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 convex functions. Some extensions of convexity. Linear support and separation of sets. Cones and polyhedral convexity. Alternative theorems. Subgradient and subdifferential.
2. Uncostrained Optimization and Optimization over a Convex Set
Outline of monodimensional optimization. The Newton method. Gradient methods. The conjugate directions method and the conjugate gradient method. The feasible directions methods. Linearization methods: the method of Frank and Wolfe. The gradient projection method.
3. Multipliers Theory
Necessary conditions. Equality constraints; Lagrange multipliers. Inequality constraints; Kuhn-Tucker multipliers. Regularity conditions and constraints qualification. John multipliers. The perturbation function. The meaning of the multipliers; economic intepretation. Sufficient conditions. Saddle point of a function. Saddle point conditions and minimum points.
4. Lagrangian Duality
The dual problem. Dual variables and Lagrange multipliers. Weak and strong duality theorems. Convex and nonconvex problems. Some applications.
|M.Minoux||Mathematical Programming: theory and algorithms||John Wiley and Sons||1986||0471901709|
|D.M.Bertsekas||Nonlinear Programming||Athena Scientific||2004||1886529140|
Written and oral final examination.
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