Hyperplane {x|a Tx = b}

Half space {x|a Tx ≤ b}

Polyhedron P = {x|a T j x ≤ bj , j = 1, · · · , m})

Norms (non-negative, positive definite, homogeneous, triangle inequalities)

Convergence (Xk→X*)

Basic concepts of geometry

Supporting and separating hyperplanes

Convex set separation theorem (projection theorem, supporting hyperplane theorem, hyperplane separation theorem and its converse theorem)

Separation of affine sets and convex sets

strict separation

Meaning and theorem (closed compact set of two polyhedrons)

Strict separation and selection theorem of points and convex sets (application to generalized convex programming)

The meaning of normal cone: convex solid closed tip

Generalized inequalities and their properties

Minimal elements and minimal elements and their set descriptions

Dual properties of minimal elements

Dual properties of minimal elements

Dual cones and their properties

first-order conditions

second-order conditions

Summary: The photogenic graph of a convex function is a convex set

Norms are all convex (Chebyshev norm)

The maximum value R^n is convex upward

Geometric mean concave function in the domain

non-negative weighted sum

composite radiation mapping

Convexity of point-wise maximum and point-wise supremum

Understanding of conjugate functions (affine, exponential, and negative entropy functions are derived based on point-wise supremums)