IE 511: Integer Programming, Spring 2019 31 Jan, 2019 Lecturer: Karthik Chandrasekaran Lecture 6: Faces, Facets Scribe: Setareh Taki Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. Recall that we are attempting to get a minimal description of a polyhedron. Recap Definition 1. Consider Ax b. An inequality is redundant if it is implied by other inequalities in Ax b. An irredundant system has no redundant inequality. We will formally prove that an irredundant system gives a minimal inequality description for a polyhedron. Definition 2. Let P = {x : Ax b}. 1. An inequality c T x δ is valid for P if c T x δ x P. 2. {x : c T x = δ} is a supporting hyperplane of P if δ = max{c T x : Ax b} and c 0, i.e., c T x δ is valid for P and {x : c T x = δ} P. Example: See Figure 6.1 for valid inequalities and supporting hyperplanes. Figure 6.1: Hyperplanes corresponding to inequalities 1 and 2 are supporting but the hyperplane corresponding to inequality 3 is not. 6.1 Faces We now define the notion of a face of a polyhedron. Definition 3. A set F P is a face of P if either F = P or if F is the intersection of P and a supporting hyperplane of P. Example: See Figure 6.2. 6-1
Figure 6.2: Example for faces of a polyhedron: Left polyhedron has 11 faces two of which are marked. Right polyhedron has only two faces: {(x, y) : y = 0} and the polyhedron itself. Note that by definition, a face of a polyhedron is also a polyhedron. Next we will see a convenient way to understand faces of a polyhedron from its inequality description. Theorem 4. [Characterization of faces] Let P = {x : Ax b} and let F P. Then, F is a face of P iff F and F = {x : x P, A x = b } for some subsystem A x b. Proof. = : Let F be a face of P. Then F = P {x : c T x = δ} for some c 0 and δ = max{c T x : x P } where δ is finite. By duality theorem we have that δ = min{y T b : y T A = c T, y 0}. Let y be an optimal solution for min{y T b : y T A = c T, y 0}. Then y 0 because c 0. Let A x b be a subsystem of Ax b corresponding to the positive components of y. Since y 0, the subsystem is non-empty. By complementary slackness conditions we have that x is optimal for max{c T x : x P } iff x P and A x = b. Also, x F iff x is optimal for max{c T x : x P }. Therefore, F = {x : x P, A x = b }. = : Suppose F = {x P : A x = b } for some subsystem A x b of Ax b and F. Let c be the sum of the linearly independent rows of A and δ be the sum of the corresponding rows of b. Then c 0. We will show the following two claims which together imply that F is a face of P. Claim 4.1. Proof. Let x P. We have that F = {x P : c T x = δ}. If x F, then c T x = δ by the choice of c and δ. Hence, x {x P : c T x = δ} Suppose x / F. Then A x b as x P. Since x F, there exists an inequality a T i x b i in the system A x b for which a T i x < b i and it implies that c T x < δ. Hence x {x P : c T x = δ}. Claim 4.2. {x : c T x = δ} is a supporting hyperplane of P. 6-2
Proof. Since c T x δ is a non-negative combination of the inequalities in A x b, we have that c T x δ is valid for P. Also, since F = {x P : c T x = δ} by previous claim and F, we have P {x : c T x = δ}. So, {x : c T x = δ} is supporting hyperplane of P. The first claim implies that F = {x P : c T x = δ} and the second claim implies that {x : c T x = δ} is a supporting hyperplane of P. Hence, F is a face of P completing the proof of the theorem. The above characterization of faces tells us that all faces of a polyhedron P = {x : Ax b} are obtained by turning some of the inequalities in the system Ax b into equations. In particular, this implies that the number of faces in a polyhedron is finite. Corollary 4.1. A polyhedron has finite number of faces. The characterization of faces also implies that the face of a face of a polyhedron is also a face of the original polyhedron and vice-versa. Corollary 4.2. Let F be a face of P and F F. Then F is a face of P iff F is a face of F. 6.2 Facets We note that faces do not give a minimal description of a polyhedron as c T x δ may not be needed to describe P (e.g., see Figure 6.3). Figure 6.3: A polyhedron and one of its face However, we will next see that maximal faces give a minimal description of a polyhedron. Definition 5. A facet is a maximal face distinct from P. Example: See Figure 6.4. Figure 6.4: Facet Let us see how to obtain the facets of a polyhedron from the inequality description of the polyhedron. Recall that A = x b = are the implicit equalities of the system Ax b and A + x b + are the remaining inequalities of Ax b. 6-3
