Syllabus

• First and second derivatives of a function of several real variables. 
• Extrema of functions: Lagrange multipliers.
• Extrema of functions: consideration of the second derivatives 
• Extrema of functions: convex functions (general definition without differentiation, characterisation of convex functions, local and global minima of convex functions, quadratic functions and positive definite matrices).
• Newton’s method 
• General results on nonlinear optimisation problems 
• Relaxation and gradient methods for unconstrained problems
• Conjugate gradient methods for unconstrained problems
• Relaxation, gradient and penalty-function methods for constrained problems 
• The Kuhn-Tucker conditions  
• Lagrangians and saddle points. Introduction to duality