245documents trouvés
Lp-norms, Log-barriers and Cramer transform in Optimization
J.B.LASSERRE, E.S.ZERON
MAC, CINVESTAV-IPN Mexico
Revue Scientifique : Set-Valued Analysis, Vol.18, N°3-4, pp.513-530, Octobre 2010 , N° 10037
Lien : http://hal.archives-ouvertes.fr/hal-00444812/fr/
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Abstract
We show that the Laplace approximation of a supremum by Lp-norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem P and its dual appear naturally when using this simple approximation technique for the value function g of P or its Legendre-Fenchel conjugate. In addition, minimizing the LBF of the dual is just evaluating the Cramer transform of the Laplace approximation of g. Finally, this technique permits to sometimes define an explicit dual problem in cases when the Legendre-Fenchel conjugate of g cannot be derived explicitly from its definition.
A new look at nonnegativity on closed sets
J.B.LASSERRE
MAC
Conférence invitée : Modern Trends in Optimization and Its Application Workshop I: Convex Optimization and Algebraic Geometry, Los Angeles (USA), 28 Septembre - 1 Octobre 2010 , N° 10918
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124332On the moment-sos approach in optimization
J.B.LASSERRE
MAC
Conférence invitée : Congresso Nacional de Estadistica e Inverstigacion Operativa, coruna (Espagne), 14-17 Septembre 2010 , N° 10919
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124334Certificates of convexity for basic semi-algebraic sets
J.B.LASSERRE
MAC
Revue Scientifique : Applied Mathematics Letters, Vol.23, N°8, pp.912-916, Août 2010 , N° 10073
Lien : http://hal.archives-ouvertes.fr/hal-00356714/fr/
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120476On representations of the feasible set in convex optimization
J.B.LASSERRE
MAC
Revue Scientifique : Optimization Lettrers, Vol.4, N°1, pp.1-5, Juillet 2010 , N° 09662
Lien : http://hal.archives-ouvertes.fr/hal-00430141/fr/
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121977A "joint+marginal" algorithm for polynomial optimization
J.B.LASSERRE
MAC
Conférence invitée : International Worshop on High Performance Optimization Techniques (HPOPT 2010), Tilburg (Pays Bas), 14-16 Juin 2010, 1p. (Résumé) , N° 10921
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124338A "joint+marginal" algorithm in 0/1 programming
J.B.LASSERRE
MAC
Conférence invitée : European Worshop on Mixed Integer Nonlinear Programming, Marseille (France), 12-16 Avril 2010 , N° 10914
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124324A "joint+marginal" algorithm for polynomial optimization
J.B.LASSERRE, T.PHAN THANH
MAC
Rapport LAAS N°10186, Mars 2010, 6p.
Lien : http://hal.archives-ouvertes.fr/hal-00463095/fr/
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120962A "joint+marginal" algorithm for 0-1 nonlinear programs
J.B.LASSERRE
MAC
Conférence invitée : Programmation Non Linéaire en Nombres Entiers; 23èmes Journée JFRO, Paris (France), 19 Mars 2010, 1p. (Résumé) , N° 10916
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124328A "joint+marginal" approach to parametric polynomial optimization
J.B.LASSERRE
MAC
Revue Scientifique : SIAM Journal on Optimization, Vol.20, N°4, pp.1995-2022, Mars 2010 , N° 09315
Lien : http://hal.archives-ouvertes.fr/hal-00384400/fr/
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Abstract
Given a compact parameter set $Y\subset R^p$, we consider polynomial optimization problems $(P_y$) on $R^n$ whose description depends on the parameter $y\inY$. We assume that one can compute all moments of some probability measure $\varphi$ on $Y$, absolutely continuous with respect to the Lebesgue measure (e.g. $Y$ is a box or a simplex and $\varphi$ 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 $P_y$. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions, like e.g. their $\varphi$-mean. In addition, using this knowledge on moments, the measurable function $y\mapsto x^*_k(y)$ 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 $\varphi(\{y:x^*_k(y)=1\})$, 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 $L_1(\varphi)$ (resp. almost uniform) convergence to the optimal value function.