Some continuous functions related to corner polyhedra, II - NYU

Mathematical Programming 3 {1972) 359-389. North-Holland Publishing Company

SOME CONTINUOUS FUNCTIONS RELATED TO CORNER POLYHEDRA, II Ralph E. GOMORY and Ellis L. JOHNSON IBM Watson Research Center, Yorktown Heights, iV. Y., 10598, U.S.A. Received 3 March 1972 Revised manuscript received 11 October 1972

The group problem on the unit interval is developed, with and without continuous variables. The connection with cutting planes, or valid inequalities, is reviewed. Certain desirable properties of valid inequalities, such as minimality and extremality are developed, and the connection between valid inequalities for P(I, u 0) and P+(I, u 0) is developed. A class of functions is shown to give extreme valid inequalities for P+ (I, u0) and for certain subsets UofI. A method is used to generate such functions. These functions give faces of certain corner polyhedra. Other functions.which do not immediately give extreme valid inequalities are altered to construct a class of faces for certain corner polyhedra. This class of faces grows exponentially as the size of the group grows.

1. Review o f the problems This paper follows a previous paper [4] but will be self-contained except for proofs of some theorems from [4]. 1.1. The problems P(U, u0) and P+(U, u0) Let I be the group formed by the real numbers on the interval [0, 1 ] with addition modulo 1. Let U be a subset of I and let t be an integervalued function on U such that (i) t(u) >_ 0 for all u ~ U, and (ii) t has a finite support, that is t ( u ) > 0 only for a finite subset U t o f U . We say that the function t is a solution to the problem P(U, u 0), for u o ~ / ~ {0~, if u t(u)=u o . u~U

(1.1)

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Here, o f course, addition and multiplication are taken modulo I. Let T(U, u o ) denote the set of all such solutions t to P(U, u0). Correspondingly, the problem P+_(U, u0) has solutions t' = (t, s+, s - ) satisfying u t(u) + Y(s +) -

7(s-)

= uo ,

(1.2)

u~U

where t is, as before, a non-negative integer valued function on U with a finite support, where s ÷, s - are non-negative real numbers, and where 5r(x) denotes the element of I given by taking the fractional part of a real number x. Let T+_(U, u0) denote the set of solutions t' = (t, s ÷, s - ) to P+(U, u0). The notation u ~ I will mean that u is a member of the group I so that arithmetic is always modulo 1. If we want to consider u as a point on the real line with real arithmetic, we will write [ul. Thus, lul and 9r(x) are mappings in opposite directions between I a n d the reals and, in fact, 5r(lui) = u but x and lY(x)l may differ by an integer.

1.2. Inequalities 1.2.1. Valid inequalities For any problem P(U, u0), we have so far defined the solution set T(U, uo). A valid inequality for the problem P(U, u0) is a real-valued function lr defined for all u ~ I such that rr(0)=0,

n(u)>_0, u ~ I ,

(1.3)

and

rr(u) t(u)>_ 1,

t 6 T(U,u o).

(1.4)

u~U

For the problem P+(U, u0) , rr' = (rr, rr+, 7r- ) is a valid inequality for P+__(U, u o) when ~r is a real-valued function on I satisfying ( 1.3), and rr+ , n - are non-negative real numbers such that

rr(u) t(u)+rr+s+ +rr-s- >_ l , t'~T+_(U, uo).

(1.5)

uEU

A valid inequality (rr, rr+, rr- ) for P+_ (I, u o) can be used to give a valid inequality for P(U, u o) or P+_(U, Uo) for any subset U o f / . For example,

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ETr(u) t(u) >_ 1 is clearly true for any t ~ T(U, u o) since that t can be extended to a function t' belonging to T(I, u o) by letting t'(u) = 0 for u ~/~U. Thus, the problem P+_(L u0) acts as a master problem forall cyclic group problems in the same way that the master problem in [3] was a group problem with all group elements present. This fact is the main reason for studying the case U = I in such detail in Section 2. However, the next two properties of valid inequalities do not necessarily carry over to subsets U of I. 1.2.2. Minimal valid inequalities A valid inequality lr for P(U, u 0) is a minimal valid inequality for P(U, u0) if there is no other valid inequality p for P(U, u 0) satisfying p(u) < ~r(U), where p(U) < lr(U) is defined to mean p(u) O.

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366

Thus Theorem 1.9 says that valid inequalities can be obtained simply by connecting the points (gn, ~r(gn)) by straight line segments. 1.8. Valid inequalities for P+ (U, u o )

From valid inequalities for P+__(Gn, u 0), a different method for generating valid inequalities for P+__(U, Uo) is available. This m e t h o d will be referred to as the two-slope fill-in: Theorem 1.10 [4, Theorem 3.3]. L e t lr' = Or, 7r+, rr-) be an e x t e n d e d subadditive f u n c t i o n on G n. Define 7r(u) for u c I \ G n by

7r(u) = min(rc(L(u)) + 7r+ lu - L(u)l,

7r(R(u)) + lr- IR(u) - ul} .

