Operat ions Research Letters 11 (1992)293-302 June 1992 North-Holland

Lagrangean duality for facial programs with apphcat ons to integer and complementarity problems

Chris t ian La r sen * Department of Management, Unit~ersity of Odense, Odense, Denmark

Jo rgen T ind Department of Operations Research, University of Aarhus, Aarhus, Denmark

Received June 199(I Revised October 1991

In this paper we consider how to close the duality gap when Lagrangean duality is applied to a facial constraint. We generalize some results by Giannessi and Niccolucci. This enables us to close the duality gap for a class of mathematical programming problems with complementari ty constraints. Finally we discuss the connections to general duality theory and provide a simple way to close the duality gap for a general mathematical program that violates the Slater condition.

Lagrangean duality; facial disjunctive programming; linear complementari ty

1. Introduction

Lagrangean duality is a concept with an extensive use in mathematical programming. It is useful in the design of solution algorithms, particularly in the development of optimality tests for solution candidates.

We shall in the present context consider a general class of mathematical programming problems with only one constraint, but with a so-called facial structure. This class induces after some reformulation the class of 0-1 programming problems as well as problems with complementari ty constraints. Also general mathematical programs violating the well known Slater condition can be reformulated into this class. We will show how to close the duality gap for this selected class.

The present approach has been addressed for the case of 0-1 programming by Raghavachari [9] and Kalantari and Rosen [6]. Giannessi and Niccolucci [5] considered a general case subject to some bounds in the behaviour of the objective function and the constraint function. We shall here generalize their result while focusing on the application of the general result to the class of facial disjunctive program- ming problems and the class of problems with a complementari ty constraint.

The paper is organized as follows. Section 2 establishes the general result about the closure of the duality gap and demonstrates that an earlier result in Giannessi and Niccolucci [5] turns out as a subcase. Section 3 contains a discussion of a facial disjunctive program reformulated into a program with one constraint. It includes a result about the behaviour of the constraint function leading to the t reatment of

* Supported by a grant from the Danish Natural Science Research Council. Part of this research was done while the first author was Visiting Scholar at The School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA.

Correspondence to: Prof. Christian Larsen, Dept. of Management , University of Odense, Compusvej 55, DK-5230 Odense M, Denmark.

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0-1 programming problems and problems with a complementarity constraint as performed in Section 4. In Section 5, we consider a general mathematical program where the Slater condition is not fulfilled. This problem is studied in the general duality framework due to Tind and Wolsey [13]. We describe how the previous analysis gives us a simple class of dual functions sufficient to close the duality gap. Finally Section 6 gives some concluding remarks.

2. The general result

We shall here consider the following general form of a mathematical programming problem with only one constraint:

z = m i n { f ( x ) : x ~ C, g ( x ) < 0}, (1)

where C ~ ~n and f ( ' ) and g ( ' ) are real valued functions on C. Let F = {x ~ C: g(x) < 0} be the set of feasible solutions of (1). Besides usual conditions concerning continuity, compactness and non-emptiness explicitly stated in the theorems, we will throughout the paper make the following key assumption:

A1. Vx ~ C: g(x) > O.

We call g(x) < 0 a facial constraint. A1 is a generalization of a similar condition for facial disjunctive programs considered by Balas [2] and Sherali and Shetty [12].

A general mathematical program with several constraints gl(x) < 0 . . . . , gin(x) <_ 0 can also be brought into the form (1) by using the single constraint function g(x)= max{gl(x) . . . . . gm(X)}. Thus the facial requirement A1 amounts in this formulation exactly to the case where the Slater condition: 3 x ~ C: gi(x) < 0 Vi, does not hold.

Consider the dual problem of (1):

sup{w(M): Me (2)

where

w( M) = m i n { f ( x ) + M g ( x ) : x ~ C}. (3)

In this paper we shall establish conditions for the closure of the duality gap between (1) and (2) with a finite dual multiplier, as expressed by the following:

Property 1. 3 M o >_ 0: VM >_ Mo: z = w(M).

A way to verify this property is to introduce the perturbation function

~ ( b ) = m i n { f ( x ) : x ~ C , g ( x ) <_b}. (4)

q~(b) is a nonincreasing function. As usual, when the feasible set of (4) is empty, we define ~o(b) = ~. Thus under assumption A1, q~(b) = oo for b < 0. It is well known, see for instance Minoux [7], p. 153, that Property 1 is equivalent to:

Property 2. 3M o >_ 0: VM >_ M o, Vb ~ ~: ~(b) >_ ~(0) - Mb.

