The Lovasz theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measura...
详细信息
The Lovasz theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10, . . . , 24 and we give a new proof that it grows exponentially with the dimension.
We construct and substantiate an approximate control in the form of feedback for the problem of approximate bounded synthesis with semidefinite quality criterion for a parabolic equation containing a nonlinear term th...
详细信息
We construct and substantiate an approximate control in the form of feedback for the problem of approximate bounded synthesis with semidefinite quality criterion for a parabolic equation containing a nonlinear term that depends regularly on a small parameter.
We propose an algorithm for the global optimization of continuous minimax problems involving polynomials. The method can be described as a discretization approach to the well known semi-infinite formulation of the pro...
详细信息
We propose an algorithm for the global optimization of continuous minimax problems involving polynomials. The method can be described as a discretization approach to the well known semi-infinite formulation of the problem. We proceed by approximating the infinite number of constraints using tools and techniques from semidefinite programming. We then show that, under appropriate conditions, the SDP approximation converges to the globally optimal solution of the problem. We also discuss the numerical performance of the method on some test problems.
Let K be a totally real number field with Galois closure L. We prove that if f is an element of Q[x(1), . . . , x(n)] is a sum of m squares in K[x(1), . . . , x(n)], then f is a sum of 4m . 2([L:Q]+1) (([L:Q])(2) (+ 1...
详细信息
Let K be a totally real number field with Galois closure L. We prove that if f is an element of Q[x(1), . . . , x(n)] is a sum of m squares in K[x(1), . . . , x(n)], then f is a sum of 4m . 2([L:Q]+1) (([L:Q])(2) (+ 1)) squares in Q[x(1), . . . , x(n)]. Moreover, our argument is constructive and generalizes to the case of commutative K-algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semidefinite programming problems.
Principal component analysis (PCA) is a classical method for dimensionality reduction based oil extracting the dominant eigenvectors of the Sample covariance matrix. However, PCA is well known to behave poorly inw the...
详细信息
Principal component analysis (PCA) is a classical method for dimensionality reduction based oil extracting the dominant eigenvectors of the Sample covariance matrix. However, PCA is well known to behave poorly inw the "large p, small n" setting, in which the problem dimension p is comparable to or larger than the sample size n. This paper studies PICA ill this high-dimensional regime, but under the additional assumption that the maximal eigenvector is sparse, say, with at most k nonzero components. We consider a spiked covariance model in which a base matrix is perturbed by adding a k-sparse maximal eigenvector, and we analyze two computationally tractable methods for recovering the Support set of this maximal eigenvector, as follows: (a) a simple diagonal thresholding method. which transitions from success to failure as a function of the resealed sample size theta(dia)(n, p, k) = n/[k(2) log(p - k)];and (b) a more sophisticated semidefinite programming (SDP) relaxation, which succeeds once the resealed sample Size theta(sdp)(n, p, k) = n/[k log(p - k)] is larger than a critical threshold. In addition, we prove that no method, including the best method which has exponential-time complexity, can Succeed in recovering the Support if the order parameter theta(sdp)(n, p, k) is below a threshold. Our results thus highlight,in interesting trade-oft between computational and statistical efficiency in high-dimensional inference.
Linear programming bounds provide ail elegant method to prove optimality and uniqueness of an (n, N, t) spherical code. However, this method does not apply to the parameters (4, 10, 1/6). We use semidefinite programmi...
详细信息
Linear programming bounds provide ail elegant method to prove optimality and uniqueness of an (n, N, t) spherical code. However, this method does not apply to the parameters (4, 10, 1/6). We use semidefinite programming bounds instead to show that the Petersen code, which consists of the midpoints of the edges of the regular simplex in dimension 4, is the unique (4, 10, 1/6) spherical code. (C) 2008 Elsevier Inc. All rights reserved.
We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function w...
详细信息
We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parameterization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.
We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap (log n)(Omega(1)). This is achieved by exhibiting n-point metric spaces of negative type who...
详细信息
ISBN:
(纸本)9780769538501
We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap (log n)(Omega(1)). This is achieved by exhibiting n-point metric spaces of negative type whose L-1 distortion is (log n)(Omega(1)). Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to L-1 when restricted to cosets of the center.
A semialgebraic set is a set described by a boolean combination of real polynomial inequalities in several variables. A linear matrix inequality (LMI) is a condition expressing that a symmetric matrix whose entries ar...
详细信息
ISBN:
(纸本)9781605586090
A semialgebraic set is a set described by a boolean combination of real polynomial inequalities in several variables. A linear matrix inequality (LMI) is a condition expressing that a symmetric matrix whose entries are affine-linear combinations of variables is positive semidefinite. We call solution sets of LMIs spectrahedra and their linear images semidefinite representable. Every spectrahedron satisfies a condition called rigid convexity, and every semidefinite representable set is convex and semialgebraic. Helton, Vinnikov and Nie recently showed in several seminal papers [21, 4, 3;2] that the converse statements are true in surprisingly many cases and conjectured that they remain true in general. This shows the need for symbolic algorithms to compute LMI descriptions of convex semialgebraic sets. Once such a description is computed, it makes the corresponding semialgebraic set amenable to efficient numerical computation. Indeed, spectrahedra are the feasible sets in semidefinite programming (SDP), just in the same way as (convex closed) polyhedra are the feasible sets in linear programming (LP). The aim of this tutorial talk is to convince the audience that a symbolic interface between semialgebraic geometry and SDP has to be developed, and to initiate in the basic theory of LMI representations known so far. This theory is based to a large extent on determinantal representations of polynomials and on positivity certificates involving sums of squares.
A scheme for globally stabilized controller design was proposed based on stability criterion of density function and solved via SOS (Sum of Squares) decomposition tool. In order to deduce the complexity of the control...
详细信息
ISBN:
(纸本)9781424447947
A scheme for globally stabilized controller design was proposed based on stability criterion of density function and solved via SOS (Sum of Squares) decomposition tool. In order to deduce the complexity of the controller, it was designed with minimum monomial terms which approximated by minimizing the norm l coefficients of controller monomials. Guaranteed convergence rate performance was considered for locally stabilized controller, and finally, a switching control switched between global and local controllers was addressed to enhance system performance. The proposed control scheme achieved effective performance as illustrated in numerical example.
暂无评论