Exploiting Sparsity in the Matrix-Dilation Approach to Robust Semidefinite Programming

Oishi, Yushi; Isaka, Yusuke

doi:10.1109/acc.2007.4282187

Cited by 6 publications

(25 citation statements)

References 41 publications

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“…Since |V |≤| V 0 |, the size of the new approximate problem is not larger than that of the conventional one. Note here that the difference between |V | and |V 0 | is apparent when |S| is small and d i 's are large [13].…”

Section: A Constructing a Reduced-size Approximate Problemmentioning

confidence: 97%

“…In order to prove the statement, we need some results of the matrixdilation approach [11], [12], [13]. The key idea of the proof is based on a relationship between a reduced-size version of the SOS approach and that of the matrix-dilation approach.…”

Section: A Proof On the Main Theoremmentioning

confidence: 98%

“…We first state some graph theoretic concepts recapped from [13]. These concepts are necessary for constructing smallsize monomial bases, which leads to construction of a smallsize approximate problem.…”

Section: A Constructing a Reduced-size Approximate Problemmentioning

confidence: 99%

“…The discrepancy is apparent when the number of monomials is small and the degrees d i 's are large. Note that, this idea has been used in [13] to reduce the size of the approximate problem based on the matrix-dilation approach.…”

Section: Introductionmentioning

confidence: 99%

“…Early results on such an approximate approach are found in [1], [2], [5]. In the case of polynomial parameter dependence, asymptotically exact approximate approaches were proposed with the Kalman-Yakubovich-Popov lemma [14], [3], with Pólya's lemma [19], with the sum-of-squares (SOS) technique [15], [21], [7], and with the matrix dilation [11], [12], [13]. Here, the asymptotic exactness means that the approximation error can be made arbitrary small by considering an approximate problem of larger size, that is, an SDP with more number of variables and/or an LMI constraint of larger size.…”

Section: Introductionmentioning

confidence: 99%

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Exploiting sparsity in the sum-of-squares approximations to robust semidefinite programs

Jennawasin

2009

2009 American Control Conference

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This paper aims to improve computational complexity in the sum-of-squares approximations to robust semidefinite programs whose constraints depend polynomially on uncertain parameters. By exploiting sparsity, the proposed approach constructs sum-of-squares polynomials with smaller number of monomial elements, and hence gives approximate problems with smaller sizes. The sparse structure is extracted by a special graph pattern. The quality of the approximation is improved by dividing the parameter region, and can be expressed in terms of the resolution of the division. This expression shows that the proposed approach is asymptotically exact in the sense that, the quality can be arbitrarily improved by increasing the resolution of the division.

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Section: A Constructing a Reduced-size Approximate Problemmentioning

confidence: 97%

Section: A Proof On the Main Theoremmentioning

confidence: 98%

Section: A Constructing a Reduced-size Approximate Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Exploiting sparsity in the sum-of-squares approximations to robust semidefinite programs

Jennawasin

2009

2009 American Control Conference

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A region-dividing technique for constructing the sum-of-squares approximations to robust semidefinite programs

Jennawasin¹,

Oishi

2007

2007 46th IEEE Conference on Decision and Control

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In this paper, we present a novel approach to robust semidefinite programs, whose coefficient matrices depend polynomially on uncertain parameters. The procedure to construct an approximate problem for a given robust semidefinite program is based on the sum-of-squares representation of a positive semidefinite polynomial matrix. In contrast to the conventional sum-of-squares approach, quality of the approximation is improved by dividing the parameter region into several subregions. The optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer. An advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division. Numerical examples on polynomial optimizations are presented to show usefulness of the present approach. We also consider exploitation of sparsity of the given problem with respect to the uncertain parameters, in order to construct a reduced-size approximate problem.

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Asymptotic exactness of parameter-dependent lyapunov functions: An error bound and exactness verification

Oishi

2007

2007 46th IEEE Conference on Decision and Control

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This paper provides an approximate approach to a robust semidefinite programming problem with a functional variable and shows its asymptotic exactness. This problem covers a variety of control problems including a robust stability/performance analysis with a parameter-dependent Lyapunov function. In the proposed approach, an approximate semidefinite programming problem is constructed based on the division of the set of parameter values. This approach is asymptotically exact in the sense that, as the resolution of the division becomes higher, the optimal value of the constructed approximate problem converges to that of the original problem. Our convergence analysis is quantitative. In particular, this paper gives an a priori upper bound on the discrepancy between the optimal values of the two problems. Moreover, it discusses how to verify that an optimal solution of the approximate problem is actually optimal also for the original problem.Keywords-parameter-dependent Lyapunov functions, robust semidefinite programming, approximation error, exactness verification, matrix dilation, linear matrix inequalities.

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Exploiting Sparsity in the Matrix-Dilation Approach to Robust Semidefinite Programming

Cited by 6 publications

References 41 publications

Exploiting sparsity in the sum-of-squares approximations to robust semidefinite programs

Exploiting sparsity in the sum-of-squares approximations to robust semidefinite programs

A region-dividing technique for constructing the sum-of-squares approximations to robust semidefinite programs

Asymptotic exactness of parameter-dependent lyapunov functions: An error bound and exactness verification

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