Convex SetsConvex Optimization > Convex sets | Convex functions | Convex optimization problems | Algorithms | DCP
DefinitionsA subset ![]() Subspaces and affine sets, such as lines, planes and higher-dimensional ‘‘flat’’ sets, are obviously convex, as they contain the entire line passing through any two points, not just the line segment. That is, there is no restriction on the scalar Examples: A set is said to be a convex cone if it is convex, and has the property that if Operations that preserve convexityIntersectionThe intersection of a (possibly infinite) family of convex sets is convex. This property can be used to prove convexity for a wide variety of situations. Examples: Affine transformationIf a map ![]() is convex. In particular, the projection of a convex set on a subspace is convex. Example: Projection of a convex set on a subspace. Separation theoremsSeparation theorems are one of the most important tools in convex optimization. They convex the intuitive idea that two convex sets that do not intersect can be separated by a straight line. There are many versions of separation theorems. One of them is the separating hyperplane theorem: Theorem: Separating hyperplane
If Another result involves the separation of a set from a point on its boundary: Theorem: Supporting hyperplane
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