Optimization Theory and Applications

Inhaltsverzeichnis

• § 1 Introduction, Examples, Survey.
• 1.1 Optimization problems in elementary geometry.
• 1.2 Calculus of variations.
• 1.3 Approximation problems.
• 1.4 Linear programming.
• 1.5 Optimal Control.
• 1.6 Survey.
• 1.7 Literature.
• § 2 Linear Programming.
• 2.1 Definition and interpretation of the dual program.
• 2.2 The FARKAS-Lemma and the Theorem of CARATHEODORY.
• 2.3 The strong duality theorem of linear programming.
• 2.4 An application: relation between inradius and width of a polyhedron.
• 2.5 Literature.
• § 3 Convexity in Linear and Normed Linear Spaces.
• 3.1 Separating convex sets in linear spaces.
• 3.2 Separation of convex sets in normed linear spaces.
• 3.3 Convex functions.
• 3.4 Literature.
• § 4 Convex Optimization Problems.
• 4.1 Examples of convex optimization problems.
• 4.2 Definition and motivation of the dual program. The weak duality theorem.
• 4.3 Strong duality, KUHN-TUCKER saddle point theorem.
• 4.4 Quadratic programming.
• 4.5 Literature.
• § 5 Necessary Optimality Conditions.
• 5.1 GATEAUX and FRECHET Differential.
• 5.2 The Theorem of LYUSTERNIK.
• 5.3 LAGRANGE multipliers. Theorems of KUHN-TUCKER and F. JOHN type.
• 5.4 Necessary optimality conditions of first order in the calculus of variations and in optimal control theory.
• 5.5 Necessary and sufficient optimality conditions of second order.
• 5.6 Literature.
• § 6 Existence Theorems for Solutions of Optimization Problems.
• 6.1 Functional analytic existence theorems.
• 6.2 Existence of optimal controls.
• 6.3 Literature.
• Symbol Index.