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5B5880 Convexity and duality in Rn

This course is primarily intended for graduate students in optimization and systems theory, but other students are also welcome.

Summary of contents

The course deals with finite dimensional convex analysis. The emphasis of the course is on the fundamental theory but its implications for solving convex minimization problems by iterative methods are also highlighted.

An outline of the material covered in the course:

  • Convex sets
    • operations on convex sets,
    • closed convex sets, and hulls
    • relative interior, extreme points, exposed faces,
    • cones (asymptotic, tangent, normal),
    • projections onto convex sets,
    • separations of convex sets.
  • Convex functions
    • dilation, perspectives, and infimal convolusion,
    • local and global behavior,
    • first and second order differentiability.
  • Sublinearity and support funtions.
  • Subdifferentials
    • definitions,
    • local properties,
    • calculus rules,
    • monotonicity and continuity.
  • Constrained convex minimization
    • constraint qualifications,
    • Lagrange multipliers,
    • saddle points.
  • Conjugacy.


J-B. Hiriart-Urruty and C. Lemaréchal Convex Analysis and Minimization Algorithms I + (chapter X in) II, Springer Verlag.


The examination is by homework assignments and an oral final exam.

Optimeringslära och systemteori, KTH
Krister Svanberg, e-post