Nonlinear programming

Lecturer:

Čestmír Burdík (gar.)

Tutor:

Čestmír Burdík (gar.)

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:

Basic course of Calculus and Linear Algebra (in particular, the courses 01MA1, 01MAA2-4, 01LAP, 01LAA2 held at the FNSPE CTU in Prague).

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:

Knowledge: mathematical basis convex optimization. Abilities: able to use nonlinear optimization algorithms in practice.

Study materials:

Key references: [1] Stephen Boyd and Lieven Vandenberghe, Convex optimization, Cambridge University Press 2004.

Recommended references: [2] Fletcher: Practical methods of optimization, Wiley, 2000. [3]Luenberger: Linear and Nonlinear Programming, Addison-Wesley, 1984. [3] Stoer, Witzgall: Convexity and Optimization in Finite Dimensions, Springer-Verlag, Berlin, 1970. [4]Bazaraa, Sherali, Shetty: Nonlinear Programming: Theory and Algorithms, Wiley, 1993, L mat 1360

Note:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans: