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134
On the Convergence of Monte Carlo Maximum Likelihood Calculations
 Journal of the Royal Statistical Society B
, 1992
"... Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family f h ` : ` 2 \Theta g of nonnegative integrable functions, maximum likelihood estimates in the family obtained by normalizing the the ..."
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Cited by 58 (3 self)
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Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family f h ` : ` 2 \Theta g of nonnegative integrable functions, maximum likelihood estimates in the family obtained by normalizing the the functions to integrate to one can be approximated by Monte Carlo, the only regularity conditions being a compactification of the parameter space such that the the evaluation maps ` 7! h ` (x) remain continuous. Then with probability one the Monte Carlo approximant to the log likelihood hypoconverges to the exact log likelihood, its maximizer converges to the exact maximum likelihood estimate, approximations to profile likelihoods hypoconverge to the exact profile, and level sets of the approximate likelihood (support regions) converge to the exact sets (in Painlev'eKuratowski set convergence). The same results hold when there are missing data (Thompson and Guo, 1991, Gelfand and Carlin, 19...
Optimization Problems with perturbations, A guided tour
 SIAM REVIEW
, 1996
"... This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and app ..."
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Cited by 46 (10 self)
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This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and approximate solutions in situations where the set of Lagrange multipliers may be unbounded, or even empty. We give rather complete results for nonlinear programming problems, and describe some partial extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.
Theory and implementation of numerical methods based on RungeKutta integration for solving optimal control problems
, 1996
"... ..."
Duality between Probability and Optimization
 In Proceedings of the workshop "Idempotency
, 1997
"... this paper. The link between the weak convergence and the epigraph convergence used in convex analysis is done. The Cramer transform used in the large deviation literature is defined as the composition of the Laplace transform by the logarithm by the Fenchel transform. It transforms convolution into ..."
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Cited by 17 (8 self)
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this paper. The link between the weak convergence and the epigraph convergence used in convex analysis is done. The Cramer transform used in the large deviation literature is defined as the composition of the Laplace transform by the logarithm by the Fenchel transform. It transforms convolution into infconvolution. Probabilistic results about processes with independent increments are then transformed into similar results on dynamic programming equations. Cramer transform gives new insight on the Hopf method used to compute explicit solutions of some HJB equations. It also explains the limit theorems obtained directly as the image of the classic limit theorems of probability. Bibliographic notes are given at the end of the paper. 2 Cost Measures and Decision Variables
A handbook of Γconvergence
 in “Handbook of Differential Equations – Stationary Partial Differential Equations
"... The notion of Γconvergence has become, over the more than thirty years after its introduction by Ennio De Giorgi, the commonlyrecognized notion of convergence for variational problems, and it would be difficult nowadays to think of any other ‘limit ’ than a Γlimit when talking about asymptotic an ..."
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Cited by 15 (9 self)
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The notion of Γconvergence has become, over the more than thirty years after its introduction by Ennio De Giorgi, the commonlyrecognized notion of convergence for variational problems, and it would be difficult nowadays to think of any other ‘limit ’ than a Γlimit when talking about asymptotic analysis in a general variational setting (even though special convergences may fit better specific problems, as Moscoconvergence, twoscale convergence, G and Hconvergence, etc.). This short presentation is meant as an introduction to the many applications of this theory to problems in Partial Differential Equations, both as an effective method for solving asymptotic and approximation issues and as a means of expressing results that are derived by other techniques. A complete introduction to the general theory of Γconvergence is the bynowclassical book by Gianni Dal Maso [85], while a userfriendly introduction can be found in my book ‘for beginners ’ [46], where also simplified onedimensional versions of many of the problems in this article are treated. These notes are addressed to an audience of experienced mathematicians, with some background and interest in Partial Differential Equations, and are meant to direct the reader to what I regard as the most interesting features of this theory. The style of the exposition is how I would present the subject to a colleague in a neighbouring field or to an interested PhD student: the issues that I think
Consistency of Minimizers and the SLLN for Stochastic Programs
, 1996
"... A general strong law of large numbers for stochastic programs is established. It is shown that solutions and approximate solutions may not be consistent with the strong law in general, but consistency holds locally, or when the decision space is compact. An additional integrability condition implies ..."
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Cited by 15 (1 self)
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A general strong law of large numbers for stochastic programs is established. It is shown that solutions and approximate solutions may not be consistent with the strong law in general, but consistency holds locally, or when the decision space is compact. An additional integrability condition implies the uniform consistency of approximate solutions. The results are applied in the context of linear recourse models.  2  1. Introduction The paper examines relations between solutions of a stochastic optimization problem, and the solutions of large sampled versions of the problem. We consider an abstract stochastic program of the form () minimize x2X E P (d) \Gamma f(x; ) \Delta where E P (d) is the expectation operator with respect to the probability measure P over the space \Xi of random elements. The decision space here is taken as a metric space. For a given sequence 1 ; : : : ; n of realizations of the random variable we form the deterministic problem () minimize x2X 1...
