On the semidefinite representation of real functions applied to symmetric matrices

Sagnol, Guillaume

doi:10.1016/j.laa.2013.08.021

Cited by 33 publications

(35 citation statements)

References 14 publications

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“…7.3). Sagnol (2013) showed that each criterion in the Kiefer's class of optimality criteria defined by (4) is SDr for all rational values of δ ∈ (−∞, −1] and general SDP formulations exist. This result is also applicable to the case where δ → 0.…”

Section: Semidefinite Programmingmentioning

confidence: 99%

Adaptive grid semidefinite programming for finding optimal designs

Duarte

Wong²,

Dette

2017

Stat Comput

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Section: Semidefinite Programmingmentioning

confidence: 99%

Adaptive grid semidefinite programming for finding optimal designs

Duarte

Wong²,

Dette

2017

Stat Comput

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“…where |ψ ABC = 1 √ 2 |0 B |00 AC + 1 √ 2 |1 B (cos(θ)|01 AC + sin(θ)|10 AC ) (14) Figure 4 shows I(A : C|B) and the relative entropy of recovery for the range θ ∈ [0, π/2]. We have also included the quantum Rényi divergence (for α = 7/8) defined bȳ D α (ρ σ) = 1 α − 1 log Tr ρ α σ 1−α which is a lower bound on the relative entropy D and which can be optimized exactly using semidefinite programming for rational α [FS17,Sag13]. We see that there is a range of values of θ where the relative entropy of recovery is greater than or equal the conditional mutual information.…”

Section: Relative Entropy Of Recovery and Conditional Mutual Informationmentioning

confidence: 99%

Efficient optimization of the quantum relative entropy

Fawzi

2018

J. Phys. A: Math. Theor.

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Many quantum information measures can be written as an optimization of the quantum relative entropy between sets of states. For example, the relative entropy of entanglement of a state is the minimum relative entropy to the set of separable states. The various capacities of quantum channels can also be written in this way. We propose a unified framework to numerically compute these quantities using off-the-shelf semidefinite programming solvers, exploiting the approximation method proposed in [Fawzi, Saunderson, Parrilo, Semidefinite approximations of the matrix logarithm, arXiv:1705.00812]. As a notable application, this method allows us to provide numerical counterexamples for a proposed lower bound on the quantum conditional mutual information in terms of the relative entropy of recovery.

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“…Hence, our approach gives an outer approximation of the set of information matrices, which is SDP representable. As shown by the interesting works [20,18], the criterion φ q is also SDP representable in the case where q is rational. It proves that our procedure (depicted in Algorithm 1) makes use of two semidefinite programs and it can be efficiently used in practice.…”

Section: Algorithm 1: Approximate Optimal Designs On Semi-algebraic Setsmentioning

confidence: 90%

Approximate optimal designs for multivariate polynomial regression

Castro¹,

Gamboa²,

Henrion³

et al. 2019

Ann. Statist.

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We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semi-algebraic) design spaces. We use the moment-sum-of-squares hierarchy of semidefinite programming problems to solve numerically the approximate optimal design problem. The geometry of the design is recovered via semidefinite programming duality theory. This article shows that the hierarchy converges to the approximate optimal design as the order of the hierarchy increases. Furthermore, we provide a dual certificate ensuring finite convergence of the hierarchy and showing that the approximate optimal design can be computed numerically with our method. As a byproduct, we revisit the equivalence theorem of the experimental design theory: it is linked to the Christoffel polynomial and it characterizes finite convergence of the moment-sum-of-square hierarchies.MSC 2010 subject classifications: Primary 62K05, 90C25; secondary 41A10, 49M29, 90C90, 15A15.This paper introduces a general method to compute approximate optimal designs-in the sense of Kiefer's φ q -criteria-on a large variety of design spaces that we refer to as semi-algebraic imsart-generic ver.

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On the semidefinite representation of real functions applied to symmetric matrices

Cited by 33 publications

References 14 publications

Adaptive grid semidefinite programming for finding optimal designs

Adaptive grid semidefinite programming for finding optimal designs

Efficient optimization of the quantum relative entropy

Approximate optimal designs for multivariate polynomial regression

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