(1.24) Then 7r' is an e x t e n d e d subadditive f u n c t i o n on I, and p' defined by

p' = (~, ~+, ~- )/~r(u o) is a valid inequality for P+(U, u0) provided rr(Uo) > O.

Theorem 1.8 shows how to compute faces for P+_(Gn, Uo) and Theorem 1.10 shows how to use them to generate valid inequalities for any U. Table 2 of [4] was obtained using Theorem 1.8, and we will frequently refer to the two-slope fill-in of those faces.

2. The problems P(L Uo ) and P+_(I; u o ) 2.1. Problem definitions

Let the set U now be the entire interval I. The problem P(/, u o) involves the congruence u t(u) = u o ,

(2.1)

uEI

and P+_(L u0) has the constraint u t(u) + 7 ( s +) - 8r(s - ) = u o , u~I

(2.2)

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367

where t is a non-negative integer valued function on I having finite support. The present section intends to reveal something about the extreme valid inequalities for those problems. Such information could be useful in dealing with problems involving subsets of I. The relation to P(U, u o) is the same as the relation between the master polyhedra and the corner polyhedra corresponding to subsets of a group [ 3 ] . Here, every finite cyclic group G n is a subset of I. In particular, if rr is a valid inequality for P(I, Uo) , then trivially rr is also a valid inequality for P(U, Uo) for every subset U of I, including all cyclic groups U-- G n or subset U of G n . Furthermore, if rr' is a valid inequality for P+_(/, u 0), then lr is a valid inequality for P(U, u0) , (rr, rr+) is a valid inequality for P+(U, u0) , (~r, rr- ) is a valid inequality for P_ (U, uo), and rr' = (rr, lr+, 7r- ) is a valid inequality for P+_(U, u o ) for any subset U o f I. The property of being a valid inequality is hereditary, that is, if rr is a valid inequality for P(& u 0), then it is also valid for any P(S', u o) with S' c S. Subadditivity for a valid inequality is also hereditary. However, minimality and extremeness are n o t hereditary properties. That is, rr can be a minimal or extreme valid inequality for P(U, Uo) and still not be for P(U', Uo) with U' c U. 2.2. Properties and relations b e t w e e n P(I, u o ) and P+_(L u o ) Property 2.1. I f 7r' = (Tr, rr+ , 7r- ) is a valid inequality f o r P+_(I, u o), then 7r is a valid inequality f o r P(I, Uo). P r o o f if rr is not a valid inequality for P(/, u 0), then there is a t satisfying (1) with Err(u) t(u) < 1. Clearly (t, 0, 0) solves (2.2) as well, contradicting rr' being a valid inequality for P+_(I, u 0 ), and completing the proof.

Recall that we define lul as the real number corresponding to u ~ I. We can then define right and left limits, lim , u~u 0

lim , utu 0

as the point [ul approaches lu01 on the real line from the right ( l u l > lu 0 I) or from the left (lul < [u 0 I), respectively.

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Property 2.2. I f zr is a valid inequality for P(I, Uo) and if l + = lim rr(u)/lul ,

l- = lira {zr(u)/(1 - lul))

u~O

u~l

both exist (that is, i f zr has right and left "derivatives" at 0 and 1, respectively), then rr'=(rr, l + , l - ) is a valid inequality f o r P+_(I, uo). P r o o f Suppose t' = (t, s +, s - ) solves (2.2) but 7r(u) t ( u ) + I+ s + + l -

s-

= 1 -

e,

e > 0.

u~I

We can assume that o n l y one o f s ÷, s - is positive, say s + > 0 and s - = O, since otherwise b o t h s ÷ and s - could be reduced until one reaches zero. Choose an integer M large e n o u g h that l+ -

7r(CY(s+/M)) s +/m

I
0 such t h a t

~(u - v) ~ (/3 + e) (lul - Iol) for l u l > Ivl and l u l -

I v l ~ 6. F o r such u, v E I ,

~(u) _ p(v) .

But here the left-hand side is equal to 7r(v) by our assumption of case (i) above. Hence ~r(v) >_ p(v), contradicting p(v) > 7r(v). The proof is thus completed. We can apply this theorem to Table 2 of the appendix of [4]. Corresponding to each extreme valid inequality for P+ (Gn, u o), n = 1.... , 6 , we can easily give the set Uc on which ~r(u)+ 7r(u0 - u ) = 1, u E S c. Then for any set U, G n c U c Uc, the inequality given by Theorem 1.10 is an extreme valid inequality for P+(U, u0). For Go, G1, G2, G3, G 4 and G6, Uc = I for all extreme valid inequalities 0f P+_(Gn, u 0 ); the first exception occurs at G s . There are four exceptions for G s among the 6 faces given by [4, Table 2] and the reflections. These exceptions are discussed further following Corollary 4.4. The unique extreme valid inequality for P+ (Go, u 0), where G o is the subset consisting o f only the point 0, is o f particular interest. It is readily seen that this inequality 7r' when used in conjunction with a mapping ~0 gives Gomory's mixed integer cut [ 1, p. 528]. We see at once that for this 7r', lr(u) + 7r(u0 - u) = 1 for all u so that lr' is an extreme valid in-

