Nondegeneracy of Polyhedra and Linear Programs

Transcription

1 Computational Optimization and Applications 7, (1997) c 1997 Kluwer Academic Publishers. Manufactured in The Netherlands. Nondegeneracy of Polyhedra and Linear Programs YANHUI WANG AND RENATO D.C. MONTEIRO School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Received December 23, 1994; Revised September 25, 1995 Abstract. This paper deals with nondegeneracy of polyhedra and linear programming (LP) problems. We allow for the possibility that the polyhedra and the feasible polyhedra of the LP problems under consideration be non-pointed. (A polyhedron is pointed if it has a vertex.) With respect to a given polyhedron, we consider two notions of nondegeneracy and then provide several equivalent characterizations for each of them. With respect to LP problems, we study the notion of constant cost nondegeneracy first introduced by Tsuchiya [25] under a different name, namely dual nondegeneracy. (We do not follow this terminology since the term dual nondegeneracy is already used to refer to a related but different type of nondegeneracy.) We show two main results about constant cost nondegeneracy of an LP problem. The first one shows that constant cost nondegeneracy of an LP problem is equivalent to the condition that the union of all minimal faces of the feasible polyhedron be equal to the set of feasible points satisfying a certain generalized strict complementarity condition. When the feasible polyhedron of an LP is nondegenerate, the second result shows that constant cost nondegeneracy is equivalent to the condition that the set of feasible points satisfying the generalized condition be equal to the set of feasible points satisfying the same complementarity condition strictly. For the purpose of giving a preview of the paper, the above results specialized to the context of polyhedra and LP problems in standard form are described in the introduction. Keywords: linear programming, polyhedron, nondegeneracy, constant cost face, complementary slackness 1. Introduction This paper deals with the subject of nondegeneracy of polyhedra and linear programming (LP) problems. Nondegeneracy is a subject worth of intensive investigation due to its application in several branches of mathematical programming and has already been studied in several papers in the literature. These include papers dealing with cycling and termination of the simplex method and with the study of sensitivity and parametric analysis (Adler and Monteiro [1], Akgül [2], Aucamp and Steinberg [3], Beale [5], Bland [6], Charnes [7], Dantzig [8], Gal [10, 11], Greenberg [12], Hoffman [15], Magnanti and Orlin [16], Megiddo [17], Monteiro and Mehrotra [18], Ward and Wendell [29], Williams [30], Wolfe [31]), with the convergence of the affine scaling interior point algorithm (Barnes [4], Dikin [9], Hall and Vanderbei [14], Monteiro and Tsuchiya [19], Monteiro et al. [20], Tsuchiya [24 26], Vanderbei et al. [28], Vanderbei and Lagarias [27]), and etc. The paper by Güler et al. [13] surveys the theoretical and practical issues related to degeneracy in the context of interior point methods for linear programming. The work of these authors was based on research supported by the National Science Foundation under grant DMI and the Office of Naval Research under grants N and N

2 222 WANG AND MONTEIRO Recall that the LP problem optimize {c T x Ax = b, x 0}, where x, c IR n, b IR m and A is an m n-matrix, is said to be primal nondegenerate if every feasible point x has at least m positive components, and strongly primal nondegenerate if every x IR n satisfying Ax = b has at least m nonzero components (see Murty [21], page 121). These two concepts depend on A and b only, and hence only on the feasible polyhedron of the LP problem. The above LP problem is said to be dual nondegenerate if s(y) = c A T y has at least n m nonzero components for every dual feasible solution y IR m, and strongly dual nondegenerate (see Murty [21], page 253) if the same property holds for every y IR m. Note also that the two types of dual nondegeneracy depend only on A and c, and hence only on the dual feasible polyhedron. Hence, it is natural to think of the above notions of nondegeneracy as being concepts associated with polyhedra. In the first part of the paper (Section 3), we study the concept of nondegeneracy of a general (not necessarily pointed) polyhedron. (A polyhedron is said to be pointed if it contains a vertex.) Two notions of nondegeneracy (corresponding to the polyhedron being nondegenerate and/or strongly nondegenerate) are defined and then several equivalent conditions for each type of nondegeneracy are given. Most of the results derived in Section 3 are well known in the context of pointed polyhedra, but are scattered throughout the literature. Our goal here is to provide a unified treatment of this subject and to generalize it to the context of not necessarily pointed polyhedra. The results of Section 3 are not only interesting in their own right but are also needed for a full understanding of the subject of Section 4. In the second part of the paper (Section 4), we discuss the concept of constant cost nondegeneracy (or simply, CC-nondegeneracy) of an LP problem whose feasible region is allowed to be a non-pointed polyhedron. Tsuchiya [25] refers to this concept as dual nondegeneracy, a term which is not appropriate since it is already used to refer to a different but related concept. Consider the LP problem optimize {b T y y P}, where P IR m is a (not necessarily pointed) polyhedron. A nonempty face F of P is said to be a constant cost face of the LP problem optimize {b T y y P} if b T y is constant over F. When the reference to the LP problem is understood, we simply say that F is a constant cost face. The LP problem optimize {b T y y P} is said to be CC-nondegenerate if every constant cost face is a minimal face of P. (A nonempty face is said to be minimal if it does not properly contain any other nonempty face.) One of the main results of Section 4, namely Theorem 4.4, states that CC-nondegeneracy of the LP problem optimize {b T y y P} is equivalent to the condition that the union of all minimal faces of P be equal to the set of feasible points satisfying a certain generalized strict complementarity condition. Moreover, when the polyhedron P is nondegenerate, we show that CC-nondegeneracy is equivalent to the following condition: every point satisfying the generalized complementarity condition must also satisfy it strictly (see Theorem 4.6). We give below a preview of the main results of the paper when specialized to the context of polyhedra and LP problems in standard form. First, we introduce the following notation. We assume for the remaining part of this section that A is an m n matrix and b is an m-vector. Given x IR n, we let σ(x) {i x i 0}. Ifα {1,...,m}and β {1,...,n}, we let A αβ denote the submatrix [A ij ] i α, j β. If α ={1,...,m} we denote A αβ simply by A β and if β ={1,...,n}we denote A αβ by A α, or simply, A α. Given a vector x IR p and an

