} methods. (3) Dikin and Karmarkar geodesic descent methods. For those directions, we make computable the theoretical Luenberger geodesic descent method, since we are able to explicit very accurate expressions of the corresponding geodesics. Convergence results are given.
- OSTI ID:
- 36352
- Report Number(s):
- CONF-9408161-; TRN: 94:009753-0692
- Resource Relation:
- Conference: 15. international symposium on mathematical programming, Ann Arbor, MI (United States), 15-19 Aug 1994; Other Information: PBD: 1994; Related Information: Is Part Of Mathematical programming: State of the art 1994; Birge, J.R.; Murty, K.G. [eds.]; PB: 312 p.
- Country of Publication:
- United States
- Language:
- English
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