(Geometric multiplier). The pair is a called a "geometric multiplier" for the problem ( Primal problem ) if and

The following statement directly follows from the definitions ( Primal problem ),( Geometric multiplier

(Visualization lemma). Assume .

1. The hyperplane in with normal that passes through also passes through .

2. Among all hyperplanes with normal that contain the set in the upper half-space, the highest level of intersection with the axis is given by .

(Geometric multiplier property). Let be a geometric multiplier then is a global minimum of the problem ( Primal problem ) if and only if and

Note that implies and and the definition ( Geometric multiplier ) requires . Hence, Hence, and .

Let be a global minimum of the problem ( Primal problem ) then By the definition ( Geometric multiplier ), Therefore, and .

The statement is proven similarly in the other direction.

(Lagrange multiplier). The pair is called "Lagrange multiplier of the problem ( Primal problem ) associated with the solution " if and

The following statement is a consequence of the proposition ( Local minimum of a sum ) and definitions.

Assume that the problem ( Primal problem ) has at least one solution .

1. Let and are either convex or smooth , are smooth, is closed and is convex then every geometric multiplier is a Lagrange multiplier.

2. Let and are convex, are affine and is closed and convex then the sets of Lagrange and geometric multiplier coincide.