3 NONDEGENERACY OF POLYHEDRA AND LINEAR PROGRAMS 223 index set α {1,...,p}, we denote the subvector [x i ] i α by x α and the vector [ x i ] p i=1 by x. Given two vectors x IR p and s IR p, we denote the vector [x i s i ] p i=1 by x s. Ifαis a finite set then α denotes its cardinality. The Euclidean norm is denoted by. The polyhedron {x Ax = b, x 0} is said to be nondegenerate if it satisfies any one of the equivalent conditions of the result below. Theorem 1.1. The following statements are equivalent: (a) for any x {x Ax = b, x 0}, σ(x) m(that is, x has at least m positive components); (b) for any x {x Ax = b, x 0}, the rows of A σ(x) are linearly independent (that is, the submatrix of A consisting of the columns corresponding to the positive components of x has full row rank); (c) for any vertex x {x Ax = b, x 0}, σ(x) =m(that is, every vertex has exactly m positive components); (d) for any vertex x {x Ax = b, x 0}, the submatrix A σ(x) is nonsingular (that is, the submatrix of A consisting of the columns corresponding to the positive components of x is a basis of A); (e) for any c IR n and any x {x Ax = b, x 0}, the set arg min{ x s A T y+s= c,(y,s) IR m IR n } contains exactly one point; (f) for any c IR n and any constant cost face F of optimize {c T x Ax = b, x 0}, the set {(y, s) A T y + s = c and s x = 0 for some x F} contains exactly one point; (g) for any c IR n, if the L P problem max{b T y A T y c} has an optimal solution then it has a unique optimal solution. The polyhedron {x Ax = b, x 0} is said to be strongly nondegenerate if it satisfies any one of the equivalent conditions of the result below. Theorem 1.2. The following statements are equivalent: (a) for any x IR n such that Ax = b, σ(x) m; (b) for any x IR n such that Ax = b, the rows of A σ(x) are linearly independent; (c) for any x IR n such that Ax = b and rank(a σ(x) ) = m, σ(x) =m; (d) for any x IR n such that Ax = b and rank(a σ(x) ) = m, the submatrix A σ(x) is nonsingular; (e) for any c IR n and any x IR n satisfying Ax = b, the set arg min{ x s A T y+s =c, (y,s) IR m IR n } contains exactly one point; (f) for any c IR n and any set of the form A {x IR n A σ x σ = b, x σ c = 0} where σ {1,...,n}and c T x is constant on A, the set {(y, s) IR m +n x Asuch that A T y+ s = c, s x = 0} contains exactly one point; (g) for all c IR n, every constant cost face of the L P problem optimize {b T y A T y c} is a vertex. Note that in the results above we have not assumed that the polyhedron {x Ax =b, x 0} is nonempty.

4 224 WANG AND MONTEIRO Regarding CC-nondegeneracy of an LP problem, we have the following two main results for standard form LP problems. In the following two results, c denotes an n-vector. Theorem 1.3. The LP problem optimize {c T x x P} where P {x Ax = b, x 0} is CC-nondegenerate if and only if the set of all vertices of P is equal to the set { x P (y, s) such that } AT y + s = c,. x s = 0, and x + s >0. Theorem 1.4. Assume that P {x Ax = b, x 0} is nondegenerate. Then, the LP problem optimize {c T x x P} is CC-nondegenerate if and only if the two sets { x P (y, s) such that } AT y + s = c, x s = 0 and { x P (y, s) such that } AT y + s = c,. x s = 0, and x + s >0 are equal. We end this introduction by pointing out the relationship between CC-nondegeneracy of an LP problem in standard form and the strong dual nondegeneracy defined above. For the sake of future reference, we repeat the definition below. Definition 1. The LP problem optimize {c T x Ax = b, x 0} is said to be strongly dual nondegenerate if for every (y, s) IR m IR n such that A T y + s = c, the vector s has at least n m nonzero components. Using a more general version of Theorem 1.2, namely Theorem 3.9, the following equivalence can be proved under the assumption that rank(a) = m: the LP problem optimize {c T x Ax =b, x 0}is strongly dual nondegenerate if and only if the LP problem optimize {c T x Ax =b, x 0} is CC-nondegenerate for every b IR m (see Corollary 3.10). 2. Notation and terminology In this section, we introduce some additional notation which will be used in the remaining part of the paper. If M is a matrix then Null(M) denotes the null space of M and Range(M) denotes the range space of M. In our study of nondegeneracy of polyhedra and LP problems, we consider the following polyhedron P {y IR m H I y c I, H E y = c E }, (1) where H IR n m, c IR n and I ={1,...,n 1 }, E ={n 1 +1,...,n}.

