An oracle-based projection and rescaling algorithm for linear semi-infinite feasibility problems and its application to SDP and SOCP

Muramatsu, Masakazu; Kitahara, Tomonari; Lourenço, Bruno F.; Okuno, Takayuki; Tsuchiya, Takashi

doi:10.48550/arxiv.1809.10340

Cited by 2 publications

(3 citation statements)

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“…On the other hand, once we find that P ∞ (A) and P 1,∞ (A) have no feasible solution whose minimum eigenvalue is greater than ε, the next issue is to find an x ∈ K \ intK satisfying A(x) = 0, or to find the smallest dimensional symmetric cone K min satisfying KerA ∩ intK min = ∅ and K min K. It has been shown that the smallest dimensional symmetric cone K min can be detected by using a feasible solution of D(A) [15], and several algorithms have been proposed to find a feasible solution of D(A) in a finite number of iterations [5,10].…”

Section: Discussionmentioning

confidence: 99%

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A New Extension of Chubanov's Method to Symmetric Cones

Kanoh¹,

Yoshise²

2021

Preprint

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We propose a new variant of Chubanov's method for solving the feasibility problem over the symmetric cone by extending Roos's method (2018) for the feasibility problem over the nonnegative orthant. The proposed method considers a feasibility problem associated with a norm induced by the maximum eigenvalue of an element and uses a rescaling focusing on the upper bound of the sum of eigenvalues of any feasible solution to the problem. Its computational bound is (i) equivalent to Roos's original method (2018) and superior to Lourenço et al.'s method (2019) when the symmetric cone is the nonnegative orthant, (ii) superior to Lourenço et al.'s method (2019) when the symmetric cone is a Cartesian product of second-order cones, and (iii) equivalent to Lourenço et al.'s method (2019) when the symmetric cone is the simple positive semidefinite cone, under the assumption that the costs of computing the spectral decomposition and the minimum eigenvalue are of the same order for any given symmetric matrix.We also conduct numerical experiments that compare the performance of our method with existing methods by generating instance in three types: (i) strongly (but ill-conditioned) feasible instances, (ii) weakly feasible instances, and (iii) infeasible instances. For any of these instances, the proposed method is rather more efficient than the existing methods in terms of accuracy and execution time.

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Section: Discussionmentioning

confidence: 99%

“…Proof. The first inequality of ( 16) follows from (10) in the proof of Proposition 3.5. The second inequality is obtained by evaluating q ℓ,i (α…”

Section: Finite Termination Of the Basic Proceduresmentioning

confidence: 98%

A New Extension of Chubanov's Method to Symmetric Cones

Kanoh¹,

Yoshise²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…[96], applied to the SDP (44). For an explicit treatment, using a substantially more complicated arguments, see [97]. We only have to notice that the number of localising constraints is polynomial in the dimension.…”

Section: Computability and Convergence Ratesmentioning

confidence: 99%

Quantum Optimal Control via Magnus Expansion and Non-Commutative Polynomial Optimization

Marecek,

Vala

2020

Preprint

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Quantum optimal control has numerous important applications ranging from pulses in magnetic resonance imagining to laser control of chemical reactions and quantum computing. Our objective is to address two major challenges that have limited the success of applications of quantum optimal control so far: non-commutativity inherent in quantum systems and non-convexity of quantum optimal control problems involving more than three quantum levels. Methodologically, we address the non-commutativity of the control Hamiltonian at different times by the use of Magnus expansion. To tackle the non-convexity, we employ non-commutative polynomial optimisation and non-commutative geometry. As a result, we present the first globally convergent methods for quantum optimal control.

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An oracle-based projection and rescaling algorithm for linear semi-infinite feasibility problems and its application to SDP and SOCP

Cited by 2 publications

References 19 publications

A New Extension of Chubanov's Method to Symmetric Cones

A New Extension of Chubanov's Method to Symmetric Cones

Quantum Optimal Control via Magnus Expansion and Non-Commutative Polynomial Optimization

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