Cones & Dual Cones#


Definition of a Cone#

A cone VRn is a set of vectors that is closed under both vector addition and nonnegative-scalar multiplication.

Closure under vector addition

u+vVu,vV

Closure under nonnegative-scalar multiplication

cvVvV,cR0


Conical Combination#

A conical combination is just a linear combination constrained to have nonnegative coefficients.

i=1naixiwhereai0


Cones in Rn#

Space

Cones

R

{0}, rays, R

R2

{0}, rays, lines, wedges, half planes, R2

R3

{0}, rays, lines, 2D wedges, half planes, planes, 3D wedges, half solids, R3

R4

{0}, rays, lines, 2D wedges, half planes, planes, 3D wedges, half solids, solids, 4D wedges, half hyper solids, R4

Cones in R#

The zero vector {0} is a cone.

All rays are cones.

construction of a ray r={cvcR0,vR0}

R is a cone.

Cones in R2#

The zero vector#

The zero vector {0} is a cone.

Rays#

construction of a ray r={cvcR0,vR20}

All rays are cones.

Lines#

A line is usually constructed as ={cvcR,vR20}. But a cone is closed under nonnegative-scalar multiplication, not scalar multiplication. So a line is constructed as the union of two rays pointing in opposite directions.

construction of a line ={c1v1+c2v2c1,c2R0,v1,v2R20,v2=v1}={c1v+c2(v)c1,c2R0,vR20}

All lines are cones.

Wedges#

construction of a wedge w={c1v1+c2v2c1,c2R0,v1,v2R20} for independent v1,v2

If v1,v2 are independent, then V is a wedge.

If v1,v2 are dependent, then v2=cv1 and this construction is reduced to that of either a ray or a line. If c>0, then V is a ray. If c<0, then V is the union of two rays pointing in opposite directions and so is a line.

All wedges are cones.

Half Planes#

construction of a half plane H={c1v1+c2v2+c3v3c1,c2,c3R0,v1,v2,v3R20,v2=v1}={c1v1+c2(v1)+c3v3c1,c2,c3R0,v1,v3R20} for independent v1,v3

If v1,v3 are independent, then V is a half plane.

All half planes are cones.

R2#

Cones in R3#

the zero vector {0}

rays r={cvcR0,vR30}

lines ={c1v+c2(v)c1,c2R0,vR30}

2D wedges w={c1v1+c2v2c1,c2R0,v1,v2R30} for independent v1,v2

half planes h={c1v1+c2(v1)+c3v3c1,c2,c3R0,v1,v3R30} for independent v1,v3

Planes#

A plane is usually constructed as p={c1v1+c2v2c1,c2R,v1,v2R20} for independent v1,v2. But a cone is closed under nonnegative-scalar multiplication, not scalar multiplication. So a plane requires that v2 be moved out of the half plane constructed by v1,v2,v3 bounded by the line given by v2=v1.

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.

h1={c1v1+c2(v1)+c3v3c1,c2,c3R0,v1,v3R30} for independent v1,v3

h2={c1v1+c2(v1)+c3(v3)c1,c2,c3R0,v1,v3R30} for independent v1,v3

p=h1h2={c1v1+c2(v1)+c3v3+c4(v3)cR0,vR30} for independent v1,v3

3D wedges#

Construction: Tetrahedron (Trigonal Cone)#

w={c1v1+c2v2+c3v3cR0,vR30} for independent v1,v2,v3

Construction: n-gonal Cone#

w={c1v1+c2v2+c3v3+...+cnvncR0,vR30} for independent v1,v2,v3

  • 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 R3.

Construction: Cylinders (Fans?)#

h1={c1v1+c2(v1)+c3v2cR0,vR30} for independent v1,v2

h2={c1v1+c2(v1)+c3v3cR0,vR30} for independent v1,v3

c=h1h2={c1v1+c2(v1)+c3v2+c4v3cR0,vR30} for independent v1,v2,v3

Half Solids#

Construction: Union of a plane and a ray#

r={cv5cR0,vR30}

p={c1v1+c2(v1)+c3v3+c4(v3)cR0,vR30} for independent v1,v3

s=rp={c1v1+c2(v1)+c3v3+c4(v3)+c5v5cR0,vR30} for independent v1,v3,v5

R3#


Dual Cone#

If VRn is a cone, then its dual is

V={uRnuv0vV}


If V is a cone, then its dual V is also a cone.

