Convex sets
2.4 Generalized inequalities
2.4.1 Proper cones and generalized inequalities
A cone K⊆ Rn is called a proper cone if it satisfies the following:
• K is convex.
• K is closed.
• K is solid, which means it has nonempty interior.
• K is pointed, which means that it contains no line (or equivalently, x ∈ K, − x ∈ K =⇒ x = 0).
A proper cone K can be used to define a generalized inequality, which is a partial ordering on Rn that has many of the properties of the standard ordering on R.
We associate with the proper cone K the partial ordering on Rn defined by x¹Ky ⇐⇒ y − x ∈ K.
We also write xºK y for y¹K x. Similarly, we define an associated strict partial ordering by
x≺K y ⇐⇒ y − x ∈ int K,
and write x ÂK y for y ≺K x. (To distinguish the generalized inequality ¹K from the strict generalized inequality, we sometimes refer to ¹K as the nonstrict generalized inequality.)
When K = R+, the partial ordering ¹K is the usual ordering ≤ on R, and the strict partial ordering ≺K is the same as the usual strict ordering < on R.
So generalized inequalities include as a special case ordinary (nonstrict and strict) inequality in R.
Example 2.14 Nonnegative orthant and componentwise inequality. The nonnegative orthant K = Rn+ is a proper cone. The associated generalized inequality¹K corre-sponds to componentwise inequality between vectors: x¹K y means than xi ≤ yi, i = 1, . . . , n. The associated strict inequality corresponds to componentwise strict inequality: x≺Ky means than xi< yi, i = 1, . . . , n.
The nonstrict and strict partial orderings associated with the nonnegative orthant arise so frequently that we drop the subscript Rn+; it is understood when the symbol
¹ or ≺ appears between vectors.
Example 2.15 Positive semidefinite cone and matrix inequality. The positive semidef-inite cone Sn+is a proper cone in Sn. The associated generalized inequality¹K is the usual matrix inequality: X ¹K Y means Y − X is positive semidefinite. The inte-rior of Sn+ (in Sn) consists of the positive definite matrices, so the strict generalized inequality also agrees with the usual strict inequality between symmetric matrices:
X≺K Y means Y − X is positive definite.
Here, too, the partial ordering arises so frequently that we drop the subscript: for symmetric matrices we write simply X ¹ Y or X ≺ Y . It is understood that the generalized inequalities are with respect to the positive semidefinite cone.
Example 2.16 Cone of polynomials nonnegative on [0, 1]. Let K be defined as K ={c ∈ Rn| c1+ c2t +· · · + cntn−1≥ 0 for t ∈ [0, 1]}, (2.15) i.e., K is the cone of (coefficients of) polynomials of degree n−1 that are nonnegative on the interval [0, 1]. It can be shown that K is a proper cone; its interior is the set of coefficients of polynomials that are positive on the interval [0, 1].
Two vectors c, d∈ Rnsatisfy c¹Kd if and only if
c1+ c2t +· · · + cntn−1≤ d1+ d2t +· · · + dntn−1 for all t∈ [0, 1].
Properties of generalized inequalities
A generalized inequality¹K satisfies many properties, such as
• ¹K is preserved under addition: if x¹Ky and u¹K v, then x + u¹K y + v.
• ¹K is transitive: if x¹K y and y¹Kz then x¹K z.
• ¹K is preserved under nonnegative scaling: if x ¹K y and α ≥ 0 then αx¹K αy.
• ¹K is reflexive: x¹Kx.
• ¹K is antisymmetric: if x¹K y and y¹K x, then x = y.
• ¹K is preserved under limits: if xi¹K yifor i = 1, 2, . . ., xi→ x and yi→ y as i→ ∞, then x ¹Ky.
The corresponding strict generalized inequality≺K satisfies, for example,
• if x ≺K y then x¹K y.
• if x ≺K y and u¹K v then x + u≺K y + v.
• if x ≺K y and α > 0 then αx≺K αy.
• x 6≺Kx.
• if x ≺K y, then for u and v small enough, x + u≺K y + v.
These properties are inherited from the definitions of¹K and ≺K, and the prop-erties of proper cones; see exercise 2.30.
2.4.2 Minimum and minimal elements
The notation of generalized inequality (i.e., ¹K, ≺K) is meant to suggest the analogy to ordinary inequality on R (i.e.,≤, <). While many properties of ordinary inequality do hold for generalized inequalities, some important ones do not. The most obvious difference is that ≤ on R is a linear ordering: any two points are comparable, meaning either x ≤ y or y ≤ x. This property does not hold for other generalized inequalities. One implication is that concepts like minimum and maximum are more complicated in the context of generalized inequalities. We briefly discuss this in this section.
We say that x∈ S is the minimum element of S (with respect to the general-ized inequality ¹K) if for every y∈ S we have x ¹K y. We define the maximum element of a set S, with respect to a generalized inequality, in a similar way. If a set has a minimum (maximum) element, then it is unique. A related concept is minimal element. We say that x∈ S is a minimal element of S (with respect to the generalized inequality ¹K) if y ∈ S, y ¹K x only if y = x. We define maxi-mal element in a similar way. A set can have many different minimaxi-mal (maximaxi-mal) elements.
We can describe minimum and minimal elements using simple set notation. A point x∈ S is the minimum element of S if and only if
S⊆ x + K.
Here x + K denotes all the points that are comparable to x and greater than or equal to x (according to ¹K). A point x∈ S is a minimal element if and only if
(x− K) ∩ S = {x}.
Here x− K denotes all the points that are comparable to x and less than or equal to x (according to¹K); the only point in common with S is x.
For K = R+, which induces the usual ordering on R, the concepts of minimal and minimum are the same, and agree with the usual definition of the minimum element of a set.
Example 2.17 Consider the cone R2+, which induces componentwise inequality in R2. Here we can give some simple geometric descriptions of minimal and minimum elements. The inequality x¹ y means y is above and to the right of x. To say that x∈ S is the minimum element of a set S means that all other points of S lie above and to the right. To say that x is a minimal element of a set S means that no other point of S lies to the left and below x. This is illustrated in figure 2.17.
Example 2.18 Minimum and minimal elements of a set of symmetric matrices. We associate with each A∈ Sn++an ellipsoid centered at the origin, given by
EA={x | xTA−1x≤ 1}.
We have A¹ B if and only if EA⊆ EB. Let v1, . . . , vk∈ Rnbe given and define
S ={P ∈ Sn++| vTiP−1vi≤ 1, i = 1, . . . , k},
PSfrag replacements x1
x2
S1
S2
Figure 2.17 Left. The set S1 has a minimum element x1 with respect to componentwise inequality in R2. The set x1+ K is shaded lightly; x1 is the minimum element of S1 since S1 ⊆ x1+ K. Right. The point x2 is a minimal point of S2. The set x2− K is shown lightly shaded. The point x2 is minimal because x2− K and S2intersect only at x2.
which corresponds to the set of ellipsoids that contain the points v1, . . . , vk. The set S does not have a minimum element: for any ellipsoid that contains the points v1, . . . , vk we can find another one that contains the points, and is not comparable to it. An ellipsoid is minimal if it contains the points, but no smaller ellipsoid does.
Figure 2.18 shows an example in R2 with k = 1.