5 NONDEGENERACY OF POLYHEDRA AND LINEAR PROGRAMS 225 The polyhedron P is said to be pointed if it contains a vertex. Throughout the remaining of the paper, we let l denote the dimension of the lineality space Null(H) of P. It is well known that P is pointed if and only if l = 0. Given y IR m and a set IR m, we let ν(y) {i I H i y=c i }and ν( ) {i I H i y=c i, y }. For a face F of P, ri(f)denotes its relative interior. For any y IR m, we denote the corresponding slack vector by s(y) c Hy. When the variable y is understood, we denote s(y) simply by s. Also, we denote s(ŷ) by ŝ, s(ȳ) by s, and etc. For y P, F(y) denotes the smallest face of P containing y. Finally, for any set α I, α c denotes the set I \α. For the purpose of future reference, we now make the following simple observations. Given y P and two faces F and F of P,wehave: ν(y) = ν(f(y)); (2) y F F(y) F; (3) y ri F F(y) = F; (4) F F ν(f) ν(f ). (5) Given a nonempty face F of P, there always exists a y F such that s ν(f) = 0, s ν c (F) > 0. (6) 3. Nondegeneracy of a polyhedron In this section, we discuss two notions of nondegeneracy of a (not necessarily pointed) polyhedron. We then provide several equivalent conditions for these two types of nondegeneracy. The results of this section are not only interesting in their own right but are also needed for the discussion of Section 4. Most of the results derived in this section, when specialized to the context of pointed polyhedra, are well known in the research community but are scattered throughout the literature. Hence, one of our goals is to provide a unified treatment of this subject. In what follows, P denotes the polyhedron defined in (1). A nonempty face F of P is called a minimal face if it does not have any nonempty face properly contained in it. The following result gives the main properties of minimal faces that are used in our presentation. For its proof, we refer the reader to Chapter 8 of Schrijver [23]. Proposition 3.1. Let l denote the dimension of the linearity space of P (hence, rank(h) = m l) and let F be a nonempty face of P. Then, F is a minimal face of P if and only if rank(h ν(f) E ) = m l, in which case ν(f) E m l. Lemma 3.2. Given any y P, there exists a minimal face F of P such that ν(y) ν(f). Proof: Let y P be given. We claim that if F(y) is not a minimal face then there exists ŷ P such that ν(ŷ) properly contains ν(y). It is easy to see that using this claim a

6 226 WANG AND MONTEIRO finite number of times, we can construct a point ȳ such that F F(ȳ) is a minimal face and ν(f)=ν(ȳ) ν(y), thereby showing that the lemma holds. To show the claim, assume that F(y) is not a minimal face. It follows from Proposition 3.1 and (2) that rank(h ν(y) E )<rank(h) = m l, or equivalently, that Null(H ν(y) E ) properly contains Null(H). Then, there exists d Null(H ν(y) E ) such that H r d 0 for some r ν c (y). By multiplying d by 1 if necessary, we may assume without loss of generality that H r d > 0. Let λ min{(c i H i y)/h i d i such that H i d > 0} and let ŷ = y + λd. It is now easy to see that ν(ŷ) ν(y) {r}, from which the claim follows. Given b, ȳ IR m, define the following sets E b (ȳ) arg min { s I x I H T I x I +H T E x E =b, x IR n }, X b (ȳ) { x IR n H T I x I + H T E x E = b, s I x I = 0 }. (7) where s = s(ȳ). Observe that E b (ȳ) whenever b Range(H T I H T E ) and that E b(ȳ) = X b (ȳ) whenever X b (ȳ). The proof of the following lemma is left to the reader. Lemma 3.3. Let b IR m and ȳ P be given. Then the following statements are equivalent: (a) F(ȳ) is a constant cost face of the LP problem optimize {b T y y P}; (b) b Range(Hν(ȳ) T HT E ); (c) X b (ȳ). Lemma 3.4. Let b IR m, a face F of P and ȳ ri(f) be given. Then X b (ȳ) X b (y) for every y F. Proof: We have ν(ȳ) ν(y) for every y F. Hence the implication s I x I = 0 s I x I = 0 holds for every x IR n, where s = s(ȳ) and s = s(y). This clearly implies the lemma. The following theorem, which is the first main result of this section, gives several equivalent characterizations of the notion of nondegeneracy of a polyhedron. The first four conditions are primal-type characterizations with appealing geometric meanings while the other three conditions are dual type characterizations. Theorem 3.5. The following statements are equivalent: (a) for any y P, ν(y) E m l(there are at most m l active hyperplanes at any feasible point); (b) for any y P, the set {H i i ν(y) E} is linearly independent (the normal vectors to the active hyperplanes at any feasible point are linearly independent); (c) for any minimal face F of P, ν(f) E =m l(there are exactly m l hyperplanes containing a minimal face); (d) for any minimal face F of P, the set {H i i ν(f) E} is linearly independent (the normal vectors to the hyperplanes containing a minimal face are linearly independent); (e) for any b Range(HI T HE T ) and any ȳ P, E b(ȳ) contains exactly one point;

