Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite singularities

Guo, Feng; Wang, Li; Guang-ming, Zhou

doi:10.1007/s10898-013-0047-0

Cited by 14 publications

(7 citation statements)

References 27 publications

(108 reference statements)

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“…As shown in [28], Lasserre's hierarchy of relaxations, in combination with Jacobian representations, always has finite convergence, under some nonsingularity conditions. This result has been improved in [14,Theorem 3.9] under weaker conditions. Flat truncation can be used to detect the convergence (cf.…”

Section: 1mentioning

confidence: 85%

Bilevel Polynomial Programs and Semidefinite Relaxation Methods

Nie¹,

Wang²,

Ye³

2017

SIAM J. Optim.

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Abstract. A bilevel program is an optimization problem whose constraints involve the solution set to another optimization problem parameterized by upper level variables. This paper studies bilevel polynomial programs (BPPs), i.e., all the functions are polynomials. We reformulate BPPs equivalently as semi-infinite polynomial programs (SIPPs), using Fritz John conditions and Jacobian representations. Combining the exchange technique and Lasserre type semidefinite relaxations, we propose a numerical method for solving bilevel polynomial programs. For simple BPPs, we prove the convergence to global optimal solutions. Numerical experiments are presented to show the efficiency of the proposed algorithm.

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Section: 1mentioning

confidence: 85%

Bilevel Polynomial Programs and Semidefinite Relaxation Methods

Nie¹,

Wang²,

Ye³

2017

SIAM J. Optim.

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“…In Section 3.2 we show that the strong positivity condition is generic since it is implied by a particular generic algebraic condition on S considered in [26,27,57].…”

Section: Existence Of Copositive Certificates Of Non-negativitymentioning

confidence: 97%

“…Closedeness at infinity is one of the sufficient conditions for hierarchies of relaxations to PO problems proposed in [26,27,57] to converge to the optimal value [see, e.g., 57, Thm 2.5, condition (d)]. In [27,57], this condition is shown to hold generically. To connect closedness at infinity with strong positivity, we introduce the horizon cone of S ⊆ R n [74],…”

Section: Genericity Of Strong Positivitymentioning

confidence: 99%

The copositive way to obtain certificates of non-negativity over general semialgebraic sets

Kuryatnikova¹,

Vera²,

Zuluaga³

2019

Preprint

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A certificate of non-negativity is a way to write a given function so that its non-negativity becomes evident. Certificates of non-negativity are fundamental tools in optimization, and they underlie powerful algorithmic techniques for various types of optimization problems. We propose certificates of non-negativity of polynomials based on copositive polynomials. The certificates we obtain are valid for generic semialgebraic sets and have a fixed small degree, while commonly used sums-of-squares (SOS) certificates are guaranteed to be valid only for compact semialgebraic sets and could have large degree. Optimization over the cone of copositive polynomials is not tractable, but this cone has been well studied. The main benefit of our copositive certificates of non-negativity is their ability to translate results known exclusively for copositive polynomials to more general semialgebraic sets. In particular, we show how to use copositive polynomials to construct structured (e.g., sparse) certificates of non-negativity, even for unstructured semialgebraic sets. Last but not least, copositive certificates can be used to obtain not only hierarchies of tractable lower bounds, but also hierarchies of tractable upper bounds for polynomial optimization problems.

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“…As is shown in [6,18], U is not closed at ∞ because there exists a point (0, 0, 1) ∈ U but (0, 0, 1) / ∈ closure(U 0 ). Since for any x ∈ [1, 2], g(x, (0, 0, 1)) = −x < 0, we have M = ∅.…”

Section: Consequently We Havementioning

confidence: 99%

Semidefinite relaxations for semi-infinite polynomial programming

Wang

Guo

2013

Comput Optim Appl

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Abstract. This paper studies how to solve semi-infinite polynomial programming (SIPP) problems by semidefinite relaxation method. We first introduce two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints. Then we propose an exchange algorithm with SDP relaxations to solve SIPP problems with compact index set. At last, we extend the proposed method to SIPP problems with noncompact index set via homogenization. Numerical results show that the algorithm is efficient in practice.

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Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite singularities

Cited by 14 publications

References 27 publications

Bilevel Polynomial Programs and Semidefinite Relaxation Methods

Bilevel Polynomial Programs and Semidefinite Relaxation Methods

The copositive way to obtain certificates of non-negativity over general semialgebraic sets

Semidefinite relaxations for semi-infinite polynomial programming

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