basis for convex optimization
Published:
Notable Concepts
- line / (closed) line segments: \theta * x1 + (1 - \theta) * x2, theta in R / in [0, 1]
- affine set / convex set: \theta * x1 + (1 - \theta) * x2, theta in R / in [0, 1]
- cone: theta * x, theta in R+ (nonnegative real values)
- hyperplane / halfspace: a^T x = b or a^T (x - x0) = 0 / a^T x < b, a^T x > b or a^T (x - x0) < 0, a^T (x - x0) > 0
- balls / ellipsoids: (x - xc)^T P (x - xc) <= r^2 , P = I or other kinds of positive definite solution.
- polyhedron: the solution set of a finite number of linear equalities and inequalities
- simplex: conv hull of k+1 affinely independent points, and its affine dimensionality is k. such as line segment, triangle (with interior), tetrahedron (with interior). As well as, unit simplex (x >= 0, 1^T * x <= 1), or probability simplex (x >= 0, 1^T * x = 1)
- partial order, minimum (最小, for each s in S, s <= s0, or, S included in s0+K), minimal (极小, if s in S s.t. s<= s0, then s = s0, or, Intersection(so-K, S) = {x} )
- supporting hyperplane
Optimization Problems
- min f_0, s.t. f_i <= 0, h_j = 0.