7 NONDEGENERACY OF POLYHEDRA AND LINEAR PROGRAMS 227 (f) for any b IR m and any constant cost face F of the LP problem optimize {b T y y P}, the set {X b (y) y F} contains exactly one point; (g) for any b IR m, if the linear program minimize subject to c T I x I + c T E x E HI T x I + HE T x E = b x I 0, x E unrestricted, (8) has an optimal solution, then it has a unique optimal solution. Proof: (a) (c): The forward implication follows from Proposition 3.1 and the reverse implication follows from Lemma 3.2. (c) (d): Follows immediately from Proposition 3.1. (b) (d): The forward implication is trivial. The reverse implication follows from Lemma 3.2. (b) (e): Assume (b) holds and let b Range(HI T HE T ) and ȳ P be given. Define (B, N) (ν(ȳ), ν c (ȳ)). Since s B = 0, we have s I x I = s N x N. Hence, due to (7), if x E b (ȳ) then x N is an optimal solution of minimize { s N x N 2 H T N x N b Range ( H T B E)}. (9) Since s N > 0, (9) is a strictly convex quadratic program. Hence, x N is uniquely determined. Moreover, by (b), the columns of HB E T are linearly independent. These two observations together with the fact that HB E T x B E + HN T x N = b implies that x B E is also uniquely determined. (e) (f): Let b IR m, a constant cost face F of optimize {b T y y P} and y F be given. By Lemma 3.3, it follows that X b (y). This implies that E b (y) = X b (y), and hence, by (e), it follows that X b (y) contains exactly one point for every y F. This together with Lemma 3.4 implies that {X b (y) y F} contains exactly one point. (f) (g): Assume (f) holds and let b IR m be such that (8) has an optimal solution. Let D denote the optimal face of (8). It follows from the duality theorem that the dual of (8), namely the problem maximize {b T y y P}, has a nonempty optimal face F. By the complementary slackness theorem, we know that D X b (y) for any y F. This fact, (f) and the fact that F is obviously a constant cost face of maximize {b T y y P} implies that D contains exactly one point. (g) (b): Let ȳ P be given. We will show that the set {H i i ν(ȳ) E} is linearly independent. Indeed, let b Hν(ȳ) T 1 + HT E 1. Clearly, x = ( x ν(ȳ), x ν c (ȳ), x E ) (1,0,1) is a feasible solution of (8) with b = b which, together with ȳ, satisfies the strict complementarity condition. Hence, it follows that the optimal face D of problem (8) with b = b is given by D ={x IR n H T ν(ȳ) x ν(ȳ) + H T E x E = b, x ν(ȳ) 0, x ν c (ȳ) = 0}. Since x D and x satisfies x ν(ȳ) > 0, it follows that the dimension of D is equal to the dimension of Null (H T ν(ȳ) HT E ). Since (g) holds by assumption, it follows that (8) with b = b

8 228 WANG AND MONTEIRO has a unique optimal solution, and hence its optimal face has dimension zero. Thus, we conclude that the dimension of Null (Hν(ȳ) T HT E ) is equal to zero, or equivalently, the set {H i i ν(ȳ) E} is linearly independent. In view of Theorem 3.5, we introduce the following definition. Definition 2. P is said to be nondegenerate if any one of the equivalent conditions of Theorem 3.5 holds. We remark that nondegeneracy of a polyhedron is a notion that depends not only on the polyhedron itself but also on its representation as a system of linear equalities and inequalities. We can also define a stronger notion of nondegeneracy as follows. Definition 3. P is said to be strongly nondegenerate if ν(y) E m lfor every y {y IR m H E y = c E }. Note that strong nondegeneracy is a condition that depends on H, c and the index set E only. More specifically, if P is strongly nondegenerate then any other polyhedron of the form {y IR m A E y = c E, A I y? c I } is strongly nondegenerate, where? denotes a vector of and symbols. Similar to the concept of nondegeneracy, there are several equivalent ways to express the strong nondegeneracy of a polyhedron. In what follows, we discuss this issue. For the purpose of stating the next result, we need to introduce the following set: M {y IR m H E y = c E, rank(h ν(y) E ) = m l}. Since, by Proposition 3.1, {F F is a minimal face of P} ={y P rank(h ν(y) E ) = m l}, it follows that M P = {F Fis a minimal face of P}. Hence, the set M is a natural extension of the set {F F is a minimal face of P}. A nonempty set A is said to be an affine set associated with P if A ={y IR m H E y = c E, H ν y = c ν } for some index set ν I. It can be shown that for every affine set A there exists ȳ A such that ν(ȳ) = ν(a). We state the following lemmas whose proofs are similar to those of Lemma 3.2, Lemma 3.3 and Lemma 3.4, and hence are left to the reader. Lemma 3.6. ν(ȳ). Given any y IR m satisfying H E y = c E, there exists ȳ M such that ν(y) Lemma 3.7. Let b IR m and an affine set A IR m be given. Then the following statements are equivalent:

