Cones & Dual Cones#
Definition of a Cone#
A cone
Closure under vector addition
Closure under nonnegative-scalar multiplication
Conical Combination#
A conical combination is just a linear combination constrained to have nonnegative coefficients.
Cones in #
Space |
Cones |
---|---|
Cones in #
The zero vector
All rays are cones.
construction of a ray
Cones in #
The zero vector#
The zero vector
Rays#
construction of a ray
All rays are cones.
Lines#
A line is usually constructed as
construction of a line
All lines are cones.
Wedges#
construction of a wedge
If
If
All wedges are cones.
Half Planes#
construction of a half plane
If
All half planes are cones.
#
Cones in #
the zero vector
rays
lines
2D wedges
half planes
Planes#
A plane is usually constructed as
All planes are cones.
Plane Construction: Union of two half planes with two independent generating vectors#
One way to construct a plane is to take the union of two half planes with a common boundary line and which share two independent generating vectors.
3D wedges#
Construction: Tetrahedron (Trigonal Cone)#
Construction: -gonal Cone#
Pyramidal Cone
Pentagonal Cone
Hexagonal Cone
Heptagonal Cone
Octogonal Cone
Nonogonal Cone
Decagonal Cone
…
Circular Cone
Construction: Circular Cones#
Rotate two independent vectors around their bisector to generate a circular cone in
Construction: Cylinders (Fans?)#
Half Solids#
Construction: Union of a plane and a ray#
#
Dual Cone#
If
If
Proof
Examples of dual cones in #
Wedge
Summary of cones in and their duals#
Cone |
Dual |
---|---|
ray |
half plane |
line |
perpendicular line |
wedge |
wedge |
half plane |
ray |
[PROPOSITION]
If
PROOF
Suppose
Since
Let
Then
The dual cone is always closed; so, the dual of the dual is closed.
The only truly open cone in
There are cones which are neither open nor closed because they need not contain all their boundary points to satisfy the closure requirements that define a cone. But the dual is defined by weak inequalities that necessarily result in a closed set; so, the dual of the dual includes the primal and all the boundary points of the primal, which is called the closure of the primal.
[EXAMPLE OF UNCLOSED CONE]
If
The dual cone
[EXAMPLE OF UNCLOSED CONE]
If
The dual cone
[FEASIBLE DIRECTION & CONE OF FEASIBLE DIRECTIONS]
Consider a feasible point
Regardless of its magnitude,
For any feasible point
[PROPOSITION]
PROOF
Follows from the convexity of the feasible set.
[PROPOSITION]
If
That is, it is a necessary condition for a minimum at a feasible point
PROOF
Starting at
Therefore
which, by definition, places
This is an important test for a minimum.
[EXERCISE]
Choose some feasible point
(an interior point; a boundary point; a vertex).Determine the cone
of feasible directions at .Determine the dual cone
.Verify that any function
whose gradient has a minimum at .Verify that any function
whose gradient does not have a minimum at .
[EXAMPLE]
Suppose
Then the cone of feasible directions is
In other words, only a constant function has a minimum at an interior point.
[PROPOSITION]
If the feasible set of an LP is defined by greater-than-or-equal-to constraints
then at any regular point
of the active constraints at
[PROPOSITION]
Let
Then the dual cone
PROOF
Because
which means that
and
The condition
If
If
[PROPOSITION] special case of KKT Karush Kuhn Tucker conditions
This enables solving the primal minimization problem by finding all feasible points
[COROLLARY]
The test
applies to a maximization problem with less-than-or-equal-to constraints, just like it
applies to a minimization problem with greater-than-or-equal-to constraints.
PROOF
and if the constraints are
then
are the boundaries of
The other two cases
maximization problem with greater-than-or-equal-to constraints
minimization problem with less-than-or-equal-to constraints
require a sign change of either RHS or LHS or an equivalent reformulation of the contraints.
Terms#
[W] Cone
[W] Cone, topology
[W] Conical Combination
[W] Conical Hull
[W] Conical Surface
[W] Convex Cone
[W] Half Line
[W] Half Plane
[W] Half Space
[W] Hypercone
[W] Normal Fan
[W] Polyhedral Complex
[W] Ray
[W] Simplicial Complex