Proof

(c1u)v=c1(uv)(c10)(uv0)(c1u)v0(u1v0)(u2v0)(u1+u2)v=u1v+u2v0


Examples of dual cones in R2#

Wedge

S={c1v1+c2v2ci0}


Summary of cones in R2 and their duals#

Cone

Dual

{0}

R2

ray

half plane

line

perpendicular line

wedge

wedge

half plane

ray

R2

{0}


[PROPOSITION]

If V is a closed cone, then V=V. That is, the dual of the dual is the primal.

PROOF

vVuv0uVVV

Suppose vV and let wV be a vector with minimal angle θ with v. In other words, if wV and θ is the angle between v and w, then θθ.

Since V is closed, Vc is open θ>0.

Let uVuw=0.

Then uv<0vVVV

The dual cone is always closed; so, the dual of the dual is closed.

The only truly open cone in Rn is Rn itself, which is both open and closed.

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 V is the union of the zero vector along with the interior of a namesake cone, then V is a cone.

The dual cone V picks up boundary vectors, as does the dual of the dual.

[EXAMPLE OF UNCLOSED CONE]

If V is the union of the zero vector along with the interior of a half plane cone, then V is a cone.

The dual cone V is the ray normal to the half plane. The dual fo the dual includes the full half plane.


[FEASIBLE DIRECTION & CONE OF FEASIBLE DIRECTIONS]

Consider a feasible point x in the control space of an LP and let vx be a vector with origin at x.

Regardless of its magnitude, vx can be considered to define a direction. Call the direction defined by vx a feasible direction if there exists another feasible point y in the direction vx from x, at any positive distance from x not necessarily at the terminus of vx.

For any feasible point x, define the set Vx={vxvxrepresents a feasible direction fromx}

[PROPOSITION]

Vx is a cone.

PROOF

Follows from the convexity of the feasible set.

[PROPOSITION]

If F is the feasible set of an LP and the objective function z has a minimum at x, then

zVx

That is, it is a necessary condition for a minimum at a feasible point x that the gradient of the objective function be in the dual cone of the cone of feasible directions at the point x.

PROOF

Starting at x, z gets smaller on the set F there is a feasible direction in which z has a negative directional derivative; that is, a feasible direction vxVx such that zvx<0.

Therefore

z(x)=minFzzvx0vxVx

which, by definition, places z in the dual cone Vx at x.

This is an important test for a minimum.

[EXERCISE]

  1. Choose some feasible point x (an interior point; a boundary point; a vertex).

  2. Determine the cone Vx of feasible directions at x.

  3. Determine the dual cone Vx.

  4. Verify that any function z whose gradient zVx has a minimum minz at x.

  5. Verify that any function z whose gradient zVx does not have a minimum minz at x.

[EXAMPLE]

Suppose x is an interior point of the feasible set.

Then the cone of feasible directions is Rn and the dual is just the zero vector.

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

g1(x)b1,...,gk(x)bk

then at any regular point x the boundaries of the dual cone Vx are the gradients

g1,...,gk

of the active constraints at x

[PROPOSITION]

Let x be a feasible point in the control space of an LP and let s be the nonnegative vector of slack variables at x.

Then the dual cone

Vx={i=1nyigiy0ys=0}

PROOF

Because y and s are each nonnegative

ysyisi=0i

which means that

yi>0si=0

and

si>0yi=0

The condition ys=0 stipulates that only the gradients gi of the active constraints si=0 can be given positive coefficients yi>0.

If x is a regular point, then there is a unique solution for y.

If x is not a regular point, then there are redundant active constraints for which si=0 and these provide extra, unnecessary gradients yielding multiple solutions for y. But these extra gradients are already in the dual cone which is not changed by their inclusion in the generating set.

[PROPOSITION] special case of KKT Karush Kuhn Tucker conditions

zis at a minimum at a feasible pointxiyigihas a solutiony=(y1,...,yk)satisfyingy,s0ys=0

This enables solving the primal minimization problem by finding all feasible points x at which the objective gradient is in the dual cone of the cone of feasible directions at x.

[COROLLARY]

The test

z=iyigiwherey0ys=0

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

zhas a maximum atxzhas a minimum atx

and if the constraints are

g1(x)b1,...,gk(x)bk

then

g1,...,gksi=0

are the boundaries of Vx

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