#convex-analysis

• 24 June 2026 · 6 min

  Convex Duality and the KKT Conditions

  How a constrained minimum is certified. The Lagrangian and its dual function, weak duality, strong duality under Slater's condition from the supporting hyperplane theorem, and the Karush-Kuhn-Tucker conditions that characterise the optimum of a convex program.

  • optimization
  • convex-analysis
  • duality

• 23 June 2026 · 6 min

  Convex Sets and Functions

  The geometry that makes minimisation tractable. Convex sets and functions and their first and second-order characterisations, the separating and supporting hyperplane theorems from projection onto a closed convex set, and the fact that a convex function's local minimum is global.

  • optimization
  • convex-analysis
  • real-analysis