Convex Optimization 2026, 31 - Projection on a Set

Define the distance between the a point and a closed set in the same space as the infimum of the distances of that point and any point of the set under a norm. Given that the set is nonempty, this infimum exists. That point which defines the infimum is considered the projection of the point on the set. For closed and convex sets and strictly convex norm, that relation is uniquely identifying, and if for every result. Conversely, if for every point there is a unique Euclidean projection onto the set C, then C is closed and convex. Note P_C: ℝⁿ → ℝⁿ to denote any function for which P_C(x₀) is a projection of x₀ on C as ||x₀ - P_C(x₀)|| = dist(x₀, C). P_C then is a projection on C. The projection and distance of points w.r.t. a convex set C can be cast as a convex optimization problem, as C itself can be described as a set of linear inequalities and convex inequalities. The resulting problem presupposes a nonempty C (8.1.1). A point can be separated from a convex set by finding a hyperplane orthogonal to the point's projection onto the set. The hyperplane is built as (P_C(x₀) - x₀)^T(x - (1/2)(x₀ + P_C(x₀))) = 0, given the norm used in the projection. This doesn't function equally well for all norms, occasionally requiring the Lagrange duality (8.1.2).