9 NONDEGENERACY OF POLYHEDRA AND LINEAR PROGRAMS 229 (a) b T y is constant over the set A; (b) b Range(H T ν(a) HT E ); (c) X b (y) for every y A. Lemma 3.8. Let b IR m and an affine set A IR m be given. Let ȳ A be such that ν(ȳ) = ν(a). Then X b (ȳ) X b (y) for every y A. We are now ready to state the second main result of this section. It gives several equivalent characterizations of the notion of strong nondegeneracy of a polyhedron. Theorem 3.9. The following statements are equivalent. (a) for any y IR m satisfying H E y = c E, ν(y) E m l (i.e., P is strongly nondegenerate); (b) for any y IR m satisfying H E y = c E, the set {H i i ν(y) E}is linearly independent; (c) for any y M, ν(y) E =m l; (d) for any y M, {H i i ν(y) E} is linearly independent; (e) for any b Range(HI T HE T ) and any ȳ IRm satisfying H E ȳ = c E, E b (ȳ) contains exactly one point; (f) for any b IR m and any nonempty affine set A IR m over which b T y is constant, the set {X b (y) y A} contains exactly one point; (g) for all b IR m, every constant cost face of the LP problem optimize x IR n s.t. is a vertex. c T I x I + c T E x E HI T x I + HE T x E = b x I 0, x E unrestricted, (10) Proof: The equivalences and the implications (= (a) (c), (c) (d), (b) (d), (b) (e) (f) can be proved using similar arguments as the ones used to prove the same equivalences and implications of Theorem 3.5, except that now Lemma 3.6, Lemma 3.7 and Lemma 3.8 are used instead of Lemma 3.2, Lemma 3.3 and Lemma 3.4. We next prove that (f) (g) and (g) (b). (f) (g): Assume (f) holds and let b IR m be given. Suppose that D is a constant cost face of (10). D can be written as D ={x IR n H T x = b, x B 0, x N = 0 }, for some index sets B I and N = I \B such that x B > 0 for at least one x D. Since D is a constant cost face of (10), Lemma 3.3 implies that c B 0 H B c N Range I N H N, c E 0 H E

10 230 WANG AND MONTEIRO where I N is the N N identity matrix. This implies that there exists a y IR m such that H B y = c B, H E y = c E. Define the affine set A ={y IR m H B y = c B, H E y = c E }. Clearly, D X b (y) for any y A. Lemma 3.7 and the fact that HB T x B + HE T x E = b, for x D, imply that b T y is constant over A. Hence, by (f), we conclude that D contains exactly one point, that is, that D is a vertex. (g) (b): Let ȳ satisfying H E ȳ = c E be given. We will show that the set {H i i ν(ȳ) E} is linearly independent. Indeed, let b Hν(ȳ) T 1 + HT E 1 and let D { x IR n H T ν(ȳ) x ν(ȳ) + H T E x E = b, x ν(ȳ) 0, x ν c (ȳ) = 0 }. Clearly, x = ( x ν(ȳ), x ν c (ȳ), x E ) = (1,0,1) D. Since c ν(ȳ) E = H ν(ȳ) E ȳ, we know that c ν(ȳ) E Null(Hν(ȳ) E T ), and hence D is a constant cost face of (10) with b = b. By condition (g), we know that D is a vertex. This implies that the set {H i i ν(ȳ) E} is linearly independent. Corollary Let A IR m n, b IR m, c IR n and assume that rank(a) = m. Then the LP problem optimize {c T x Ax = b, x 0} is strongly dual nondegenerate if and only if the LP problem optimize {c T x Ax = b, x 0}is CC-nondegenerate for every b IR m. Proof: The assumption rank(a) = m implies that the polyhedron {y IR m A T y c} is pointed. By this observation, Definition 1 and Definition 3, we conclude that strong dual nondegeneracy of the LP problem optimize {c T x Ax = b, x 0} is equivalent to the condition that the polyhedron {y IR m A T y c} be strongly nondegenerate. From the equivalence of (a) and (g) of Theorem 3.9 and the fact that every minimal face of {x Ax =b, x 0} is a vertex, it follows that the latter condition is equivalent to the condition that the LP problem optimize {c T x Ax = b, x 0} be CC-nondegenerate for every b IR m. 4. CC-nondegeneracy of a linear program In this section, we discuss the notion of CC-nondegeneracy with respect to the LP problem optimize b T y subject to y P, (11) where b IR m and P is the polyhedron defined in (1). Problem (11) is allowed to be either a maximization or a minimization problem. The main results of this section are Theorem 4.4 and Theorem 4.6. Theorem 4.4 gives a characterization of CC-nondegeneracy of (11) for general feasible polyhedron P while Theorem 4.6 gives an alternative characterization that holds only when P is nondegenerate. Throughout this section we consider the following two sets: C {y P X b (y) } = { y P x IR n such that HI T x I + HE T x E = b, x I s I = 0 } ; { SC y P x IRn such that HI T x I + HE T x } E = b,. x I s I = 0, and x I +s I >0

