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How to convexify the intersection of a second order cone and a nonconvex quadratic
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Journal Article
How to convexify the intersection of a second order cone and a nonconvex quadratic
2017
Overview
A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown—by several authors using different techniques—that the convex hull of the intersection of an ellipsoid,
E
, and a split disjunction,
(
l
-
x
j
)
(
x
j
-
u
)
≤
0
with
l
<
u
, equals the intersection of
E
with an additional second-order-cone representable (SOCr) set. In this paper, we study more general intersections of the form
K
∩
Q
and
K
∩
Q
∩
H
, where
K
is a SOCr cone,
Q
is a nonconvex cone defined by a single homogeneous quadratic, and
H
is an affine hyperplane. Under several easy-to-verify conditions, we derive simple, computable convex relaxations
K
∩
S
and
K
∩
S
∩
H
, where
S
is a SOCr cone. Under further conditions, we prove that these two sets capture precisely the corresponding conic/convex hulls. Our approach unifies and extends previous results, and we illustrate its applicability and generality with many examples.
Publisher
Springer Berlin Heidelberg,Springer Nature B.V
