Semidefinite programming for gradient and Hessian computation in maximum entropy estimation

Lasserre, Jean-Bernard

doi:10.1109/cdc.2007.4434063

Cited by 4 publications

(10 citation statements)

References 15 publications

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“…Solving such dual problems (usually by Newton's method) turns to be the basic technique to this aim. The main effort is then to deal with the computational cost of approximating multiple integrals (like T t j e i∈I λ i t i ρ(t)dt if a ≡ 1, for instance) needed for the gradient of L [23], [27], [28], [29], [8], [1], [18].…”

Section: Resultsmentioning

confidence: 99%

Multivariate truncated moments problems and maximum entropy

Ambrozie

2013

Anal.Math.Phys.

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Section: Resultsmentioning

confidence: 99%

Multivariate truncated moments problems and maximum entropy

Ambrozie

2013

Anal.Math.Phys.

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“…As M → ∞, u ⋆ M → u weakly (see Proposition 1) but not pointwise, i.e., one cannot guarantee u ⋆ M (x, y) → u(x, y) on Ω. Nevertheless, as reported in [Lasserre(2007)] the maximum entropy estimation may provide accurate pointwise approximation of the unknown function to be recovered on certain segments of the domain Ω. We next illustrate on a variety of PDE problems and OCPs, that indeed good pointwise approximation can be obtained on some parts of the domain Ω.…”

Section: Methodsmentioning

confidence: 95%

“…In this section we briefly introduce the maximum entropy estimation of [Borwein et al(1991), Lasserre(2007), Lasserre(2009)], our second tool to find smooth approximations for solutions of nonlinear differential equations. The maximum entropy estimation is concerned with the following problem: Let u ∈ L 1 (Ω) be nonnegative and partially known by the finite vector m of moments up to order M of the associated Borel measure dµ := u dx dy on Ω.…”

Section: Maximum Entropy Estimationmentioning

confidence: 99%

“…In the case the domain Ω is simple, we may compute them by quadrature and cubature formulas. For more difficult domains one may use the procedure described in [Lasserre(2007), Bertsimas et al(2008)]. Concerning the behavior of u ⋆ M as M → ∞ and its relationship with u, one has the following weak convergence result from [Borwein et al(1991)].…”

Section: Maximum Entropy Estimationmentioning

confidence: 99%

“…In this paper we present a novel approach to obtain smooth approximations for solutions of differential equations and of problems involving differential equations such as optimal control problems. Namely, an approximate solution is obtained by applying the maximum entropy estimation of [Borwein et al(1991), Lasserre(2007)] with a finite number of moments of a Borel measure associated with a discrete approximation of the solution of the differential equation. For linear differential equations, an SDP relaxation based method was proposed in [Bertsimas et al (2006)] to generate contracting sequences of lower and upper bounds for the moments.…”

Section: Introductionmentioning

confidence: 99%

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Moment and SDP relaxation techniques for smooth approximations of problems involving nonlinear differential equations

Mevissen

Lasserre

Henrion

2011

IFAC Proceedings Volumes

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Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of problems involving nonlinear differential equations. Given a system of nonlinear differential equations, we apply a technique based on finite differences and sparse SDP relaxations for polynomial optimization problems (POP) to obtain a discrete approximation of its solution. In a second step we apply maximum entropy estimation (using moments of a Borel measure associated with the discrete solution) to obtain a smooth closed-form approximation. The approach is illustrated on a variety of linear and nonlinear ordinary differential equations (ODE), partial differential equations (PDE) and optimal control problems (OCP), and preliminary numerical results are reported.

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A “Joint+Marginal” Approach to Parametric Polynomial Optimization

Lasserre¹

2010

SIAM J. Optim.

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Abstract. Given a compact parameter set Y ⊂ R p , we consider polynomial optimization problems (Py) on R n whose description depends on the parameter y ∈ Y. We assume that one can compute all moments of some probability measure ϕ on Y, absolutely continuous with respect to the Lebesgue measure (e.g. Y is a box or a simplex and ϕ is uniformly distributed). We then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions x * (y) of Py, as y ∈ Y. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions, like e.g. their ϕ-mean. In addition, using this knowledge on moments, the measurable function y → x * k (y) of the k-th coordinate of optimal solutions, can be estimated, e.g. by maximum entropy methods. Also, for a boolean variable x k , one may approximate as closely as desired its persistency ϕ({y : x * k (y) = 1}, i.e. the probability that in an optimal solution x * (y), the coordinate x * k (y) takes the value 1. At last but not least, from an optimal solution of the dual semidefinite relaxations, one provides a sequence of polynomial (resp. piecewise polynomial) lower approximations with L 1 (ϕ) (resp. almost uniform) convergence to the optimal value function. IntroductionRoughly speaking, given a set parameters Y and an optimization problem whose description depends on y ∈ Y (call it P y ), parametric optimization is concerned with the behavior and properties of the optimal value as well as primal (and possibly dual) optimal solutions of P y , when y varies in Y. This a quite challenging problem and in general one may obtain information locally around some nominal value y 0 of the parameter. There is a vast and rich literature on the topic and for a detailed treatment, the interested reader is referred to e.g. Bonnans and Shapiro [4] and the many references therein. Sometimes, in the context of optimization with data uncertainty, some probability distribution ϕ on the parameter set Y is available and in this context one is also interested in e.g. the distribution of the optimal value, optimal solutions, all viewed as random variables. In particular, for discrete optimization problems where cost coefficients are random variables with joint distribution ϕ, some bounds on the expected optimal value have been obtained. More recently Natarajan et al. [17] extended the earlier work in [3] to even 1991 Mathematics Subject Classification. 65 D15, 65 K05, 46 N10, 90 C22.

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Semidefinite programming for gradient and Hessian computation in maximum entropy estimation

Cited by 4 publications

References 15 publications

Multivariate truncated moments problems and maximum entropy

Multivariate truncated moments problems and maximum entropy

Moment and SDP relaxation techniques for smooth approximations of problems involving nonlinear differential equations

A “Joint+Marginal” Approach to Parametric Polynomial Optimization

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