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R.E. Gomory, E.L. Johnson

equality for P+_(U, u0) for any set S c I provided 0 ~ S, which is actually not a restriction since 0 can always be adjoined to S without changing the problem. 3.2. Some extreme inequalities for P( Gm , u o) When the inequalities rr' given by the two-slope fill-in (Theorem 1.10) satisfy rr(u) + rr(u 0 - u) -- 1, then the theorem just proven says thatTr' is an extreme valid inequality for P+_(L u0). By Property 2.9, rr is an extreme valid inequality for P(I, u o). For subsets U of I, we k n o w that rr is a valid inequality for P(U, Uo) , but we do not know that lr is extreme for P(U, Uo). The following theorem establishes that result for some U and, in fact, applies for any extreme valid inequality for P(I, u o), not just those given b y the two-slope fill-in.

Theorem 3.2. I f lr is an extreme valid inequality for P(L Uo) and consists' o f straight line segments connected at values u belonging to a regular grid G m with u o ~ Gm, then 7r is an extreme valid inequality for P(U, Uo) whenever U is a subset o f I including G m . Proof. Since lr is extreme for P(/, u 0 ), it cannot be written as ½p + ½o for different valid inequalities p, o for P(I, u o). Certainly rr is a valid inequality for P(U, Uo), and if it is not extreme for P(U, Uo), then lr = ½p +½o for different valid inequalities p, a for P(U, Uo). If both p and o are valid inequalities for P(I, u 0), a contradiction is reached. However, both can be extended to valid inequalities for P(L u 0) by the straight-line fill-in from Gin as in Theorem 1 10. Furthermore, such a construction maintains rr = ½p + ½a on all o f / s i n c e rr also consists of straight line segments joined at points of. Gin. The proof is thus completed. This theorem enables us to construct some extreme valid inequalities (faces) of the polyhedra P(G, go) of [ 3 ]. It is of particular interest when one extreme inequality of P+_(G n , u O) gives rise to many slight variants, all of which are extreme for P(L u 0) and all of which in turn give rise to apparently unrelated faces of P(G, Uo). Before showing that possibility, we digress to give some results related to the two-slope construction of Theorem I. 10.

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377

3.3. E x t r e m a l i t y o f two-slope f u n c t i o n s Theorem 3.3. L e t zr be a continuous f u n c t i o n on I consisting o f a finite n u m b e r o f straight line segments, each line s e g m e n t having a slope 7r+ > 0 or else -Tr- < O. I f 7r is a subadditive f u n c t i o n on I with zr(u o) = I f o r some u o c L then zr is e x t r e m e among the subadditive valid inequalities p f o r P(I, u o ) which have p(u o ) = 1. P r o o f The t h e o r e m asserts that if rr = l p + ½o, where p and o are subadditive valid inequalities for P(I, u 0) w i t h p(u o) = o(u 0) = 1, t h e n p(u) = o(u) for all u ~ I. We k n o w f r o m T h e o r e m 1.4 t h a t 7r is a subadditive valid inequality for P(L u0 ). Suppose t h a t rr = ½p + ½o for subadditive valid inequalities p, o for P(I, u 0) w i t h p(u o) = o(u 0) = 1. Since zr has a right-hand derivative 7r+ at 0, limsup {p(u)/lul) 0 be small e n o u g h t h a t u + 8 is in the same interval and that 6' itself lies in the very first interval. Then, zr(u) + rr(8)= rr(u + 8) by the fact t h a t 7r has the same slope zr+ on (0, 8) and (u, u + 6). Hence ½p(u) + ~ o(u) + ~ p ( 8 ) + ½~(8) = ½p(u + 8) + ~ a ( u + 8) , or

½(p(u) + p ( 6 ) - p ( u + 8)) + ½(a(u) + a(8) - o(u + 6)) = O .

