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LAGRANGIAN-CONIC RELAXATIONS, PART I: A UNIFIED FRAMEWORK AND ITS APPLICATIONS TO QUADRATIC OPTIMIZATION PROBLEMS

Title
LAGRANGIAN-CONIC RELAXATIONS, PART I: A UNIFIED FRAMEWORK AND ITS APPLICATIONS TO QUADRATIC OPTIMIZATION PROBLEMS
Authors
Arima, NaohikoKim, SunyoungKojima, MasakazuToh, Kim-Chuan
Ewha Authors
김선영
SCOPUS Author ID
김선영scopus
Issue Date
2018
Journal Title
PACIFIC JOURNAL OF OPTIMIZATION
ISSN
1348-9151JCR Link
Citation
PACIFIC JOURNAL OF OPTIMIZATION vol. 14, no. 1, pp. 161 - 192
Keywords
Lagrangian-conic relaxationcompletely positive programming relaxationdoubly nonnegative relaxationconvexificationquadratic optimization problemsexploiting sparsity
Publisher
YOKOHAMA PUBL
Indexed
SCIE WOS
Document Type
Article
Abstract
In Part I of a series of study on Lagrangian-conic relaxations, we introduce a unified framework for conic and Lagrangian-conic relaxations of quadratic optimization problems (QOPs) and polynomial optimization problems (POPs). The framework is constructed with a linear conic optimization problem (COP) in a finite dimensional Hilbert space, where the cone used is not necessarily convex. By imposing a copositive condition on the COP, we establish fundamental theoretical results for the COP, its (convex hull) conic relaxations, its Lagrangian-conic relaxations, and their duals. A linearly constrained QOP with complementarity constraints and a general POP can be reduced to the COP satisfying the copositivity condition. Thus the conic and Lagrangian-conic relaxations of such a QOP and POP can be discussed in a unified manner. The Lagrangian-conic relaxation takes a particularly simple form involving only a single equality constraint together with the cone constraint, which is very useful for designing efficient numerical methods. As demonstration of the elegance and power of the unified framework, we present the derivation of the completely positive programming relaxation, and a sparse doubly nonnegative relaxation for a class of a linearly constrained QOPs with complementarity constraints. The unified framework is applied to general POPs in Part II.
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자연과학대학 > 수학전공 > Journal papers
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