11 NONDEGENERACY OF POLYHEDRA AND LINEAR PROGRAMS 231 C is the set of feasible points satisfying the complementarity condition while SC is the set of feasible points satisfying the same condition strictly. The main of this section is to show that CC-nondegeneracy of (11) is equivalent to the condition SC = {F Fis minimal face of P} (see Theorem 4.4) and, when the feasible polyhedron P is nondegenerate, to the condition C = SC (see Theorem 4.6). We start with the following result which follows as an immediate consequence of Lemma 3.3. Proposition 4.1. C = {F Fis a constant cost face of (11)}. The next result establishes an important relationship between the set SC and the maximal constant cost faces of problem (11). First we need the following definition. Definition 4. A face F of P is said to be a maximal constant cost face (of problem (11)) if it is a constant cost face and is not properly contained in any other constant cost face. Theorem 4.2. {ri F F is maximal constant cost face} SC. (12) Proof: Let F be a maximal constant cost face for (11) and let ȳ ri F. We will show that ȳ SC. By (4), we have F = F(ȳ), and hence F(ȳ) is a maximal constant cost face. Hence, by the equivalence of statements (a) and (b) of Lemma 3.3 we conclude that there exists an x IR n such that HB T x B + HE T x E = b, x N = 0, where B ν(ȳ) and N I \B. Clearly, if x B > 0 then ȳ SC since s N > 0, s B = 0. Hence, assume that x B 0 and let B + ={i B x i > 0}, and B 0 = B\B + ={i B x i =0}. Consider the problem maximize b T y subject to H B+ y = c B+ H B0 y c B0 H N y c N H E y = c E (13) and its dual problem minimize subject to c T B + x B+ + c T B 0 x B0 + c T N x N + c T E x E H T B + x B+ + H T B 0 x B0 + H T N x N + H T E x E = b x B0 0, x N 0, x B+, x E unrestricted. (14) It is easy to verify that x and ȳ are optimal solutions to problems (14) and (13), respectively. Since every pair of primal and dual linear programs has a pair of optimal solutions satisfying strict complementarity, we conclude that there exist optimal solutions ˆx and ŷ of (14) and

12 232 WANG AND MONTEIRO (13), respectively, such that ˆx B0 ŝ B0 =0, ˆx B0 +ŝ B0 >0, (15) ˆx N ŝ N =0, ˆx N +ŝ N >0. (16) Since every pair of optimal solutions of (13) and (14) satisfies the complementarity condition, we have ˆx N s N =0. Since s N > 0, we obtain ˆx N = 0. It then follows from (16) that ŝ N > 0, and hence ν(ŷ) B = ν(ȳ). By (5), we conclude that F(ȳ) F(ŷ). Since ŷ is an optimal solution of (13), it follows that the face F(ŷ) is contained in the optimal face of (13) and hence that F(ŷ) is a constant cost face. Hence, we conclude that F(ȳ) = F(ŷ) since F(ȳ) F(ŷ) and F(ȳ) is a maximal constant cost face. Thus, ν(ȳ) = ν(ŷ), and since B 0 ν(ȳ) we conclude that ŝ B0 = 0. It then follows from (15) that ˆx B0 > 0. Using the fact that x B+ > 0, x B0 =0 and ˆx B0 > 0, we conclude that there exists a δ (0, 1] sufficiently small such that λ ˆx B + (1 λ) x B > 0 for all 0 <λ δ. Defining x = δ ˆx + (1 δ) x and noting that x N =ˆx N =0, we have H I x I + H E x E = b, x B > 0 and x N = 0. Clearly, this shows that ȳ SC. The following simple example shows that the two sets in (12) may differ. Example 4.3. Consider the LP problem given by maximize y 1 + y 2 subject to y 1 + y 2 0, y 1 + y 2 2, y 2 1. (17) Let F 1 denote the face of the feasible polyhedron in which the first constraint is active. Clearly, the only maximal constant cost face is F 1. It is easy to verify that SC = F 1 and hence that the two sets in (12) differ. The next result provides a characterization of CC-nondegeneracy of the LP problem (11). Theorem 4.3. Assume that C. Then, problem (11) is CC-nondegenerate if and only if SC = {F Fis minimal face of P}, (18) in which case, we have SC = C. Proof: We first prove the only if part. So assume that problem (11) is CC-nondegenerate. Using Proposition (11), Theorem 4.1 and the CC-nondegeneracy of problem (11), we obtain C = {F Fis a constant cost face} = {F Fis a minimal face} = {ri F F is a minimal face} = {ri F F is a maximal constant cost face} SC. (19)