By subadditivity, each o f p ( u ) + p(6) - p(u + 6) and o(u) + a(6) - a(u + 6) is non-negative. Since t h e y sum to zero, each m u s t be zero. Hence lira {(p(u + 6) - p(u))/16 840

l} = l i r a

(p(6)/18

I} = p +

,

a~0

lim ((o(u + 6) - a(u))/161} = lim {a(6)/J61) = a + . 6~0

~0

R.E. Gornory, E.L. Johnson

378

Similarly, we can show t h a t the left-hand derivatives o f p and a at u are p+ and o + . Therefore, p (resp. a) has a c o n s t a n t derivative p+ (resp. o +) on the interval, and so it is a straight line with this slope. A similar result is obtained for any x on an interval where 7r has slope - r r - . Here one works w i t h subadditivity t h r o u g h the inequality p ( - 6 ) + p(u + 5) >p(u), and concludes t h a t b o t h the left and right derivatives at u are p - . Hence b o t h p and a are of the same f o r m as r w i t h t w o slope straight line segments over the same intervals. We n o w show t h a t p+ = o + = lr + and p - = ~ - = l r - . L e t ~L be the total length o f the intervals on which the slope o f 7r is rr+ and which lie to the left of u 0 . Similarly, let l~ be the length of those intervals to the right of u 0 on which rr has slope 7r+, and let l~ and l~ be the corresponding l e n g t h s o f intervals on which 7r has slope - r r - . Since rr(u 0 ) = 1, 1r+ l ~ - n -

l~ = 1,

zr+ l ~ - z r -

l~ = -

1,

and the same equations hold for p+, p - and o +, o - . But these two equations have o n l y the solution rr+, rr- because in order for t h e m to have more t h a n one solution, one e q u a t i o n would have to be a linear multiple o f the other. But then ~L + FR = 0 and - l ~ - l~ = 0, implying t h a t all of FL, l~, l~ and l~ are zero. Hence p+ = lr+, p - = # - , and a + = rr+, O-- = ? r - .

We have two i m m e d i a t e corrolaries. Corollary 3.4. I f 7r meets the conditions Of T h e o r e m 1.10 and i f 7r(u) + rr(u o - u) = 1 f o r all u E L then 7r is an e x t r e m e valid inequality f o r P(I, u0). P r o o f If ~r(u) + 7r(u 0 - u) = 1 for all u E I, t h e n by T h e o r e m 1.6, 7r is a minimal valid inequality for P(I, u 0 ). The subadditive valid inequalities p for which p(u o) = 1 i n c l u d e the minimal valid inequalities by T h e o r e m s 1.2 and 1.6. Since by T h e o r e m 1.10, rr is e x t r e m e among those inequalities, Ir c a n n o t be written as a mid-point o f two other m i n i m a l valid inequalities. By [4, L e m m a 1.4] and the m i n i m a l i t y o f rr, Ir is an extreme valid inequality for P(L u0). Corollary 3.5. I f lr m e e t s the c o n d i t i o n s o f T h e o r e m 1.10, i f it(u) + 7r(u o - u) = 1 f o r all u c L and i f 1r+, - l r - are the t w o slopes o f ,r with lr+ > O, It- > O, then lr' = Or, Ir+ , It- ) is an e x t r e m e valid inequality f o r P+(I, u0).

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Continuous f u n c t i o n s related to corner polyhedra, H

Proof

This is immediate from Corollary 3.4 and Property 2.8.

Before leaving this section, let us point out the difference between Theorem 3.1 and Corollary 3.4. Corollary 3.4 and Theorem 3.2 would prove that rr of Theorem 3.1 is extreme when U includes the cyclic group including all of the break points of It, not just G n . However, Theorem 3.1 applies only to those rr constructed using the two-slope fill-in, whereas Corollary 3.4 applies to arbitrary two-slope functions which are subadditive and minimal.

4. Generating extreme inequalities and exponential growth for faces of some P(G, u o) We begin by discussing some of the possibilities for creating extreme inequalities for P(I, u o) from extreme inequalities of P+_(G n, u o) when the condition rr(u) + rr(u 0 - u) = 1 does n o t hold for all u ~ I for the rr constructed by the two-slope fill-in. By way of background, we observe that the rr given by the two-slope fill-in o f Theorem 1.10 does satisfy rr(u) + rr(u 0 - u) = 1 when u ~ G n . This fact is a consequence of ( 1 . 2 0 ) - ( 1 . 2 2 ) because they imply rc(gi) = min {rr(L(u o )) - rr(L(u o ) - N ) , rr(R(uo))

- ~r(R(u o ) -

&))

= rain (1 - zr+ (lu o - L ( u o )1) - rc(L(u o) - g i ) , 1 -

7r- ([R(u o) - u o I) - lr(R(u o)

- gi)}

.

Hence rr(gi) + min{Ir(L(u0) - g i ) + rr+(lu0 - L ( u 0 ) l ) , 7r(R(uo) - g i ) + l r - ( I R ( u o ) - U o I)) = 1.

By the construction o f rr on I \ G n and by L ( u 0 - gi) = L ( u o ) - gi and R ( u o - gi) = R ( u o ) - gi, the minimum in the equation above is precisely rr(uo - gi). Since lr(u) + Ir(u 0 - u) = 1 for u ~ Gn, equality also clearly holds for u = u o - gi, gi ~ Gn" These points are located between consecutive grid points L ( u ) , R ( u ) in the same relative position as u 0 is between L ( u o) and R ( u o ) .

