Lectures:
An introduction to the calculus of variations. Linear spaces, funkcionls and their differentials (Fréchet, Gateaux).
Euler equation and the solution of the classical problems of variational calculus.
Unconstrained minimization. One-dimensional minimization of unimodular functions.
Conditions of minimum, the Newton method and its modification. Gradient methods, method of conjugate gradients.
Constrained minimization. Karush-Kuhn-Tucker conditions of optimality.
Penalization and barrier methods for constrained minimization. Feasible direction method (SLP) and active set strategy for bound constrained problems.
Duality in convex programming. Saddle points, Uzawa algorithm and augmented Lagrangians.
Linear programming, simplex method.
Non-smooth optimization, subgradients and optimality conditions.
Global optimization, genetic and evolutionary algorithms, simulated annealing, tabu search.
Software.
Exercises:
Introduction to the MATLAB programming.
Implementation of the golden section and Fibonacci series methods.
Implemenation of the Newton-like methods.
Implementation of the gradient based method.
Implementation of the conjugate gradient method.
Implementation of the penalty methody for equality constrained minimization.
Implementation of the feasible direction method (SLP).
Implementation of the active set method for bound constrained quadratic programming.
Implementation of the augmented Lagrangian metod.
Implementation of algorithms for global optimization.
Solution of selected engeneering problems using optimization software.
Projects:
Comparing performance of the methods for unconstrained optimization using a numerical example (max 10 marks).
Comparing performance of the methods for constrained optimization using a numerical example (max 10 marks).
Solution of a selected engineering problem (max 10 marks).
Computer labs:
Introduction to the MATLAB programming.
Implementation of the golden section and Fibonacci series methods.
Implemenation of the Newton-like methods.
Implementation of the gradient based method.
Implementation of the conjugate gradient method.
Implementation of the penalty methody for equality constrained minimization.
Implementation of the feasible direction method (SLP).
Implementation of the active set method for bound constrained quadratic programming.
Implementation of the augmented Lagrangian metod.
Implementation of algorithms for global optimization.
Solution of selected engeneering problems using optimization software.
An introduction to the calculus of variations. Linear spaces, funkcionls and their differentials (Fréchet, Gateaux).
Euler equation and the solution of the classical problems of variational calculus.
Unconstrained minimization. One-dimensional minimization of unimodular functions.
Conditions of minimum, the Newton method and its modification. Gradient methods, method of conjugate gradients.
Constrained minimization. Karush-Kuhn-Tucker conditions of optimality.
Penalization and barrier methods for constrained minimization. Feasible direction method (SLP) and active set strategy for bound constrained problems.
Duality in convex programming. Saddle points, Uzawa algorithm and augmented Lagrangians.
Linear programming, simplex method.
Non-smooth optimization, subgradients and optimality conditions.
Global optimization, genetic and evolutionary algorithms, simulated annealing, tabu search.
Software.
Exercises:
Introduction to the MATLAB programming.
Implementation of the golden section and Fibonacci series methods.
Implemenation of the Newton-like methods.
Implementation of the gradient based method.
Implementation of the conjugate gradient method.
Implementation of the penalty methody for equality constrained minimization.
Implementation of the feasible direction method (SLP).
Implementation of the active set method for bound constrained quadratic programming.
Implementation of the augmented Lagrangian metod.
Implementation of algorithms for global optimization.
Solution of selected engeneering problems using optimization software.
Projects:
Comparing performance of the methods for unconstrained optimization using a numerical example (max 10 marks).
Comparing performance of the methods for constrained optimization using a numerical example (max 10 marks).
Solution of a selected engineering problem (max 10 marks).
Computer labs:
Introduction to the MATLAB programming.
Implementation of the golden section and Fibonacci series methods.
Implemenation of the Newton-like methods.
Implementation of the gradient based method.
Implementation of the conjugate gradient method.
Implementation of the penalty methody for equality constrained minimization.
Implementation of the feasible direction method (SLP).
Implementation of the active set method for bound constrained quadratic programming.
Implementation of the augmented Lagrangian metod.
Implementation of algorithms for global optimization.
Solution of selected engeneering problems using optimization software.