Nonlinear Optimization

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

Convex optimization has the application in many areas of natural sciences. The lecture includes the basics of the theory convex analysis and develops algorithms for unconstrained optimization and optimization with equality-constraints. The duality theory is studied and interior point method is formulated to be applied to inequality-constraint problems.

Requirements:

Syllabus of lectures:

1. Affine and convex set, operation that preserves convexity, separating and supporting hyperplanes .

2. Convex function, basic properties and examples, operations that preserve convexity, the conjugate function, quasiconvex functions, log-concave and log-convex functions, convexity with respect to generalized inequalities.

3. Optimization problem in standard form, convex optimization problem, quasiconvex optimization, linear optimization, quadratic optimization, geometric programming.

4. Duality, Lagrange dual problem, weak and strong duality, optimality condition, perturbative and sensitive analysis.

5. Numerical linear algebra background, matrix structure and algorithm complexity, solving linear equation with factorized matrices, LU and Cholesky factorization, block elimination and the matrix inversion lemma.

6. Unconstrained minimization, gradient descent method, steepest descent method, Newton method, self-concondart function.

7. Equality constrained minimization, eliminating equality constraints, infesable start Newton method.

8. Interior-point methods, logarithmic barrier function and central parth, barrier method.

9. Linear complementarity problem and quadratic programming.

Syllabus of tutorials:

Study Objective:

Study materials:

Povinná literatura

1. S. Boyd and L. Vandenberghe: Convex optimization, Cambridge University Press, 2004.

Doporučená literatura

2. R. W. Cottle, M. N. Thapa: Linear and nonlinear optimization, Springer, 2017.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: