We use variational methods to provide a concise development of a number of basic results in convex and functional analysis. This illuminates the parallels between convex analysis and smooth subdifferential theory. 1. The purpose of this note is to give a concise and explicit account of the following folklore: several fundamental theorems in convex analysis such as the sandwich theorem and the F...

The following synthesis is based on notes taken in 2007 by Fernando Amat, Argyris Zymnis, John Duchi, Krishnamurthy Iyer, Vahideh H. Manshadi and Sewoong Oh. The merging might have lead to inconsistencies that I will repair in the version posted online. Excellent (and more complete) references on variational methods are the review [WJ08], as well as the papers [WJW05] and [YFW05]. Relevant mate...

In the study of graphical models, methods based on the concept of variational freeenergy bounds have been widely used for approximating functionals of probability distributions. In this paper, we provide a method based on the same principles that can be applied to problems of stochastic optimization. In particular, this method is based upon the same principles as the generalized EM algorithm. W...

In this thesis we investigate nonsmooth classical and continuum mechanics and its discretizations by means of variational numerical and geometric methods. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Next, we develop the f...

Tutorial topics • A bit of history • Examples of variational methods • A brief intro to graphical models • Variational mean field theory – Accuracy of variational mean field – Structured mean field theory • Variational methods in Bayesian estimation • Convex duality and variational factorization methods – Example: variational inference and the QMR-DT Variational methods • Classical setting: " f...

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When underlying financial variables follow a Markov jump-diffusion process, the value function of a derivative security satisfies a partial integro-differential equation (PIDE) for European-style exercise or a partial integro-differential variational inequality (PIDVI) for American-style exercise. Unless the Markov process has a special structure, analytical solutions are generally not availabl...