Nonlinear Optimization
Code  Completion  Credits  Range  Language 

01NEL  ZK  4  3P  Czech 
 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 equalityconstraints. The duality theory is studied and interior point method is formulated to be applied to inequalityconstraint 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, logconcave and logconvex 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, selfconcondart function.
7. Equality constrained minimization, eliminating equality constraints, infesable start Newton method.
8. Interiorpoint 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 timetable has been prepared for this course
 The course is a part of the following study plans: