Optimization (2007/2008)

Learning outcomes

The aim of the course is to treat nonlinear optimization problems by looking into the theoretical analysis tools and the solution methods.

1. Fundamental Concepts
Optimization problems. Convex sets and cones. Linear support and separation of sets. Convex functions. Some extensions of convexity. Subgradient and subdifferential. Alternative theorems.

2. Uncostrained Optimization, Optimization over a Convex Set
Gradient methods. The Newton method. The conjugate directions method. The feasible directions method. The reduced gradient method. The gradient projection method. Linearization methods: the method of Frank and Wolfe.

3. Multipliers Theory
Necessary conditions. Equality constraints, Lagrange multipliers. Regularity conditions. John multipliers. The general case; Karush-Kuhn-Tucker multipliers. Sufficient conditions. Saddle point conditions and minimum points.

4. Duality and Complementarity
The dual problem. Dual variables and Lagrange multipliers. Weak and strong duality. Nonlinear problems. Quadratic problems. Complementarity systems and complementarity problems.

5. Quadratic Programming
The convex case. The modified simplex method. Geometric interpretation. Other solution methods in the convex case. Some applications.

Written and oral final examination.