Property 2 says that we can find a subgradient for q~(.) at 0. For each b > 0 define A(b) as the set of optimal solutions of (4). Under the assumptions that C is compact, f ( . ) and g(-) are continuous on C and F is nonempty, A(b) is a nonempty compact set for all b > 0. Define for b > 0 the shortest distance between optimal solutions of (1) and (4) as

d(b) =min{ I I x - y l l : x ~ A ( b ) , y c A ( 0 ) } .

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Lemma 1. Assume C is a compact set, f ( ' ) and g( . ) are continuous on C and the set F is nonempty. Then the function d(" ) is continuous from the right in O.

Proof. Choose {bk}~= 0 C ~ + such that b k ~ 0. We must show that d(bk) --* O. Because lim inf k ~ d ( b k) > 0, it is enough to show that lim s u p k ~ = d ( b ~ = 0. Assume on the contrary that there exists a subsequence {bi}~= 0 of {bk} ~ 0 such that d(b i) ~ d > 0. Choose x(b i) cA(hi ) . Then f (x(bi)) < q~(0) for all i, because q~(') is nonincreasing. Because C is compact we can find a subsequence {bp}~_ o of {bi}~= 0 and a point $ c C such that x ( b p ) ~ . By continuity f ( x (bp) )~ f (£ )<q~(O) . On the o ther hand {g(x(bp)) - bp} ---, g(£) <_ O, thus f ( £ ) >_ q~(0). Therefore f ( ~ ) = q~(0), that is Y c A ( 0 ) , d(dp) ~ 0 because d(bp) <_ II x(bp) - x II, which contradicts d(b i) --* ff > O. []

L e m m a 2. Assume for all x c C: g(x ) > 0 and fmin := min{f (x) : x c C} > - ~ . I f there exists a b ° > 0 and a constant Ksuch that q~(b) > q~(O) - K b f o r all b c [0, b°], then q~(') has a subgradient at O.

Proof. q~(b) = ~ for all b < 0. For b > b °,

~p(b) >_ ~ (0 ) - (q~(O) - - fmin) > ~ ( 0 ) -- b(~(O) - fm in ) /b 0

Thus q~(b) >_ ~p(O) - max{K, (~p(O) - fmin)/b°}b for all b c I~. []

Deno te by B(x, e) the closed ball with center x and radius e. We now state:

A2. There exists constants e > 0 and H > 0 such that f ( £ ) - f ( x ) < H g ( x ) for all ~ c A ( 0 ) , and for all x c B ( £ , e) A ( C \ F ) .

Theorem 1. Assume for all x c C: g(x ) >__ O, C is a compact set, f ( ' ) and g( ' ) are continuous on C, F is a nonempty set and A2. Then Property 2 and thus Property 1 holds.

Proof. Accord ing to L e m m a 2 we want to find a b ° > 0 and constant K such that q~(b) _> q~(0) - Kb for all b c [0, b°]. Choose e > 0 and H > 0 such that assumption A2 is fulfilled. By Lemma 1 we can select b ° > 0 such that for any b c [0, b °] there exists an x ( b ) c A ( b ) and an y ( b ) c A ( O ) w i t h II x(b)- y(b)II < e, i.e. x(b) c B ( y ( b ) , e). If there exists b c ]0 , b °] with g(x(b)) = 0 we deduce from q~(0) > q~(b) =

f ( x (b ) ) > q~(0), that q~(') is horizontal close to 0, and we are finished by Lemma 2. So let g(x(b)) > 0 for all b c ]0 , b°]. Then by A2,

b >_g(x(b)) > H - ' [ f ( y ( b ) ) - f ( x ( b ) ) ] = H - ' [ q ~ ( 0 ) - ~p(b)].

Thus ~(b) > ~ ( 0 ) - Hb for all b c [0, b°]. []

Assumpt ion A2 is also necessary, independent of assumption A1, as demons t ra ted by the next theorem.

Theorem 2. Assume C is a compact set, g(. ) is continuous on C and F is a nonempty set. Then Property 2 (Property 1)implies assumption A2.

Proof. Assume that A2 does not hold. Let there be given a sequence {e~}~_ 0 such that e k $0. When A2 is not valid, we must be able to find a sequence {Y~}~=0 c_A(0) and a sequence {xk}~_ 0 with x k c B (y k, e k) n ( C \ F ) , in such a way that ( f (Yk) - - f ( x k ) ) / g ( x k ) $ ~ " Let b k = g(xk). We will first show that limk _~b k = 0.

