Advanced Optimization (QEM) 221941-D
Ćwiczenia (CW) Semestr zimowy 2024/25

Overview of key optimization techniques and their applications in economics.

Static vs. dynamic optimization. Finite vs. infinite planning horizon.

Convexity: convex sets, convex cones, convex functions.

Topology in Euclidean spaces: norm, distance, continuity, compactness, openness, closedness, boundedness.

Heine-Borel theorem. Level curves.

Unconstrained optimization. Existence results (Weierstrass theorem, coercivity).

First-order and second-order conditions.

Constrained optimization with equality constraints. Lagrange multipliers.

First-order and second-order conditions. Geometric interpretation.

Constrained optimization with inequality constraints.

Karush-Kuhn-Tucker. Geometric interpretation.

Infinite dimensional normed and metric spaces.

Norms and distances in sequence and function spaces. Pointwise and uniform convergence.

Compactness in infinite dimensional spaces. Complete spaces. Banach spaces.

Banach fixed point theorem. Blackwell theorem.

Dynamic programming with a finite planning horizon.

Framework: control and state variables, value function, policy function.

Bellman equation. Time separability, time consistency.

Solving dynamic optimization problems with backward induction.

Dynamic programming with an infinite planning horizon.

Euler equation. Transversality conditions.

Steady state. Dynamics around the steady state. Stability.

Correspondences. Upper and lower hemi-continuity. Berge theorem.

Economic applications.

Detailed elaboration of the Ramsey growth model and the optimal resource extraction model.