380

R.E. Gomory, E.L. Johnson

(a}

I I t

g i-I

I

I I"

a--~

' i I

I

u.i

u i+l

gi

g i+l

(b) I I I

I I

I I I

_1

u-gi+l

Uo-Ui+l

uj

gj

Uo'gi

Uo'Ui

Uj+l

gj+l

Uo'gi-I uj+2

Fig. 1.

Figs. l(a) and (b) illustrate the possibilities for rr o n the intervals g i - 1 , gi, gi+l and the c o m p l e m e n t a r y intervals u 0 - ui+ 2, u 0 - ui+ 1 , u o - u i, u o - u i _ l , where we let ui+ 1 = gi + Uo - L ( u o ) and u i = gi - R ( u o ) + Uo. T h e n u 0 - ui+ 1 E G n , say g! = u o - ui+ 1 and gi+l = Uo - ui ~ Gn" If, as in Fig. 1, the m a x i m u m o f rr in (gi, g i + l ) occurs at u = u i + l , t h e n 7r(u) + rr(u 0 - u ) = 1 for all u ~ (gi, gi+l)" In o r d e r to see this result, consider any interval ( u i + l , gi+l ) w h e r e ui+ 1 = gi + Uo - L ( u o ) = gi+l R ( u o ) + Uo, and the c o m p l e m e n t a r y interval (u 0 - gi+l , Uo - Ui+l ). T h e d i f f e r e n c e rr(ui+ 1 ) - r r ( g i + 1) m u s t be the same as lr(u 0 - g i + l ) rr(u 0 -- ui+ 1 ) because rr(gi+ 1 ) + 7r(u 0 -- gi+l ) = 1 and rr(u i) + rr(u o - u i) = 1. Since rr can have o n l y t w o slopes, rr m u s t be the same, e x c e p t for a c o n s t a n t d i f f e r e n c e in height, in the two intervals ( u i + l , g i + l ) and (Uo -- gi+l , UO -- Ui+I)"

T h e s e c o n d possibility is illustrated in Fig. 1 b y the interval (ui, gi) and its c o m p l e m e n t a r y interval (u 0 - gi, Uo - ui). In b o t h intervals, rr has t w o slopes and a relative m a x i m u m occurs within the interval, In this case, we m u s t have lr(u) + rr(u 0 - u) > 1 for all u w i t h i n e i t h e r interval. F o r at u = u i, rr(u) + rr(u 0 - u) = 1, b u t as u is increased, b o t h rr(u) and rr(u 0 - u) increase until o n e o f rr(u), rr(u 0 - u) reaches a maxima. T h e n rr(u) + rr(u o - u) remains c o n s t a n t as u increases since one o f

Continuous functions related to corner polyhedra, II

381

7r(u), ~r(u0 - u) is increasing while the other is decreasing at the same rate. When the other lr(u), 7r(u 0 - u) reaches its maxima, t h e n lr(u) + 7r(u 0 - u) decreases until u reaches gi and u 0 - u reached u o - gi at which point 7r(u) + 7r(u 0 - u) = 1. An interval (u i, gi) o r (gi, Ui+l ) with o n l y one slope for 7r will be called an interval o f the first type; here, ui+ 1 = gi + Uo - L ( u o ) . The c o m p l e m e n t a r y interval will also be an interval o f the first type, and for u in an interval o f the first type, 7r(u)+ ¢r(u 0 - u ) = 1. A n interval (Ui-1, gi) o r (gi, Ui+l ) with two slopes for lr will be called an interval o f the s e c o n d type. T h e n its c o m p l e m e n t a r y interval is also of the second type, and for u within an interval o f the second type, ~r(u) + r.(u 0 - u) > 1. We n o t e t h a t the intervals ( L ( u o ) , u o) and (u 0, R ( u o ) ) are o f the first type, and so are their c o m p l e m e n t a r y intervals (0, u o - L ( u o ) ) , (1 - R ( u o ) + U o, 1). An interval (ui, gi) will be its own c o m p l e m e n t if g i + gi = R ( u o ) since t h e n u o - gi = Uo - R ( u o ) + gi = ui" The interval (gi, Ui+l) will be its own c o m p l e m e n t if gi + gi = L ( u o ) since t h e n u o - gi = Uo - L ( u o ) + gi = Ui+l. These self-complementary intervals m a y be o f either the first or second type. In w h a t follows, we will exclude the self-complementary intervals in the discussion of intervals o f the second type. With this background, we can construct a f u n c t i o n ~r~ f r o m 7r which will lead to some interesting results. Let ~ = (gi, Ui+l ) be an interval o f the second t y p e and let t3 be its c o m p l e m e n t a r y interval. We assume t h a t o~ is n o t its o w n c o m p l e m e n t , so o~ ¢ ~. Then 7r(u) + 7r(u 0 - u) > 1 for u within either ~ or 13. Define ~r~ on I by

[ 7r(u), %(u)= [ l _ l r ( u 0 - u ) ,

u ~ I\e, uE~.

