SUBJECT
Nonlinear optimization
lecture
master
4
Semesters 1-4
Autumn/Spring semester
Extremal points, extremal sets. Krein-Milman theorem. Convex cones. Recession direction, recession cones. Strictly-, strongly convex functions. Locally convex functions. Local minima of the functions. Characterization of local minimas. Stationary points. Nonlinear programming problem. Characterization of optimal solutions. Feasible, tangent and decreasing directions and their forms for differentiable and subdifferentiable functions. Convex optimization problems. Separation of convex sets. Separation theorems and their consequences. Convex Farkas theorem and its consequences. Saddle-point, Lagrangean-function, Lagrange multipliers. Theorem of Lagrange multipliers. Saddle-point theorem. Necessary and sufficient optimality conditions for convex programming. Karush-Kuhn-Tucker stationary problem. Karush-Kuhn-Tucker theorem. Lagrange-dual problem. Weak and strong duality theorems. Theorem of Dubovickij and Miljutin. Specially structured convex optimization problems: quadratic programming problem. Special, symmetric form of linearly constrained, convex quadratic programming problem. Properties of the problem. Weak and strong duality theorem. Equivalence between the linearly constrained, convex quadratic programming problem and the bisymmetric, linear complementarity problem. Solution algorithms: criss-cross algorithm, logarithmic barrier interior point method.
- Béla Martos: Nonlinear Programming: Theory and Methods. Akadémiai Kiadó, Budapest, 1975.
- M. S. Bazaraa, H. D. Sherali and C. M. Shetty: Nonlinear Programming: Theory and Algorithms. John Wiley & Sons, New York, 1993.
- J.-B. Hiriart-Urruty and C. Lemaréchal: Convex Analysis and Minimization Algorithms I-II. Springer-Verlag, Berlin, 1993.
- J. P. Aubin: Mathematical Methods of Game and Economic Theor. North-Holland, Amsterdam, 1982.
- D. P. Bertsekas: Nonlinear Programming. Athena Scientific, 2004.