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Computational Mathematics for Learning and Data Analysis - AA 2024/25

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Lecture 3.2: "real" quadratic functions and how they work

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 Quadratic optimization: the real (nonseparable) case. (Epi)graph, (sub)level sets and tomography of quadratic (homogeneous) functions, the several different cases. Characterising level sets of quadratic (homogeneous) functions out of the spectral decomposition of Q. Consequence: characterising when a quadratic (homogeneous) function has minimum and maximum and where it lies. Extension to the box-constrained case and why it is *not* a good idea. Computing the "center" of the level sets of a quadratic non-homogeneous function: the simple (nonsingular) case.

Lecture 3.2: "real" quadratic functions and how they work
◄ Lecture 3.1: multivariate optimization: initial concepts, easy functions
Lecture 4.1: quadratic optimization: from optimality conditions to the gradient method ►

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          • Slides: Numerical Linear Algebra

          • Slides: Optimization

          • Optimization & Learning Lecture Notes

          • Lecture Recordings: Numerical Linear Algebra

          • Lectures Recordings: Optimization

            • FileLecture 1.1 - introduction to the course

            • FileLecture 1.2 - motivation for the course: four exam...

            • FileLecture 2.1: general notions of optimization

            • FileLecture 2.2: starting very very easy and very slow...

            • FileLecture 3.1: multivariate optimization: initial co...

            • FileLecture 3.2: "real" quadratic functions and how th...

            • FileLecture 4.1: quadratic optimization: from optimali...

            • FileLecture 4.2: the gradient method for quadratic fun...

            • FileLecture 5.1: convergence rates: from the gradient ...

            • FileLecture 5.2: sublinear convergence and where this ...

            • FileLecture 6.1: optimizing more general functions, bu...

            • FileLecture 6.2: first steps with local optimization: ...

            • FileLecture 7.1: dichotomic search, from naive to mod...

            • FileLecture 7.2: faster local optimization and the rol...

            • FileLecture 8.1: closing thoughts of univariate optimi...

            • FileLecture 8.2: theory of gradients and Hessians towa...

            • FileLecture 9.1: local first- and second-order optimal...

            • FileLecture 10.1: the gradient method with "exact" lin...

            • FileLecture 10.2: inexact line search, the Armijo-Wolf...

            • FileLecture 11.1: convergence with the A-W LS, theory

            • FileLecture 11.2: the A-W LS in practice

            • FileLecture 12.1: "extremely inexact LS": fixed stepsize

            • FileLecture 12.2: gradient twisting approaches at thei...

            • FileLecture 13.1: all around Newton's method

            • FileLecture 13.2: towards the very-large-scale, quasi-...

            • FileLecture 14.1: deflected gradient methods I - Conju...

            • FileLecture 14.2: deflected gradient methods II - Heav...

            • FileLecture 15.1: the scary world of nondifferentiable...

            • FileLecture 15.2: (convex) nondifferentiable optimizat...

            • FileLecture 16.1: better nondifferentiable approachess...

            • FileLecture 16.2: first steps on constrained optimization

            • FileLecture 17.1: algebraic representation of feasibl...

            • FileLecture 17.2: from the KKT conditions to duality

            • FileLecture 18.1: first step in constrained optimization

            • FileLecture 18.2: more (projected gradient) steps in c...

            • FileLecture 19.1: from Frank-Wolfe to the dual method

            • FileLecture 19.2: ending with a bang: the (primal-dual...

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      • Anno Accademico 2013-14

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