Semidefinite representation of convex sets
Let be a semialgebraic set defined by multivariate polynomials g i ( x ). Assume S is convex, compact and has nonempty interior. Let , and ∂ S (resp. ∂ S i ) be the boundary of S (resp. S i ). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMI...
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| Veröffentlicht in: | Mathematical programming Jg. 122; H. 1; S. 21 - 64 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin/Heidelberg
Springer-Verlag
01.03.2010
Springer Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0025-5610, 1436-4646 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | Let
be a semialgebraic set defined by multivariate polynomials
g
i
(
x
). Assume
S
is convex, compact and has nonempty interior. Let
, and ∂
S
(resp. ∂
S
i
) be the boundary of
S
(resp.
S
i
). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMIs). The set
S
is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex
S
may not be LMI representable (Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007). A question arising from Nesterov and Nemirovski (SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994), see Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007 and Nemirovski in Plenary lecture, International Congress of Mathematicians (ICM), Madrid, Spain, 2006, is: given a subset
S
of
, does there exist an LMI representable set Ŝ in some higher dimensional space
whose projection down onto
equals
S
. Such
S
is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following are the main contributions of this paper: (i) assume
g
i
(
x
) are all concave on
S
. If the positive definite Lagrange Hessian condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function
ℓ
T
x
on
S
is positive definite at the minimizer, then
S
is SDP representable. (ii) If each
g
i
(
x
) is either sos-concave ( − ∇
2
g
i
(
x
) =
W
(
x
)
T
W
(
x
) for some possibly nonsquare matrix polynomial
W
(
x
)) or strictly quasi-concave on
S
, then
S
is SDP representable. (iii) If each
S
i
is either sos-convex or poscurv-convex (
S
i
is compact convex, whose boundary has positive curvature and is nonsingular, i.e., ∇
g
i
(
x
) ≠ 0 on ∂
S
i
∩
S
), then
S
is SDP representable. This also holds for
S
i
for which ∂
S
i
∩
S
extends smoothly to the boundary of a poscurv-convex set containing
S
. (iv) We give the complexity of Schmüdgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)–(iii). |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 content type line 14 ObjectType-Article-2 ObjectType-Feature-1 content type line 23 |
| ISSN: | 0025-5610 1436-4646 |
| DOI: | 10.1007/s10107-008-0240-y |