13 NONDEGENERACY OF POLYHEDRA AND LINEAR PROGRAMS 233 On the other hand, we know that SC C. Hence, we conclude from (19) that (18) holds and SC = C. For the if part, note that Theorem 4.2 and relation (18) imply {ri F F is a maximal constant cost face} SC = {F Fis a minimal face}. This inclusion clearly implies that problem (11) is CC-nondegenerate. Theorem 4.4 shows that CC-nondegeneracy of (11) implies that SC = C. On the other hand, the reverse implication may not hold as Example 4.3 illustrates. But when the polyhedron P is nondegenerate, we will show in what follows that the condition SC = C implies that (11) is CC-nondegenerate. With this goal in mind, we introduce the following set: ( ) ( ) HB Hν(y) SC 0 = y P B ν(y) such that rank = rank, H E H E x IR n such that HI T x I + HE T x E = b, x B > 0, x B c = 0.. (20) The following theorem relates the set {ri F F is a maximal constant cost face} with the set SC 0. Its proof is postponed until the end of this section. Theorem 4.5. SC 0 {ri F F is a maximal constant cost face}. As a consequence of this theorem, we obtain the following result with respect to problem (11) when P is nondegenerate. Theorem 4.6. Assume that P is nondegenerate. Then, (a) SC = SC 0 = {ri F F is a maximal constant cost face}; (b) problem (11) is CC-nondegenerate if and only if C = SC. Proof: We first prove (a). Combining Theorem 4.2 and 4.5, we obtain SC 0 {ri F F is a maximal constant cost face} SC. (21) Since P is nondegenerate, it follows from Theorem 3.5 that the set {H i i ν(y) E} is linearly independent for any y P. This observation implies that SC = SC 0, due to the definition of these sets. Hence, (a) follows due to (21). For (b), note that the only if part follows from Theorem 4.4. For the proof of if part, assume that C = SC. This condition, statement (a) and Proposition 4.1 then imply {ri F F is a maximal constant cost face} = SC = C = {F Fis a constant cost face}. This equality obviously imply that every constant cost face must be a minimal face, that is, problem (11) is CC-nondegenerate.

14 234 WANG AND MONTEIRO We now turn our efforts towards proving Theorem 4.5. Several preliminary lemmas are needed. The first one can be proved in the same way as Proposition 3.4 of Nemhauser and Worsey [22] and hence, we leave its proof to the reader. In what follows, the following notation is used. Given two index sets B {1,...,m} and N {1,...,m} such that B N =, we denote by [B, N] the polyhedron given by [B, N] {y IR m H B y = c B, H N y c N }. Lemma 4.7. Let F be a face of P and assume that F = [B, N]. Assume that r Nis such that the ( possibly empty) set {y F H r y = c r } has dimension less than dim(f) 1. Then, F = [B, N\{r}]. Lemma 4.8. Let F and F be two nonempty faces of P such that F F and F F. Then there exists an index r ν(f)\ν(f ) such that the face F ={y F H r y=c r } satisfies that F F and dim( F) = dim(f ) 1. Proof: Let B ν(f) E, N {1,...,n}\B, B ν(f ) E and N {1,...,n}\B. By (5), we know that B B. Forr B\B, let F r {y F H r y=c r }. Obviously, F F r F, and dim(f r )<dim(f ), since otherwise we would have {y F H r y = c r }=F and hence that r ν(f ), a contradiction. Now assume for contradiction that for every r B\B, dim(f r )<dim(f ) 1 and let B\B ={r 1,...,r k }. Using Lemma 4.7, we can easily show by induction that F = [B, N \{r 1,...,r j }] for every j = 1,...,k. In particular, F = [B, N] since N = N \{r 1,...,r k }. Since F is a subface of F,it follows that F = [B N 1, N 2 ], where N 1 and N 2 are index sets such that N 1 N 2 = N and N 1 N 2 =. Clearly, B N 1 B and since B N =, we must have N 1 = and N 2 = N. Hence, it follows that F = [B, N] = F. But this contradicts the assumption that F F. Thus, we conclude there exists an index r J such that dim(f r ) = dim(f ) 1 and the result follows by letting F F r. Lemma 4.9. Let F and F be two nonempty faces of P such that F F and F F. Then there exist a face ˆF of P and an index r ν(f)\ν( ˆF) satisfying the following properties: (a) F ˆF F and dim( ˆF) = dim(f) + 1; (b) F ={y ˆF H r y=c r }; (c) rank( H ν(f) ) = rank( H ν( ˆF) {r} ). H E H E Proof: The proofs of statements (a) and (b) follow immediately from Lemma 4.8. It remains to show (c). We first prove that H r is linearly independent from the rows of H ν( ˆF) E. Assume for contradiction that there exist scalars α i with i ν( ˆF) E such that H r = α i H i. i ν( ˆF) E (22)