Fig. 2 illustrates % in this case. Let ua d e n o t e the u where lr~(u) is smallest in a. First, t w o lemmas are needed. The first applies to a n y 7r and does n o t d e p e n d on the particular c o n s t r u c t i o n here. Lemrna 4.1. L e t S be a subset o f I a n d let lr be a subadditive valid inequality f o r P(S, u o ), I f ~r(u) + lr(u o - u) >- 1 ,

~(u) + ~(v) >- ~(u + v), then lr is a valid inequality f o r P(L u 0 ).

u ElkS, u~I\S,

v~I\S,

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R.E. Gomory, E.L. Johnson

'//"

IT

I I

gi L. I~

I

Ua

Ui+l ~i ~I

Cl

l

I

uo- ui+ I

gi+l

,-

I

I

Uo'-g i u o -u i

B -----4

(o)

(b) Fig. 2.

Proof Consider any t solving P(/, u0). If t ( u ) > 0 and t ( v ) > 0 for both u and v in I~S, then we can change t by reducing t ( u ) b y 1, reducing t(v) by 1, and increasing t(u + v) by 1. The new t is still a solution, and since 7r(u) + ~r(v) >_ 7r(u + v), Z u ~flr(u) t(u) has not increased. This process can be continued until Eu ~1\s t(u) _ 1, by lr(u)+ rr(u 0 - u ) > _ 1 for u ~ I\S. The lemma is therefore proven. The second lemma applies to the particular function rr~ constructed here. It actually applies to any two-slope function 7r in an interval in which the function first decreases and then increases.

L e m m a 4.2. I f 27ra(u~) >_ 7r(2u~), then Try(u) + rr~(v) >_ 7r~(u + v) for all u, v e ~ . Proof For any u 6 a , u ¢ us, either lu[< lu~[ or lul> lull. Let us assume lul > lu~ I. The other case is similar. Then

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383

Try(u) = 7rs(u~) + 7r+(lul- lusl ) and for v ~ a, V~(u, o) = % ( u ) + % ( 0 ) - % ( u + o)

= 7rs(u s) + 7r+(lul- lull) + 7r~(v) - 7rs(u + v) = 7rs(u s ) + ~ ( v ) - ( ~ ( u + v )

- ~ + ( u + v l - lu~ + vl))

> 7rs(u~) + 7rs(v) - % ( % + v) = V~(u s , v ) ,

by 7rs(u ~ + v) + ~r+(lu + ol - I% + ol) >_ % ( % + o ) .

Similarly, we can show 7s(us, v) >_ 7a(us, us). Hence if 7s(ua, us) >_ 0, then 7s (u, v) >__0 for all u, v ~ a. These two lemmas suffice to prove the following theorem.

Theorem 4 . 3 . / f 21r~(ua) >_ 7r~(2ua), then lr~ is a valid inequality for P(I, u0). Proof By Lemma 4.1, we need only show that 7ra(u)+ 7rs(u 0 - u)>_ 1 for all u ~ ot and ~r(u) + It(o) >_ 7r(u + v) for all u, v ~ a. The first inequality is true, in fact with equality, by the construction of 7r~. The second is true by 27rs(u~) >_ lrs(2U~) and Lemma 4.2. Corollary 4.4. l f a and its complement {3are the only two intervals o f the second type, then ira is an extreme valid inequality for P(I, u O) if and Only if 2rrs(us) 7> rr(2us). Proof. If 2~r~(us) >_ 7rs(2u~) ' then by Theorem 4.3, rrs is a valid inequality for P(I, u 0). Furthermore, if a and/3 are the only two intervals of the second type, then rrs (u) + rrs (u 0 - u) = 1 for all u ~ / , so Irs is minimal. By Corollary 3.4, ~r~ is an extreme valid inequality for P(I, u 0 ). We now consider in more detail the case described in Corollary 4.4. To begin, two cases will be shown from [4, Table 2]. When n = 5 and u 0 ~ (0, ~), face 2 from [4, Table 2] is illustrated in Fig. 3. O f course, when u 0 ~ (~, 1), the reflection is also a face ofP+_(Gs, u0). Fig. 3 ac-

384

R.E. Gomory, E.L. Johnson

8

6 4

2 0

Uo

1/5

2 1 5 uQ

315 u/3

415

I

Fig. 3.

tually shows the construction of Theorem 1.10 for u 0 = N. It is easily verified directly that the two complementary intervals a and/3 are the only two on which 7r(u) + 7r(u0 - u) = 1 does not hold and that 21r~(u~ ) >_ Ir(2u~). Here, u s = 2~. Fig. 4 shows another example for n -- 5 and u 0 ~ (k, z). Its reflection is, again, another example. This figure is face 6 from [4, Table 2.2]. As in Fig. 3, a and ~ are the only two intervals of the second type, and 2rr~(u~) >_ Tr(2U~). in both Figs. 3 and 4, the role of a and/3 can be reversed, and we still have 2 % ( u ~ ) >_ 7r(2u~). In other words, if rr~ is defined analogously to rr~ with u~ = ~13 in Fig. 3 and u~ = ~ in Fig. 4, then 2~r#(u#) >_ rr(2u~). The next theorem shows that in this case a great many extreme valid inequalities can be generated which differ from 7r only in the intervals and ~.

