Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization

Waki, Hayato; Nakata, Masao; Muramatsu, Masakazu

doi:10.1007/s10589-011-9437-8

Cited by 39 publications

(47 citation statements)

References 15 publications

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“…For the triplet (B, H, G 0 ) satisfying (14), if B ∩Ḡ = ∅, we can remove at least B ∩Ḡ fromḠ and thus G \ (B ∩Ḡ) ⊆ G 0 \ B. Otherwise, we haveḠ ⊆ G 0 \ B.…”

Section: Then the Triplet (B ∩ G H ∩ G G) Satisfies (2)mentioning

confidence: 99%

A facial reduction algorithm for finding sparse SOS representations

Waki

Muramatsu

2010

Operations Research Letters

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“…For the triplet (B, H, G 0 ) satisfying (14), if B ∩Ḡ = ∅, we can remove at least B ∩Ḡ fromḠ and thus G \ (B ∩Ḡ) ⊆ G 0 \ B. Otherwise, we haveḠ ⊆ G 0 \ B.…”

Section: Then the Triplet (B ∩ G H ∩ G G) Satisfies (2)mentioning

confidence: 99%

A facial reduction algorithm for finding sparse SOS representations

Waki

Muramatsu

2010

Operations Research Letters

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“…In [15], it is shown that this problem is numerically very ill-conditioned; the general SDP solvers report wrong optimal value 1, as opposed to the optimal value 0. In fact, [15] shows that it is impossible to calculate the optimal value of (27) if we use a usual floating point precision, and proposed to use the multi-precision SDP solver SDPA-GMP [3].…”

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confidence: 99%

“…In fact, [15] shows that it is impossible to calculate the optimal value of (27) if we use a usual floating point precision, and proposed to use the multi-precision SDP solver SDPA-GMP [3].…”

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confidence: 99%

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Facial Reduction Algorithms for Conic Optimization Problems

Waki

Muramatsu

2012

J Optim Theory Appl

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133

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To obtain a primal-dual pair of conic programming problems having zero duality gap, two methods have been proposed: the facial reduction algorithm due to Borwein and Wolkowicz [1,2] and the conic expansion method due to Luo, Sturm, and Zhang [5]. We establish a clear relationship between them. Our results show that although the two methods can be regarded as dual to each other, the facial reduction algorithm can produce a finer sequence of faces including the feasible region. We illustrate the facial reduction algorithm in LP, SOCP and an example of SDP. A simple proof of the convergence of the facial reduction algorithm for conic programming is also presented.

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“…To preclude numerical troubles with SDP solvers, we advise to systematically add such a ball constraint, combined with an appropriate scaling so that all scaled variables belong to the unit sphere. Without scaling, numerical troubles can occur as well, but they are not due to the presence of a duality gap, see [1] and also the example of [9,Section 6].…”

Section: Resultsmentioning

confidence: 99%

Strong duality in Lasserre’s hierarchy for polynomial optimization

Josz

Henrion

2015

Optim Lett

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A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set K described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J. B. Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrangian of the POP). In this note, when the POP feasibility set K is compact, we show that there is no duality gap between each primal and dual SDP problem in Lasserre's hierarchy, provided a redundant ball constraint is added to the description of set K. Our proof uses elementary results on SDP duality, and it does not assume that K has an interior point.

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Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization

Cited by 39 publications

References 15 publications

A facial reduction algorithm for finding sparse SOS representations

A facial reduction algorithm for finding sparse SOS representations

Facial Reduction Algorithms for Conic Optimization Problems

Strong duality in Lasserre’s hierarchy for polynomial optimization

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