Abstract
A geometric nonconvex conic optimization problem (COP) was recently proposed by Kim, Kojima, and Toh (SIAM J. Optim., 30 (2020), pp. 1251̶1273) as a unified framework for convex conic reformulation of a class of quadratic optimization problems and polynomial optimization problems. The nonconvex COP minimizes a linear function over the intersection of a nonconvex cone K, a convex subcone J of the convex hull coK of K, and an affine hyperplane with a normal vector H. Under the assumption co(K ∩ J) = J, the original nonconvex COP in their paper was shown to be equivalently formulated as a convex conic program by replacing the constraint set with the intersection of J and the affine hyperplane. This paper further studies the key assumption co(K ∩ J) = J in their framework and provides three sets of necessary-sufficient conditions for the assumption. Based on the conditions, we propose a new wide class of quadratically constrained quadratic programs with multiple nonconvex equality and inequality constraints, which can be solved exactly by their semidefinite relaxation.
Original language | English |
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Pages (from-to) | 3194-3211 |
Number of pages | 18 |
Journal | SIAM Journal on Optimization |
Volume | 34 |
Issue number | 4 |
DOIs | |
State | Published - 2024 |
Bibliographical note
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Keywords
- convex conic reformulation
- exact semidefinite relaxation
- geometric conic optimization problem
- polynomial optimization problem
- positive semidefinite cone
- quadratically constrained quadratic program