Formulation

Summary

Formulation is a term used in various senses in various applications, both the material and the abstract or formal. Its fundamental meaning is the putting together of components in appropriate relationships or structures, according to a formula. Etymologically formula is the diminutive of the Latin forma, meaning shape. In that sense a formulation is created according to the standard for the product. Abstract applications Disciplines in which one might use the word formulation in the abstract sense include logic, mathematics, linguistics, legal theory, and computer science. For details, see the related articles. Material applications In more material senses the concept of formulation appears in the physical sciences, such as physics, chemistry, and biology. It also is ubiquitous in industry, engineering and medicine, especially pharmaceutics. Pharmacy Pharmaceutical formulation In pharmacy, a formulation is a mixture or a structure such as a capsule,

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https://en.wikipedia.org/wiki/Formulation

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This course provides an overview of key advances in continuous optimization and statistical analysis for machine learning. We review recent learning formulations and models as well as their guarantees, describe scalable solution techniques and algorithms, and illustrate the trade-offs involved.

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La réponse statique et dynamique des systèmes couplés fluide-structure résultant de l'excitation indépendante, l'excitation induite par le mouvement, et l'excitation induite par les instabilités des fluides.

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Robust Portfolio Optimization

Zheng Wei Yap

Since the 2008 Global Financial Crisis, the financial market has become more unpredictable than ever before, and it seems set to remain so in the forseeable future. This means an investor faces unprecedented risks, hence the increasing need for robust portfolio optimization to protect them against uncertainty, which is potentially devastating if unattended yet ignored in the classical Markowitz model, whose another deficiency is the absence of higher moments in its assumption of the distribution of asset returns. We establish an equivalence between the Markowitz model and the portfolio return value-at-risk optimization problem under multivariate normality of asset returns, so that we can add these excluded features into the former implicitly by incorporating them into the latter. We also provide a probabilistic smoothing spline approximation method and a deterministic model within the location-scale framework under elliptical distribution of the asset returns to solve the robust portfolio return value-at-risk optimization problem. In particular for the deterministic model, we introduce a novel eigendecomposition uncertainty set which lives in the positive definite space for the scale matrix without compromising on the computational complexity and conservativeness of the optimization problem, invent a method to determine the size of the involved uncertainty sets, test it out on real data, and explore its diversification properties. Although the value-at-risk has been the standard risk measure adopted by the banking and insurance industry since the early nineties, it has since attracted many criticisms, in particular from McNeil et al. (2005) and the Basel Committee on Banking Supervision in 2012, also known as Basel 3.5. Basel 4 even suggests a move away from the what" value-at-risk to the what-if" conditional value-at-risk' measure. We shall see that the former may be replaced with the latter or even other risk measures in our formulations easily.

EPFL2017

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Recovery of sparse signals from linear, dimensionality reducing measurements broadly fall under two well-known formulations, named the synthesis and the analysis a ́ la Elad et al. Recently, Chandrasekaran et al. introduced a new algorithmic sparse recovery framework based on the convex geometry of linear inverse prob- lems, called the atomic norm formulation. In this paper, we prove that atomic norm formulation and synthesis formulation are equiva- lent for closed atomic sets. Hence, it is possible to use the synthesis formulation in order to obtain the so-called atomic decompositions of signals. In order to numerically observe this equivalence we derive exact linear matrix inequality representations, also known as the theta bodies, of the centrosymmertic polytopes formed from the columns of the simplex and their antipodes. We then illustrate that the atomic and synthesis recovery results agree on machine precision on randomly generated sparse recovery problems.

Ieee2012

A gauge finite element method for the 2D Navier-Stokes problem

Philippe Caussignac

We set up a gauge finite element formulation of the 2D steady state incompressible Stokes problem. This formulation allows to decouple the two unknowns from the equations and to take account for the boundary conditions in a non-iterative way. A finite element discretization is used to obtain numerical convergence results. The method is then applied to the Navier-Stokes equations (C) 2007 IMACS. Published by Elsevier B.V. All rights reserved.

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Robust Portfolio Optimization

Zheng Wei Yap

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