Optimization theory - (2) Convexity 1 Convex set

• Convexity is a basic concept that must be involved in optimization theory
• The nonlinear programming model with convexity is a special and important type of model, which plays a very important role in the theoretical proof of optimization and algorithm research.

1. Convex collection

1.1 Concept and properties of convex sets

1.1.1 Several combinations of points

1.1.2 Definition of convex set

1.1.3 Common convex sets

1.1.4 Properties of convex sets

1.1.5 Cones and convex cones

1.2 Separation theorem of convex sets

1.2.1 Definition and theorem

1.2.3 Application of convex set separation theorem

1. Farkas' Lemma


2. Farkas' Lemma - Geometric Interpretation


Using Farkas' lemma, the following Gordan's theorem and alternative theorem can be derived.
3. Gordan's theorem

4. Alternative theorem