Definition 2.1 (Convex Combination) A convex combination of a set of n vectors
A set that is closed under arbitrary convex combinations is a convex set. Geometrically speaking, convex sets are those that contain all line segments that join two points inside the set. As a result, they cannot have any inward “bulges”.
The generic form of an analytic optimization problem is the following
where x is the variable of the problem,
A convex optimization problem is described as the objective is a convex function as well as the constraint set is a convex set. A non-convex optimization problem is described when it violates either one of these conditions, i.e., one that has a non-convex objective, or s non-convex constraint set, or both.