Quadratic programs with hollows
Let F be a quadratically constrained, possibly nonconvex, bounded set, and let E 1 , … , E l denote ellipsoids contained in F with non-intersecting interiors. We prove that minimizing an arbitrary quadratic q ( · ) over G : = F \ ∪ k = 1 ℓ int ( E k ) is no more difficult than minimizing q ( · ) ove...
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| Vydáno v: | Mathematical programming Ročník 170; číslo 2; s. 541 - 553 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.08.2018
Springer Nature B.V |
| Témata: | |
| ISSN: | 0025-5610, 1436-4646 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Let
F
be a quadratically constrained, possibly nonconvex, bounded set, and let
E
1
,
…
,
E
l
denote ellipsoids contained in
F
with non-intersecting interiors. We prove that minimizing an arbitrary quadratic
q
(
·
)
over
G
:
=
F
\
∪
k
=
1
ℓ
int
(
E
k
)
is no more difficult than minimizing
q
(
·
)
over
F
in the following sense: if a given semidefinite-programming (SDP) relaxation for
min
{
q
(
x
)
:
x
∈
F
}
is tight, then the addition of
l
linear constraints derived from
E
1
,
…
,
E
l
yields a tight SDP relaxation for
min
{
q
(
x
)
:
x
∈
G
}
. We also prove that the convex hull of
{
(
x
,
x
x
T
)
:
x
∈
G
}
equals the intersection of the convex hull of
{
(
x
,
x
x
T
)
:
x
∈
F
}
with the same
l
linear constraints. Inspired by these results, we resolve a related question in a seemingly unrelated area, mixed-integer nonconvex quadratic programming. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0025-5610 1436-4646 |
| DOI: | 10.1007/s10107-017-1157-0 |