I0 8 6 4

2 0

1/5

2/5

Ua 5/5 Fig. 4.

u]~4/5

I

385

Continuous functions related to corner polyhedra, II

8

7/"

6 4 2

0

uo

115

215 ua

3/5 u/~

4/5

I

Fig. 5.

Theorem 4.5. I r a and/3 are the only two c o m p l e m e n t a r y intervals o f the second type and i f lr~ and Ir~ are each valid inequalities f o r P(L u o ), then any continuous, piecewise linear f u n c t i o n p on I having only the two slopes 7r+ and 7r- satisfying

p(u) = ~ ( u ) ,

u ~ I\(~ u D,

p(u) = l - p(u o - u ) ,

u ~ ~ ,

is an e x t r e m e valid inequality f o r P(/, Uo). Fig. 5 illustrates such a function p in the example shown in Fig. 3. P r o o f We will consider only the case previously considered; that is, a = (gi, ui+l ) so that the left end-point of a is in G n . Fig. 3 is this case, but Fig. 4 is not. The case a = (u i,gi) is similar and will not be considered. First, we will show that neither o~ nor/3 is a subinterval of (0, gl) or (gn-1, 1). Since a and /3 have an element o f G n as left end-point, if either was a subinterval of (0, gl ), then it would have to be (0, u 1 ). However, this interval is o f the first type as was remarked before Lemma 4.1. Hence the only possibility is that a or/3 is (gn-1, Un)" We will now exclude that possibility. Corollary 4.4 says that rra is extreme and hence subadditive. We will show that rr then is linear on (gn-1, 1) with a slope - r r - , and hence neither a nor ~ could be (gn-1, Un)"To see that Ir is linear o n (gn-1, 1), recall that

%(gi) + %(Uo - gi) = ~r(gi) + ~r(Uo - gi) = 1 = 7r(uo)

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386

and t h a t rr~ and rr are decreasing o n (u 0 - gi, Uo - ui) because o f the shape o f rr in/3. Hence, 7ra(g i) + 7ra(u 0 -- u i ) = rrc,(R(uo) ) .

But 7ro~(uo~ ) = 7ra(gi) - ~ - - ( [ u c ~ - g i [ ) ,

and b y subadditivity, one o f the following inequalities holds:

rc~,(u~) + rrc,(u 0 - ui) >_ rr,~(R(u O) - (u~, - gq)) , rre,(gi) - rr- ([u - g/I) + rra(u 0 - u i ) >_ rcc~(R(uo) -(ue~ - g i ) ) , zr,~(R (u o )) - rr- ( luc~ - gi I) >_ rra (R (u o ) - (u,~ - gi)) • By rr~ having o n l y t w o slopes, the reverse i n e q u a l i t y also holds, and h e n c e rr~ is decreasing o n the entire interval (R(uo), R ( u o ) + g l ) " This fact and subadditivity i m p l y that zr~ is decreasing o n the e n t i r e interval (gn-1, 1), c o m p l e t i n g the p r o o f t h a t n e i t h e r ~ n o r / 3 is a subinterval o f (gn-1, 1) or (0, g l ) T o r e t u r n to the p r o o f o f the t h e o r e m , b y L e m m a 4.1 we can prove that p is a valid inequality b y showing p(u) + p(u o - u) >- 1 and p(u) + p(o) >_ p(u + v) f o r u, v ~ ~ u/3. T h e first i n e q u a l i t y is obvious f r o m the c o n s t r u c t i o n o f p. What remains is to establish p(u) + p(v) >- p(u + u) for u, u ~ a u/3. T h e r e are two cases: (i) u, v b o t h in a (or b o t h in/3), and (ii) u G a and v ~/3. In case (i), we o n l y consider u, v b o t h in a since b o t h in/3 is e x a c t l y similar. By p being c o n t i n u o u s with the same two slopes as rG,

p(u) + p(v) >_ % ( u ) + %(v) . By 7r~ being valid, and h e n c e e x t r e m e ,

%(u) + %(v) >- 7r(u + v). I f u + v ~ I \ ( a U/3), t h e n zra(u + v) = p(u + v), so p(u) + p(v) >- p(u + v). If u + v ~/3, t h e n rr (u + v) = Ir(u + v) >- p(u + v), so again p(u) + p(v) >p(u + v). T h e third subcase u + v ~ a is e x c l u d e d b y a n o t being a subinterval o f (0, g l ) o r ( g n - 1 , 1). F o r any a w h i c h is a subinterval o f (gi, gi+l ), b u t n o t (0, gl ) or (gn-1, 1), u + v q~ a w h e n u ~ ~ and v ~ a.

