IEICE TRANSACTIONS on transactions
A Homotopy Method for Numerically Solving Infinite Dimensional Convex Optimization Problems
Summary :
A homotopy method is proposed for numerically solving an infinite dimensional convex optimization problem (COP). In this method, in the first place, by introducing an auxiliary COP, which is easy to solve, the original COP is imbedded into a parametric COP. Then, the Kuhn-Tucker (K-T) equation is derived which is equivalent to the parametric COP. It is shown that this K-T equation belongs to a class of Approximation proper homotopy equations. Using this, a sequence of finite dimensional approximate equations are derived for the K-T equation. It is shown that under certain mild conditions these approximate equations can be always solved by finite dimensional homotopy method and numerical solutions for them can be identified. Moreover, using the Approximation properness of the K-T equation, it is shown that from such a sequence a subsequence can always be derived which is convergent to a solution of the K-T equation. In this paper, two types of auxiliary COP's are introduced and two types of K-T equations are considered, a fixed point homotopy type and a Newton homotopy type. Since applicability is slightly different, comparison of applicability is also discussed between them.
- Publication
- IEICE TRANSACTIONS on transactions Vol.E72-E No.12 pp.1307-1316
- Publication Date
- 1989/12/25
- Publicized
- Online ISSN
- DOI
- Type of Manuscript
- Special Section PAPER (Special Issue on the 2nd Karuizawa Workshop on Circuits and Systems)
- Category
- Nonlinear Problems
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Mitsunori MAKINO, Shin'ichi OISHI, "A Homotopy Method for Numerically Solving Infinite Dimensional Convex Optimization Problems" in IEICE TRANSACTIONS on transactions,
vol. E72-E, no. 12, pp. 1307-1316, December 1989, doi: .
Abstract: A homotopy method is proposed for numerically solving an infinite dimensional convex optimization problem (COP). In this method, in the first place, by introducing an auxiliary COP, which is easy to solve, the original COP is imbedded into a parametric COP. Then, the Kuhn-Tucker (K-T) equation is derived which is equivalent to the parametric COP. It is shown that this K-T equation belongs to a class of Approximation proper homotopy equations. Using this, a sequence of finite dimensional approximate equations are derived for the K-T equation. It is shown that under certain mild conditions these approximate equations can be always solved by finite dimensional homotopy method and numerical solutions for them can be identified. Moreover, using the Approximation properness of the K-T equation, it is shown that from such a sequence a subsequence can always be derived which is convergent to a solution of the K-T equation. In this paper, two types of auxiliary COP's are introduced and two types of K-T equations are considered, a fixed point homotopy type and a Newton homotopy type. Since applicability is slightly different, comparison of applicability is also discussed between them.
URL: https://global.ieice.org/en_transactions/transactions/10.1587/e72-e_12_1307/_p
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@ARTICLE{e72-e_12_1307,
author={Mitsunori MAKINO, Shin'ichi OISHI, },
journal={IEICE TRANSACTIONS on transactions},
title={A Homotopy Method for Numerically Solving Infinite Dimensional Convex Optimization Problems},
year={1989},
volume={E72-E},
number={12},
pages={1307-1316},
abstract={A homotopy method is proposed for numerically solving an infinite dimensional convex optimization problem (COP). In this method, in the first place, by introducing an auxiliary COP, which is easy to solve, the original COP is imbedded into a parametric COP. Then, the Kuhn-Tucker (K-T) equation is derived which is equivalent to the parametric COP. It is shown that this K-T equation belongs to a class of Approximation proper homotopy equations. Using this, a sequence of finite dimensional approximate equations are derived for the K-T equation. It is shown that under certain mild conditions these approximate equations can be always solved by finite dimensional homotopy method and numerical solutions for them can be identified. Moreover, using the Approximation properness of the K-T equation, it is shown that from such a sequence a subsequence can always be derived which is convergent to a solution of the K-T equation. In this paper, two types of auxiliary COP's are introduced and two types of K-T equations are considered, a fixed point homotopy type and a Newton homotopy type. Since applicability is slightly different, comparison of applicability is also discussed between them.},
keywords={},
doi={},
ISSN={},
month={December},}
Copy
TY - JOUR
TI - A Homotopy Method for Numerically Solving Infinite Dimensional Convex Optimization Problems
T2 - IEICE TRANSACTIONS on transactions
SP - 1307
EP - 1316
AU - Mitsunori MAKINO
AU - Shin'ichi OISHI
PY - 1989
DO -
JO - IEICE TRANSACTIONS on transactions
SN -
VL - E72-E
IS - 12
JA - IEICE TRANSACTIONS on transactions
Y1 - December 1989
AB - A homotopy method is proposed for numerically solving an infinite dimensional convex optimization problem (COP). In this method, in the first place, by introducing an auxiliary COP, which is easy to solve, the original COP is imbedded into a parametric COP. Then, the Kuhn-Tucker (K-T) equation is derived which is equivalent to the parametric COP. It is shown that this K-T equation belongs to a class of Approximation proper homotopy equations. Using this, a sequence of finite dimensional approximate equations are derived for the K-T equation. It is shown that under certain mild conditions these approximate equations can be always solved by finite dimensional homotopy method and numerical solutions for them can be identified. Moreover, using the Approximation properness of the K-T equation, it is shown that from such a sequence a subsequence can always be derived which is convergent to a solution of the K-T equation. In this paper, two types of auxiliary COP's are introduced and two types of K-T equations are considered, a fixed point homotopy type and a Newton homotopy type. Since applicability is slightly different, comparison of applicability is also discussed between them.
ER -
English