15 NONDEGENERACY OF POLYHEDRA AND LINEAR PROGRAMS 235 Then, using (22) we obtain H r y = α i H i y = i ν( ˆF ) E i ν( ˆF) E α i c i, y ˆF. (23) Since F ˆF and H r y = c r for all y F, it follows from (23) that i ν( ˆF) E α ic i = c r and that r ν( ˆF ), a contradiction. We have thus shown that H r is linearly independent from the rows of H ν( ˆF) E. Using this fact and statement (a), we obtain ( ) Hν(F) rank = n dim(f) = n dim( ˆF) + 1 H E ) ) Hν( = rank( ˆF) Hν( + 1 = rank( ˆF) {r}. H E H E Lemma X(F) = Let F be a constant cost face of P. Define the set { x IR n H I T x I + H E T x } E = b, and y F such that s I = c I H I y satisfies xi T s. (24) I = 0 Then, for any y F and any x X(F), we have x T I s I = 0. Proof: Let y F and x X(F) be given. By the definition of X(F), x satisfies H T I x I + H T E x E = b and there exists ȳ F such that x T I (c I H I ȳ) = 0. Since F is a constant cost face and y, ȳ F,wehaveb T y =b T ȳ. Using this fact and the fact that H E y = c E = H E ȳ, we obtain xi T s I = xi T (c I H I ȳ + H I ȳ H I y) = xi T H I (ȳ y) = ( b HE T x T E) (ȳ y) = b T (ȳ y) xe T H E(y ȳ) = 0. We are now ready to prove Theorem 4.5. Proof of Theorem 4.5: For any ȳ SC 0, we will show that F(ȳ) is a maximal constant cost face from which the inclusion of the theorem follows. Indeed, let ȳ SC 0 be given. Since SC 0 C, it follows that ȳ C, and hence that F(ȳ) is a constant cost face, due to Proposition 4.1. Assume for contradiction that F(ȳ) is not a maximal constant cost face, that is, there exists a constant cost face F such that F(ȳ) F and F(ȳ) F. Applying Lemma 4.9 with F = F(ȳ), we conclude that there exists a face ˆF and an index r ν(ȳ)\ν( ˆF) such that F(ȳ) ˆF F, ( ) Hν(ȳ) rank H E = rank( Hν( ˆF) {r} H E (25) ). (26)

16 236 WANG AND MONTEIRO Letting B ν( ˆF) {r} ν(ȳ) and using relation (26), the definition of SC 0 and the fact that ȳ SC o, we conclude that there exists x IR n such that H T I x I + H T E x E = b, x B > 0, x B c = 0. (27) By (6), we know there exists ŷ ˆF such that ŝ ν c ( ˆF) > 0, ŝ ν( ˆF) = 0. (28) Hence, we have x T I ŝi = x T BŝB+ x T B cŝ B c = x T ν( ˆF)ŝν( ˆF) + x rŝ r + x B T cŝ B c (29) = x r ŝ r 0, (30) where x T ν( ˆF)ŝν( ˆF) is equal to zero by (28), x B T cŝ B c is equal to zero by (27) and x rŝ r is nonzero due to (27) and (28) and the fact that r B, r ν( ˆF). On the other hand, since B ν(ȳ), relation (27) implies that x I T s I = 0. This fact, relation (27), the definition of X( ˆF) and the fact that ȳ ˆF imply that x X( ˆF). Since F is a constant cost face, it follows from (25) that ˆF is also a constant cost face. Using these two last conclusions, the fact that ŷ ˆF and Lemma 4.10, we conclude that x I T ŝi = 0, a fact that contradicts (30). The reverse inclusion in Theorem 4.5 does not hold in general. To see this, consider the LP problem of maximizing y 2 subject to the same set of constraints as the problem in Example 4.3. Then, the vertex (1, 1) is the only maximal constant cost face, but (1, 1) / SC 0. References 1. I. Adler and R.D.C. Monteiro, A geometric view of parametric linear programming, Algorithmica, vol. 8, pp , M. Akgül, A note on shadow prices in linear programming, Journal of Operations Research Society, vol. 35, pp , D.C. Aucamp and D.I. Steinberg, The computation of shadow prices in linear programming, Journal of Operations Research, vol. 33, pp , E.R. Barnes, A variation on Karmarkar s algorithm for solving linear programming problems, Mathematical Programming, vol. 36, pp , E.M.L. Beale, Cycling in the dual simplex algorithm, Navel Research Logistics Quarterly, vol. 2, pp , R.G. Bland, New finite pivoting rules for the simplex algorithm, Mathematics of Operations Research, vol. 2, pp , A. Charnes, Optimality and degeneracy in linear programming, Econometrica, vol. 20, pp , April G. Dantzig, Linear Programming and Extensions, Princeton University Press: Princeton, NJ, I.I. Dikin, Iterative solution of problems of linear and quadratic programming, Doklady Akademii Nauk SSSR, vol. 174, pp , Translated in: Soviet Mathematics Doklady, vol. 8, pp , T. Gal, Postoptimal Analysis, Parametric Programming and Related Topics, McGraw-Hill: New York, 1979.

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