As in the p roof of L e m m a 1 we must show that lim supk _~b~ = 0. Assume to the contrary that we can find a subsequence {bi}~= o of {bk}~= 0 such that b i ~ b > 0. Let {xi}~= o be the corresponding subsequence of {xk}~_ 0. Because C is compact , we can find a subsequence {xp}p= 0 of {xi}7_ 0 and a point £ c C, such

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that Xp ~ £ . By continuity of g ( . ) we have g(£) = b . Because II Yp --Xp [[ < Ep, yp ---~ ~. But g(yp) _< 0 for all p contradicts g (£ ) = b > 0. Therefore lim k ~o~b k = 0. Because f ( x k) > q~(bl,) we get that

( f ( Y k ) - - f ( X k ) ) / g ( x t , ) < (q~(0) -- ~ ( b k ) ) / b k $~.

But this implies that q~(') does not have any subgradient at 0, a contradiction. []

Instead of A2 the following two assumptions were stated in Giannessi and Niccolucci [5]. Let /3 > 0.

A3. There exists a constant L > 0 such that for all x, y • C, [ f ( x ) - f ( y ) l ~< L II x - y II ~

A4. There exists an e > 0 such that g(x) > k(e)II x - ~ [I ~ for all ~ • F and x • B(£, e) n ( C \ F ) , where k(s) > O.

Instead of expressing a direct relationship between f ( x ) and g(x) as in assumption A2, assumptions A3 and A4 claim appropriate bounds on the objective function and on the constraint function. By using the same/3 , A3 and A4 simultaneously imply A2 and the general result (Theorem 2.1) in Giannessi and Niccolucci [5] follows by Theorem 1. Clearly, in the light of assumption A2 the norm expressions in assumptions A3 and A4 could be replaced by a positive function, which should appear in both A3 and A4.

3. Bound for the constraint function

In this and the following section, we will consider the following facial disjunctive special case of (1).

min f ( x )

s.t. x • C , (5) h aix>_b~va~x<_b~, hcH,

where a h, a h • [~n\{0}, btl ', b h • ~ for all h • H , and H is a finite index set. According to Balas [2] the facial requirement for (5) is:

AI'. V x • C , h /, _ b h a l x < b I and ahx> V h • H .

If we let

g ( x ) = ~ (bh--a l l ' x ) (ahx--bh) , (6) h~H

the feasible set of (5) has the desired form {x • C: g(x) < 0}, and it fulfills assumption A1. We will also from now on assume that assumption A3 is satisfied with /3 = 1, that is f ( . ) is Lipschitz

continuous on C. In this section we will try to verify the nontrivial assumption A4 with/3 = 1. The aim of this is of course to use the result of Theorem 1 on (5).

We make the following two assumptions. The first one is:

A5. V£ • F, bh > ah~ o r a h y , > b h Vh • H.

Thus, each £ ~ F defines a unique partitioning of H via H1(2") = {h • H: ah£ = bl h} and H2(2") = {h • H: ah2 £ = b2 h} since H1(~) W H2(£) = H and H1(£) n H2(.~) = ~. For p > 0, let U(p) = {u • 0~": u = p(x - Y)/[I x - f f ]l, where ~ • F and x • C \ F } . That is, U(p) is the set of all directions from points in F to points in C \ F , normalized to length p. The second assumption is:

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A6. There exists a p > 0 such that: if 2 ~ F and H2(2) 4: ¢, then b h - ah2 > l ahu I Vh ~ H2(2), Yu e U(p) and if 2 ~ F and Hi (2) ~ ¢, then ah2 -- b h > [ahzu[ Vh c H~(£), Vu E U(p).

Due to the Cauchy-Schwarz inequality, a simple condition implying A5 and A6:

A7. Zip > 0 :V2 E F, (b~j ' - ah2)/II a~ h II >- p or (a~£ - bh2)/II a2 h II >- p Vh ~ H.

Assumption A7 says that we must have some uniform lower bound in the distance from an 2 ~ F to each of the violated disjunctive constraints.

We now state:

Theorem 3. For the feasible set of (5) assume AI', A5 and A6. Choose e ~ ]0, p[. Then the constraint function g(x) in (6) satisfies

g (x ) > ( ( p - 8 ) / p 2) inf / E [a~ul [a2hu[: u ~ U ( p ) } l l x - 2 [ ! ~ h ~ H

V2 eF , Vx ~ B ( 2 , e) n ( C \ F ) .

Proof. Choose £ ~ F and x ~B(2 , e ) n (C \F) . Let r = l[ x - 2 II < e and u = p ( x - 2 ) / r . Then x = 2 + (r/p)u. We derive

g(x) = E ( b h - a h ( 2 + ( r / p )u ) ) ( a~ (2 + ( r / p ) u ) - b h) h ~ H

= ~' ( - - ( r /p )ahu)(ah2--b h + ( r /p )ahu) h~Hj(g)

+ E (b~'--ahl2--(r/p)a~'u)(r/p)a~ u h ~ H2(.2 )

=(r/p) E [a~u[(ah~£-b~)+(r/P) E [a~u[(b~'-atl '2) h ~Hl(2) h ~H2(Y.)

