# MAI0089 Convex analysis and convex optimization theory

Number of credits: 9 hp

Examiner: Michael Patriksson

Course literature:
(i) Convex Optimization Theory by D, Bertsekas, published by Athena Scientific in 2009
(ii) Additional course material on optimization algorithms, found on the web page of (i)
(iii) Additional material from convex analysis books found in the list below.

Course contents: The main focus of the course is on the most basic notions of convex analysis, particularly as they relate to existence and duality theory for convex optimization problems. We can roughly separate the material into the following areas:
Convex sets and functions: closedness and continuity, convex and affine hulls, relative interior, closure, recession cones, hyperplanes, separation, conjugate function.
Polyhedral convexity: extreme points, polar cones, polyhedral functions.
Basic concepts of convex optimization: existence of solutions, partial minimization, saddle point and minimax theory.
Geometric duality: min common/max crossing duality, conjugate convex functions, duality, augmented Lagrangian, strong duality, existence of dual optimal solutions.
Duality and optimization: nonlinear Farkas' lemma, linear programming duality, convex programming duality, Subgradients and optimality conditions, minimax theory, theorems of the alternative, non convex problems.

Organisation: Lectures; homework assignments; student presentations of homework assignments and of additional reading material.

Examination: Active participation in exercise solution seminars. Written and orally presented summary of additional reading material.

Prerequisites: Passed courses on analysis (in one and several variables) and linear algebra.

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Last updated: 2014-04-29