Theorem 6. Suppose no inequality in A + x b + is redundant in Ax b. Then there is a one to one correspondence between facets of P and the inequalities in A + x b + given by F = {x P : a T i x = b i} for facets F and inequalities a T i x b i in A + x b +, i.e., if the non-implicit inequalities form an irredundant system, then the facets are obtained by turning exactly one of the non-implicit inequalities into an equation. Proof. We prove the theorem by proving the following two lemmas: Lemma 6.1. Each facet of P can be represented as {x P : a T i x = b i} for some a T i x b i in A + x b +. Proof. Let F be a facet of P. Then F is a face and F P. By the characterization of faces, we have that F = {x P : A x = b } for some subsystem A x b of Ax b. We may assume that A x b is a subsystem of A + x b + (otherwise F = P which contradicts F being a facet). Let a T i x b i be an inequality in A x b. Consider F = {x P : a T i x = b i}. The set F is a face of P where F P because a T i x b i is in A + x b +. We note that F F P. Moreover, F P since a T i x = b i is in A + x b + and is hence not an implicit equation. Therefore, F P and by Theorem 4, we have that F is a face of P that is distinct from P. But F is a facet, i.e., a maximal face of P that is distinct from P. So, we have that F = F = {x P : a T i x = b i}. Lemma 6.2. For each inequality a T i x b i in A + x b +, the set F = {x P : a T i x = b i} is a facet of P. Proof. Let a T i x b i be an inequality in A + x b + and let A x b be the other inequalities in A + x b +. Let F = {x P : a T i x = b i}. By characterization of faces, the set F is a face of P. Since a T i x b i is not in A = x b =, we have F P. We need to show that F is a facet of P, i.e., we need to show that the only face of P containing the face F is P. To show this, we will show the following claim. Claim 6.1. There exists x 0 such that A = x 0 = b =, A x 0 < b, a T i x0 = b i. Proof. Let x be a vector such that A = x = b =, A x < b, a T i x < b i. Since a T i x b i is not redundant in Ax b, there exists a vector x 2 such that A = x 2 = b =, A x 2 b, a T i x 2 > b i. Consequently, there exists a point x 0 on the line segment between x 1 and x 2 for which A = x 0 = b =, A x 0 < b, a T i x 0 = b i. 6-4
Claim 6.1 implies that the only face containing F is P : suppose that there exists a face F such that F F P. Then F = {x P : A x = b } for some subsystem A x b of A + x b + by the characterization of faces. Then, x 0 F. Since F F we have x 0 F. This implies that the system A x b is exactly a T i x b i and hence F = F. Therefore, F is a facet. Lemmas 6.1 and 6.2 complete the proof of the theorem characterizing facets. Theorem 6 shows that if a system has no implicit equations and is irredundant, then it is a unique minimal description of P. Corollary 6.1 (Informal). Let P = {x : Ax b}. If P is full dimensional and Ax b is irredundant then Ax b is the unique minimal representation of P (up to multiplication of inequalities by positive scalers), i.e., each inequality is necessary and the system Ax b is sufficient. Note: Theorem 6 tells that facets are necessary and sufficient to describe a polyhedron. Let us see some consequences of the characterization of facets given by Theorem 6. Definition 7. A face distinct from P is a proper face. Corollary 7.1. Each proper face of P is the intersection of facets of P. Corollary 7.2. If F is a facet of P then dim(f ) = dim(p ) 1. Proof. dim(p ) = n rank (A = x) ([ ]) A = dim(f ) = n rank = dim(p ) 1 a T i The second equality holds because a T i x = b i is not redundant. Corollary 7.2 tells us that all facets of a polyhedron have the same dimension. Note 1: To show that an inequality is necessary in the minimal description of a polyhedron, it is sufficient to argue that it is a facet. Note 2: To show that F is a facet of P, it is sufficient to prove that F is a face of P and dim(f ) = dim(p ) 1. 6.3 Minimal Faces We focused on maximal faces of a polyhedron so far. How about minimal faces? How do they look like and do they have any significance? Indeed, they play a significant role in solution techniques for linear programming. We will focus on them in the remainder of this lecture and follow-up in the next lecture. Definition 8. A minimal face is one that does not contain any other face. 6-5
Example: See Figure 6.5. Figure 6.5: Minimal Face Definition 9. An affine subspace is a set of points satisfying a finite number of linear equations. Proposition 10. Let F be a face of P. Then F a minimal face of P iff it is an affine subspace. Proof. A polyhedron Q has no proper faces Q = {x : Ax = b} (by characterization of faces) Q is an affine subspace. 6-6