A parallel splitting method for coupled monotone inclusions
"... A parallel splitting method is proposed for solving systems of coupled monotone inclusions in Hilbert spaces, and its convergence is established under the assumption that solutions exist. Unlike existing alternating algorithms, which are limited to two variables and linear coupling, our parallel met ..."
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Cited by 13 (5 self)
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A parallel splitting method is proposed for solving systems of coupled monotone inclusions in Hilbert spaces, and its convergence is established under the assumption that solutions exist. Unlike existing alternating algorithms, which are limited to two variables and linear coupling, our parallel method can handle an arbitrary number of variables as well as nonlinear coupling schemes. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of evolution inclusions, variational problems, best approximation, and network flows.
Extremal Eigenvalue Problems For Composite Membranes, I
, 1990
"... : Given an open bounded connected set\Omega ae R N and a prescribed amount of two homogeneous materials of different density, for small k we characterize the distribution of the two materials in\Omega that extremizes the kth eigenvalue of the resulting clamped membrane. We show that these extrem ..."
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Cited by 12 (3 self)
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: Given an open bounded connected set\Omega ae R N and a prescribed amount of two homogeneous materials of different density, for small k we characterize the distribution of the two materials in\Omega that extremizes the kth eigenvalue of the resulting clamped membrane. We show that these extremizers vary continuously with the proportion of the two constituents. The characterization of the extremizers in terms of level sets of associated eigenfunctions provides geometric information on their respective interfaces. Each of these results generalizes to N dimensions the now classical one dimensional work of M.G. Krein. 1 1. INTRODUCTION Within the class of fixed endpoint strings on the interval (0; 1) with density between ff and fi and mass equal to fffl + fi(1 \Gamma fl), M.G. Krein [19] was able to isolate those with the largest or smallest kth natural frequency. More precisely, denoting by k (ae) the kth Dirichlet eigenvalue of the string with density ae, and by ae fl k (ae ...
Scaling For A Random Polymer
, 1994
"... Let Q fi n be the law of the nstep random walk on Z d obtained by weighting simple random walk with a factor e \Gammafi for every selfintersection (DombJoyce model of `soft polymers'). It was proved by Greven and den Hollander (1993) that in d = 1 and for every fi 2 (0; 1) there exist ` ( ..."
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Cited by 11 (6 self)
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Let Q fi n be the law of the nstep random walk on Z d obtained by weighting simple random walk with a factor e \Gammafi for every selfintersection (DombJoyce model of `soft polymers'). It was proved by Greven and den Hollander (1993) that in d = 1 and for every fi 2 (0; 1) there exist ` (fi) 2 (0; 1) and ¯ fi 2 f¯ 2 l 1 (N) : k¯k l 1 = 1; ¯ ? 0g such that under the law Q fi n as n !1: (i) ` (fi) is the limit empirical speed of the random walk; (ii) ¯ fi is the limit empirical distribution of the local times. A representation was given for ` (fi) and ¯ fi in terms of a largest eigenvalue problem for a certain family of N \Theta N matrices. In the present paper we use this representation to prove the following scaling result as fi # 0: (i) fi \Gamma 1 3 ` (fi) ! b ; (ii) fi \Gamma 1 3 ¯ fi (d\Deltafi \Gamma 1 3 e) ! L 1 j (\Delta). The limits b 2 (0; 1) and j 2 fj 2 L 1 (R + ) : kjk L 1 = 1; j ? 0g are identified in terms of...
Theory of Cost Measures: Convergence of Decision Variables
 INRIA REPORT N
, 1995
"... Considering probability theory in which the semifield of positive real numbers is replaced by the idempotent semifield of real numbers (union infinity) endowed with the min and plus laws leads to a new formalism for optimization. Probability measures correspond to minimums of functions that we call ..."
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Cited by 11 (6 self)
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Considering probability theory in which the semifield of positive real numbers is replaced by the idempotent semifield of real numbers (union infinity) endowed with the min and plus laws leads to a new formalism for optimization. Probability measures correspond to minimums of functions that we call cost measures, whereas random variables correspond to constraints on these optimization problems that we call decision variables. We review in this context basic notions of probability theory  random variables, convergence of random variables, characteristic functions, L p norms. Whenever it is possible, results and definitions are stated in a general idempotent semiring.