research-article

Optimizing a Parametric Linear Function over a Non-compact Real Algebraic Variety

Authors:

Mohab Safey El Din,

Lihong ZhiAuthors Info & Claims

ISSAC '15: Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

Pages 205 - 212

https://doi.org/10.1145/2755996.2756666

Published: 24 June 2015 Publication History

Abstract

We consider the problem of optimizing a parametric linear function over a non-compact real trace of an algebraic set. Our goal is to compute a representing polynomial which defines a hypersurface containing the graph of the optimal value function. Rostalski and Sturmfels showed that when the algebraic set is irreducible and smooth with a compact real trace, then the least degree representing polynomial is given by the defining polynomial of the irreducible hypersurface dual to the projective closure of the algebraic set.

First, we generalize this approach to non-compact situations. We prove that the graph of the opposite of the optimal value function is still contained in the affine cone over a dual variety similar to the one considered in compact case. In consequence, we present an algorithm for solving the considered parametric optimization problem for generic parameters' values. For some special parameters' values, the representing polynomials of the dual variety can be identically zero, which give no information on the optimal value. We design a dedicated algorithm that identifies those regions of the parameters' space and computes for each of these regions a new polynomial defining the optimal value over the considered region.

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Cited By

Dai ZLi ZYang ZZhi L(2024)Whitney Stratification of Algebraic Boundaries of Convex Semi-algebraic SetsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669702(299-306)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669702
Zhi LRobertz DLinton SYokoyama K(2015)Optimization Problems over Noncompact Semialgebraic SetsProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756638(13-14)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756638

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Optimizing a Parametric Linear Function over a Non-compact Real Algebraic Variety

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cover image ACM Conferences

ISSAC '15: Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

June 2015

374 pages

ISBN:9781450334358

DOI:10.1145/2755996

Conference Chair:
Daniel Robertz
Plymouth University, United Kingdom
,
General Chair:
Steve Linton
University of St Andrews, United Kingdom
,
Program Chair:
Kazuhiro Yokoyama
Rikkyo University, Japan

Copyright © 2015 ACM.

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Published: 24 June 2015

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Chinese Fundamental Research Funds for the Central Universities
Chinese National Natural Science Foundation
National Key Basic Research Project
CNRS and INRIA
French National Research Agency

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ISSAC'15

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ISSAC'15: International Symposium on Symbolic and Algebraic Computation

July 6 - 9, 2015

Bath, United Kingdom

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ISSAC '15 Paper Acceptance Rate 43 of 71 submissions, 61%;

Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Dai ZLi ZYang ZZhi L(2024)Whitney Stratification of Algebraic Boundaries of Convex Semi-algebraic SetsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669702(299-306)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669702
Zhi LRobertz DLinton SYokoyama K(2015)Optimization Problems over Noncompact Semialgebraic SetsProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756638(13-14)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756638

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