245documents trouvés
Handbook on Semidefinite, Conic and Polynomial Optimization
J.B.LASSERRE, F.M.ANJOS
MAC, HEC Montréal
Ouvrage (éditeur) : Handbook on Semidefinite, Conic and Polynomial Optimization, Miguel F. Anjos and Jean B. Lasserre (Editors). International Series in Operations Research & Management Science Volume 166, Springer Verlag, Berlin, 2012 , N°ISBN 978-1-4614-0768-3, Janvier 2012, 957p. , N° 12238
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Abstract
Semidefinite and conic optimization is a major and thriving research area within the optimization community. Although semidefinite optimization has been studied (under different names) since at least the 1940s, its importance grew immensely during the 1990s after polynomial-time interior-point methods for linear optimization were extended to solve semidefinite optimization problems. Since the beginning of the 21st century, not only has research into semidefinite and conic optimization continued unabated, but also a fruitful interaction has developed with algebraic geometry through the close connections between semidefinite matrices and polynomial optimization. This has brought about important new results and led to an even higher level of research activity. This Handbook on Semidefinite, Conic and Polynomial Optimization provides the reader with a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization, and polynomial optimization. It contains a compendium of the recent research activity that has taken place in these thrilling areas, and will appeal to doctoral students, young graduates, and experienced researchers alike.
Positivity and optimization: beyond polynomials
J.B.LASSERRE, M.PUTINAR
MAC, Santa Barbara
Ouvrage (contribution) : Handbook on Semidefinite, Conic and Polynomial Optimization, Miguel F. Anjos and Jean B. Lasserre (Editors). International Series in Operations Research & Management Science Volume 166, 957p, Janvier 2012, pp.407-436 , N° 12782
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129095Introduction to semidefinite, conic and polynomial optimization
F.M.ANJOS, J.B.LASSERRE
HEC Montréal, MAC
Ouvrage (contribution) : Handbook on Semidefinite, Conic and Polynomial Optimization, Miguel F. Anjos and Jean B. Lasserre (Editors). International Series in Operations Research & Management Science Volume 166, 957p., Janvier 2012, pp.1-22 , N° 12780
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129091A "joint+marginal" approach in optimization
J.B.LASSERRE
MAC
Ouvrage (contribution) : Handbook on Semidefinite, Conic and Polynomial Optimization, Miguel F. Anjos and Jean B. Lasserre (Editors). International Series in Operations Research & Management Science Volume 166, 957p, Janvier 2012, pp.271-296 , N° 12781
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129093Inverse polynomial optimization
J.B.LASSERRE
MAC
Manifestation avec acte : IEEE Conference on Decision and Control and European Control Conference (CDC-ECC 2011), Orlando (USA), 12-15 Décembre 2011, pp.2794-2799 , N° 11140
Lien : http://hal.archives-ouvertes.fr/hal-00577168/fr/
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126936Level sets and non Gaussian integrals of positively homogeneous functions
J.B.LASSERRE
MAC
Rapport LAAS N°11595, Novembre 2011, 28p.
Lien : http://hal.archives-ouvertes.fr/hal-00637049/fr/
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Abstract
We investigate various properties of the sublevel set $\{x \,:\,g(x)\leq 1\}$ and the integration of $h$ on this sublevel set when $g$ and $h$are positively homogeneous functions. For instance, the latter integral reduces to integrating $h\exp(-g)$ on the whole space $R^n$ (a non Gaussian integral) and when $g$ is a polynomial, then the volume of the sublevel set is a convex function of the coefficients of $g$. In fact, whenever $h$ is nonnegative, the functional $\int \phi(g(x))h(x)dx$ is a convex function of $g$ for a large class of functions $\phi:R_+\to R$. We also provide a numerical approximation scheme to compute the volume or integrate $h$ (or, equivalently to approximate the associated non Gaussian integral). We also show that finding the sublevel set $\{x \,:\,g(x)\leq 1\}$ of minimum volume that contains some given subset $K$ is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of non Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.
On Convex optimization without convex representation
J.B.LASSERRE
MAC
Revue Scientifique : Optimization Letters, Vol.5, N°4, pp.549-556, Novembre 2011 , N° 10482
Lien : http://hal.archives-ouvertes.fr/hal-00495396/fr/
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We consider the convex optimization problem P: min { f(x): x in K} where "f" is convex continuously differentiable, and K is a compact convex set in Rn with representation {x: g_j(x) >=0, j=1,;;,m} for some continuously differentiable functions (g_j). We discuss the case where the g_j's are not all concave (in contrast with convex programming where they all are). In particular, even if the g_j's are not concave, we consider the log-barrier function phi_\mu with parameter \mu, associated with P, usually defined for concave functions (g_j). We then show that any limit point of any sequence (x_\mu) of stationary points of phi_\mu, \mu ->0, is a Karush-Kuhn-Tucker point of problem P and a global minimizer of f on K.
Min-max and robust polynomial optimization
J.B.LASSERRE
MAC
Revue Scientifique : Journal of Global Optimization, Vol.51, N°1, pp.1-10, Septembre 2011 , N° 10519
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126361Bounding the support of a measure from its marginal moments
J.B.LASSERRE
MAC
Revue Scientifique : Proceedings of the American Mathematical Society, Vol.139, N°9, pp.3375-3382, Septembre 2011 , N° 10676
Lien : http://hal.archives-ouvertes.fr/hal-00530932/fr/
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Given all moments of the marginals of a measure on Rn, one provides (a) explicit bounds on its support and (b), a numerical scheme to compute the smallest box that contains the support. It consists of solving a hierarchy of generalized eigenvalue problems associated with some Hankel matrices.
Moment and SDP relaxation techniques for smooth approximations of nonlinear differential equations
M.MEVISSEN, J.B.LASSERRE, D.HENRION
TI Tech, MAC
Manifestation avec acte : World IFAC Congress (IFAC 2011), Milan (Italie), 28 Août - 2 Septembre 2011, pp.10887-10892 , N° 10185
Lien : http://hal.archives-ouvertes.fr/hal-00462301/fr/
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