Optimization method with constraints and apparatus therefor

Details Optimization method with constraints and apparatus therefor Optimization method with constraints and apparatus therefor

2007-03-27

2007-03-27

Vincent, David (Department: 2129)

Data processing: artificial intelligence

Neural network

Learning task

C706S062000, C706S919000, C708S003000, C708S008000, C708S160000, C708S200000, C708S207000

Reexamination Certificate

active

10192158

ABSTRACT:
In a method for determining a minimum value of an optimization function under constraints given by equations, a set of points which satisfy the constraints is regarded as a Riemannian manifold within a finite-dimensional real-vector space, the Riemannian manifold is approached from an initial position within the real-vector space. An exponential map regarding a geodesic line equation with respect to a tangent vector on the Riemannian manifold ends at a finite order, an approximate geodesic line is generated as a one-dimensional orbit. An approximate parallel-translation is performed on the tangent vector on the Riemannian manifold and on the orbit generated in the orbit generating step by finite-order approximation of the exponential map regarding the parallel translation of the tangent vector. By repeating the above-described procedure from the position at which a minimum value is given until the minimum value on the orbit converges, the solution of the optimization problem with constraints is determined using a simple calculation procedure.

REFERENCES:
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“Computational Methods in Optimization”, by E. Polak, 1971, published by Academic Press.
“Optimization Techniques on Riemannian Manifolds”, S. Smith Field Institute Communications, vol. 3 1994 pp. 113-136.
“The Geometry of Algorithms With Orthogonality Constraints”, A Edelman, et al 1998 SIAM J. Matrix Anal. Appl., vol. 20, No. 2, pp. 303-353.

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