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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| Veröffentlicht in: | Mathematical programming Jg. 170; H. 2; S. 541 - 553 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.08.2018
Springer Nature B.V |
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| ISSN: | 0025-5610, 1436-4646 |
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| Abstract | 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. |
|---|---|
| AbstractList | Let F be a quadratically constrained, possibly nonconvex, bounded set, and let E1,…,El denote ellipsoids contained in F with non-intersecting interiors. We prove that minimizing an arbitrary quadratic q(·) over G:=F\∪k=1ℓint(Ek) 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 E1,…,El yields a tight SDP relaxation for min{q(x):x∈G}. We also prove that the convex hull of {(x,xxT):x∈G} equals the intersection of the convex hull of {(x,xxT):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. 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. |
| Author | Yang, Boshi Anstreicher, Kurt Burer, Samuel |
| Author_xml | – sequence: 1 givenname: Boshi orcidid: 0000-0001-6760-9054 surname: Yang fullname: Yang, Boshi email: boshiy@clemson.edu organization: Department of Mathematical Sciences, Clemson University – sequence: 2 givenname: Kurt surname: Anstreicher fullname: Anstreicher, Kurt organization: Department of Management Sciences, University of Iowa – sequence: 3 givenname: Samuel surname: Burer fullname: Burer, Samuel organization: Department of Management Sciences, University of Iowa |
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| CitedBy_id | crossref_primary_10_1007_s10107_019_01367_2 crossref_primary_10_1016_j_ejco_2021_100012 crossref_primary_10_1016_j_orl_2023_11_006 crossref_primary_10_1007_s10107_023_01985_x crossref_primary_10_1007_s10107_025_02278_1 crossref_primary_10_1137_22M1501027 crossref_primary_10_1007_s10107_020_01589_9 crossref_primary_10_1007_s13675_018_0101_2 crossref_primary_10_1137_22M1528215 crossref_primary_10_1007_s00186_020_00726_6 crossref_primary_10_1016_j_ejor_2022_11_020 crossref_primary_10_1007_s10107_019_01453_5 crossref_primary_10_1007_s10107_024_02131_x crossref_primary_10_1007_s10107_020_01560_8 crossref_primary_10_1007_s40305_019_00286_5 |
| Cites_doi | 10.1137/110826862 10.1007/BF02614438 10.1515/9781400873173 10.1016/j.disopt.2014.08.002 10.1080/10556789308805542 10.1016/S0167-6377(01)00093-1 10.1287/moor.28.2.246.14485 10.1007/s10107-013-0716-2 10.1142/5273 10.1137/S105262340139001X 10.1137/1.9781611970791 10.1007/s10589-013-9635-7 10.1007/s10107-010-0355-9 10.1137/0804009 10.1137/0805016 10.1007/978-3-642-56468-0 10.1007/s10107-015-0888-z 10.1137/050644471 10.1007/s10107-012-0609-9 10.1007/s10107-014-0749-1 10.1016/j.orl.2010.05.010 10.1137/1.9780898719857 10.1137/1.9781611973402.28 |
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| Copyright | Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017 Mathematical Programming is a copyright of Springer, (2017). All Rights Reserved. |
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| References | Burer, Anstreicher (CR7) 2013; 23 CR4 Berman, Shaked-Monderer (CR3) 2003 Buchheim, Traversi (CR5) 2015; 15 Pong, Wolkowicz (CR18) 2014; 58 Hiriart-Urruty, Lemaréchal (CR12) 2001 Burer, Letchford (CR8) 2014; 143 CR17 Beck, Eldar (CR2) 2006; 17 Sturm, Zhang (CR22) 2003; 28 Martínez (CR14) 1994; 4 CR11 CR10 Rendl, Wolkowicz (CR19) 1997; 77 Stern, Wolkowicz (CR21) 1995; 5 Burer (CR6) 2015; 151 Moré (CR15) 1993; 2 Burer, Yang (CR9) 2015; 149 Jeyakumar, Li (CR13) 2014; 147 Tunçel (CR23) 2001; 29 Nesterov, Nemirovskii (CR16) 1994 Anstreicher, Burer (CR1) 2010; 124 Rockafellar (CR20) 1970 Ye, Zhang (CR24) 2003; 14 C Buchheim (1157_CR5) 2015; 15 1157_CR17 RJ Stern (1157_CR21) 1995; 5 Y Ye (1157_CR24) 2003; 14 1157_CR4 A Beck (1157_CR2) 2006; 17 A Berman (1157_CR3) 2003 JM Martínez (1157_CR14) 1994; 4 JJ Moré (1157_CR15) 1993; 2 F Rendl (1157_CR19) 1997; 77 YE Nesterov (1157_CR16) 1994 JB Hiriart-Urruty (1157_CR12) 2001 L Tunçel (1157_CR23) 2001; 29 S Burer (1157_CR8) 2014; 143 RT Rockafellar (1157_CR20) 1970 KM Anstreicher (1157_CR1) 2010; 124 V Jeyakumar (1157_CR13) 2014; 147 1157_CR10 T Pong (1157_CR18) 2014; 58 S Burer (1157_CR6) 2015; 151 S Burer (1157_CR7) 2013; 23 S Burer (1157_CR9) 2015; 149 JF Sturm (1157_CR22) 2003; 28 1157_CR11 |
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| Snippet | 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.... Let F be a quadratically constrained, possibly nonconvex, bounded set, and let E1,…,El denote ellipsoids contained in F with non-intersecting interiors. We... |
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| SubjectTerms | Calculus of Variations and Optimal Control; Optimization Combinatorics Ellipsoids Game theory Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Quadratic programming Short Communication Theoretical |
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| Title | Quadratic programs with hollows |
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