欧几里德投影（Euclidean projection）

Euclidean projection on a set

An Euclidean projection of a point on a set is a point that achieves the smallest Euclidean distance from to the set. That is, it is any solution to the optimization problem

When the set is convex, there is a unique solution to the above problem. In particular, the projection on an affine subspace is unique.

Example: assume that is the hyperplane

The projection problem reads as a linearly constrained least-squares problem, of particularly simple form:

The projection of on turns out to be aligned with the coefficient vector . Indeed, components of orthogonal to don't appear in the constraint, and only increase the objective value. Setting in the equation defining the hyperplane and solving for the scalar we obtain , so that the projection is .

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凸集（Convex Set）：

Definitions

A subset of is said to be convex if and only if it contains the line segment between any two points in it:

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 anymore in the above condition.

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当且仅当上的子集包含任意二点直接的线段满足：

很明显，子空间和仿射集合，诸如直线，平面以及更高维度的水平集均是凸集。

Examples:

• A convex and a non-convex set.

• Convex and conic hull of a set of points.

A set is said to be a convex cone if it is convex, and has the property that if , then for every .