The Geometry of Semidefinite Programming

Pataki, Gábor

doi:10.1007/978-1-4615-4381-7_3

Cited by 73 publications

(72 citation statements)

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“…Using known results on the positive semidefinite cone (see, e.g., Pataki [19]), one can also show the following. We omit the proofs for brevity.…”

Section: All Of These Vectors Lie In K(v S) the Correspondingmentioning

confidence: 78%

On Nonconvex Quadratic Programming with Box Constraints

Burer¹,

Letchford²

2009

SIAM J. Optim.

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Abstract. Nonconvex quadratic programming with box constraints is a fundamental N P-hard global optimization problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterize their extreme points and vertices, show their invariance under certain affine transformations, and show that various linear inequalities induce facets. We also show that the sets are closely related to the Boolean quadric polytope, a fundamental polytope in the field of polyhedral combinatorics. Finally, we give a classification of valid inequalities and show that this yields a finite recursive procedure to check the validity of any proposed inequality.

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“…Using known results on the positive semidefinite cone (see, e.g., Pataki [19]), one can also show the following. We omit the proofs for brevity.…”

Section: All Of These Vectors Lie In K(v S) the Correspondingmentioning

confidence: 78%

On Nonconvex Quadratic Programming with Box Constraints

Burer¹,

Letchford²

2009

SIAM J. Optim.

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“…see, for instance, (3.2.8) and (3.2.10) in Pataki [18]. Plugging these into (22) gives (i), as required.…”

Section: But (20) Impliesmentioning

confidence: 87%

“…To establish closedness, we need to first verify that for a pair of positive semidefinite matrices x ū ,ū ∈ ri face x K , i.e., they are strictly complementary (see Alizadeh et al [1], or Pataki [18]). Ifū is of the form…”

Section: But (20) Impliesmentioning

confidence: 99%

On the Closedness of the Linear Image of a Closed Convex Cone

Pataki

2007

Mathematics of OR

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When is the linear image of a closed convex cone closed? We present very simple and intuitive necessary conditions that (1) unify, and generalize seemingly disparate, classical sufficient conditions such as polyhedrality of the cone, and Slater-type conditions; (2) are necessary and sufficient, when the dual cone belongs to a class that we call nice cones (nice cones subsume all cones amenable to treatment by efficient optimization algorithms, for instance, polyhedral, semidefinite, and p-cones); and (3) provide similarly attractive conditions for an equivalent problem: the closedness of the sum of two closed convex cones.

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“…The smaller the dimension of tan(X, K), the higher the non-smoothness ("kinkiness") of K at X . For a so-called nice cone (see Pataki 2000Pataki , 2013Roshchina 2014), the relation J (X ) ⊥ = tan(X, K) holds.…”

Section: Example 38 [Sdp Withmentioning

confidence: 99%

First order solutions in conic programming

2015

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We study the order of maximizers in linear conic programming (CP) as well as stability issues related to this. We do this by taking a semi-infinite view on conic programs: a linear conic problem can be formulated as a special instance of a linear semi-infinite program (SIP), for which characterizations of the stability of first order maximizers are well-known. However, conic problems are highly special SIPs, and therefore these general SIP-results are not valid for CP. We discuss the differences between CP and general SIP concerning the structure and results for stability of first order maximizers, and we present necessary and sufficient conditions for the stability of first order maximizers in CP. Keywords

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The Geometry of Semidefinite Programming

Cited by 73 publications

References 0 publications

On Nonconvex Quadratic Programming with Box Constraints

On Nonconvex Quadratic Programming with Box Constraints

On the Closedness of the Linear Image of a Closed Convex Cone

First order solutions in conic programming

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