Nonlinear optimization 1000-135OPN

Introduction to non-linear optimisation problems. Examples of practical models. Convex sets. Separating and supporting hyperplanes.

Convex functions. Once and twice differentiable convex functions. Gradient and sub-gradient. Quasi- and pseudo-convex functions. Sublevel sets. Minimas.

Feasible set. Feasible directions. Necessary and sufficient conditions for optimality. Lagrange function. Fritz-John necessary condition. Kuhn-Tucker necessary and sufficient conditions. Regularity conditions. Equlibrium conditions.

Dual problem and dual theorem. Saddle points of the Lagrange function, their relation to duality and Kuhn-Tucker equation. Linear complementary problem, Lemke's method, applications to quadratic programming. Solutions to quadratic programming problems.

Methods of solution of nonlinear programming problems. Unconditional minimisation of one- and multi-dimensional functions. Examples of gradient methods, conjugate gradient methods and Newton-type methods. Conditional optimisation: method of feasible directions, penalty and barrier functions, random methods.