Continuous functions related to comer polyhedra, H

387

N e x t we c o n s i d e r case (ii), u ~ a a n d v c/3. In this case t h e r e are t w o subcases: Iv] >- [u o - uf, a n d iv[ < lu o - u[. First, c o n s i d e r ivl >- fu o - ul. Since v a n d u o - u are b o t h in/3, fv-

(u 0 -

u)[- 1 - 7 t - ( i v -- (u 0 -- u ) l ) . B u t p has o n l y t w o slopes, so p(v)

>- p(u o - u ) - ~ - ( f v

-

(%

-

u)l),

p(u) + p(v) >_ p(u) + p(u o - u) - zr- (Iv - (u o - u ) l ) > 1 - 7 t - ( I v - (u o - u ) l ) , c o m p l e t i n g t h e p r o o f in this subcase. N e x t c o n s i d e r lul < lu 0 - ul. In a similar way, we can n o w s h o w t h a t

p(u + v) = 1 - zr+(l(u0 - u) - v l ) , p(v) >- p(u o - u) - lr+(l(u0 - u) - v l ) . H e n c e , as b e f o r e ,

p(u) + p(v) >_ p(u) + P(Uo - u) - ~r+(l(Uo - u) - vl) = I -

¢r+(l(u0

-

u) -

vl) = p(u + v).

H e n c e p is a valid i n e q u a l i t y f o r P(I, u 0). T o s h o w t h a t it is an e x t r e m e valid i n e q u a l i t y , we n e e d o n l y r e m a r k t h a t p ( u ) + p(u o - u ) = 1 and a p p l y C o r o l l a r y 3.4. T h e t h e o r e m is thus proven. T h e d e v e l p p m e n t h e r e can be e x t e n d e d to the case w h e r e t h e r e are several intervals o f the s e c o n d t y p e . H o w e v e r , its p r e s e n t f o r m suffices to s h o w an e x p o n e n t i a l r a t e o f g r o w t h f o r s o m e o f the p o l y h e d r a P ( G n, g) o f [ 3 ] . We s h o w this f a c t b y m e a n s o f the f o l l o w i n g e x a m p l e .

R.E. Gomory, E.L. Johnson

388

\/ p~ J

¢

Jra 2/5 __

l

i

l

l

l

l

l

I

Fig. 6.

E x a m p l e 4.6. Consider the group G n for n = 20K, K >_ 1, and let Uo = ~o c G n . We said that the function 0 in Fig. 5 gives an extreme valid inequality for P(I, Uo). The same is true for a great m a n y functions p. In Fig. 6 we illustrate the intervals a and/3 from Fig. 5. Let us restrict p to be straight lines with breaks at points k/2OK. In Fig. 6, K = 3, and we are perfectly free to let p have slope rr+ or - r r - in the 3 intervals ( ~ , ~2s) , ( 2s 26) , ( 26 ~, ~ ~ , ~ ) . The only restriction on P here is that it must have slope rr+ on as many intervals between ~ and ~-6 1o a s on which it has slope lr-. Since p has been determined on ~o to ~,1° it is given on ~o to ~ by 7r(u) + 7r(u0 - u) = 1. In general, there will be K intervals between ~o and ~ on which p can have either slope. Thus there are at least 2 K such functions p. By Theorem 3.2, each one is a face for the problem P(G20 K, ~ ) . In fact, there are more than 2K , namely ( 2 I O ! / ( K ! K [ ) , such functions p. This number results from the fact that we can choose any K of the 2K intervals between ~0 and ~1° for p to have slope rt+ . As K becomes large, this number approaches 2 2K/x/(rrK) by Stirling's approximation of n!.

There is an abundance of such examples from [4, Table 2]. In particular, for n = 7, there are several similar cases. A similar construction works as long as there are two complimentary intervals ~ and/3 with 7r~ and rr~ valid and provided the u 0 and all other break-points of lr fall on group elements.

Continuous functions related to comer polyhedra, H

389

References [ 1 ] G.B. Dantzig, Linear programming and extensions (Princeton University Press, Princeton, N.J., 1963). [2] R.E. Gomory, "An algorithm for integer solutions to linear programs", in: Recent advances in mathematical programming, Eds. R.L. Graves and P. Wolfe (McGraw-Hill, NewYork, 1963) 269-302. [3] R.E. Gomory, "Some polyhedra related to combinatorial problems", Linear Algebra and its Applications 2 (1969) 451-558. [4] R.E. Gomory and E.L. Johnson, "Some continuous functions related to corner polyhedra, I", Mathematical Programming 3 (1972) 23-86.