- ( r / p ) 2 y" (a~u)(ah2 u) h ~ H

> ( r / p ) E [ a ~ u l ( a ~ f - b ~ ) + ( r / p ) h~HI(Y~)

- ( r / P ) 2 E [a~ulla~u[ h ~ H

> - ( r / p - ( r / P ) 2) E [a~ulahu[ h ~ H

E la~u l (b~-a~Y) h~H2(Y.)

(by A6)

>>_(I]x -£] l /p ) (1-e /p ) inf{ E ]a~ul la~ul: u ~ U ( p ) } h ~ H

= ( ( p - e ) / p 2) inf{ Y'~ [ahlu[ [ a ~ u [ : u ~ U ( p ) } [ [ x - 2 [ [ . h ~ H

In the second line we used the existence of a unique partition of H into H1(2) and He(2). In the third line we used that h ~ H1(£) implies a~u < 0 as x ~ C. Similarily h ~ H2(2) implies that a~u > O. []

In order to use the bound, derived in Theorem 3, in assumption A4, and thus to satisfy all assumptions in Theorem 1, we must show that inf{Eh ~ H I a~u I I aheu I: u ~ U(p)} > 0. We will do this for two special instances of (5) in the following section.

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4. Special cases

In this section we consider two special cases of (5). First the case where (5) is a nonlinear 0-1 program and secondly the case where the feasible set of (5) is the solution set of a linear complementarity problem. In the second case we establish some new sufficient conditions for closing the duality gap, while in the first case we reobtain a result first shown by Giannessi and Niccolucci [5], Theorem 3.1.

(i) The 0-1 case. Here the realization of (5) is

min f ( x )

s.t. x ~ C, (7)

x {0, 1}".

The feasible set of (7), F, can be rewritten to

e=cnlo, 11 =Cnen , (8) j = l

where E = cony{0, 1} ~. So (7) is a facial disjunctive program, which means that assumption AI ' holds. The realization of g(x) from Section 3 is g(x)= ~ _ l X j ( 1 --Xj), and the dual problem of (7) is

s u p m i n ( f ( x ) + M ~ x j ( 1 - x j ) : x ~ C ) . (9) M>~0 j = l

First observe that assumption A7 is always true for this specific problem (with p < 1) and that Eh ~ n I a~u I I ah2 u [ = II u II 2 = p2 > 0 for all u ~ U(p). Therefore by Theorem 3 assumption A4 is true.

We can now as a direct application of Theorem 1 state the following proposition, first proved in Giannessi and Niccotucci [5] (Theorem 3.1).

Propos i t ion 1. Assume C is a closed set, F is a nonempty set and f ( . ) is Lipschitz cont&uous on C n E. Then the duality gap between (7) and (9) is closed with a finite dual multiplier.

Because E is compact and C N E plays the role as C in Theorem 1, we only have to assume C closed in Proposition 1.

(ii) The complementarity case. Here (5) is given as

min f ( x , y)

s.t. y = Q x +q, x, y > 0, (10)

xTy < O.

Q is a n × n matrix and q ~ ~n. Again this is a facial disjunctive program, where the feasible set of (10), F, can be rewritten to

F = C N ( ~'] ({(x' Y) ~ 2 n : xj<O} U {(x' Y) E~Zn: yj<O})} (11)

with C = {(x, y) e ~2+,: y = Qx + q}. For this case the realization of g(x, y) from Section 3 is g(x, y) = xTy and the dual problem of (10) is

sup m i n { f ( x , y) + MxTy: ( x, y) ~ C}. (12) M>_0

As stated earlier a prerequisite for establishing Theorem 1 is to verify Theorem 3. In the 0-1 case the

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latter was always true. For the complementarity case we have explicitly to state the tollowing specializa- tion of assumption A7:

A7. Zip > 0: V(Y, ~ ) e F , ~ j > p or y i > p V j = 1 . . . . ,n.

Assumption A7' may be considered as an extension of the nondegeneracy assumption, often used when studying the linear complementarity problem. We will now establish when the expression inf{Y~h ~ H I a~u I I a~u I: u ~ U(p)} from Theorem 3 is positive.

We describe the elements in U(p) by (u, v) where u, L' ~ Rn. For an arbitrary (u, v) ~ U(p) that is (u, v) =p((x , y ) - (~, p))/r, where r = II(x, y ) - (.~, ; ) l l , (x, y ) ~ C \ F and (£~, p ) e F , we get from

+ (r/p)v = Q(Y + (r/p)u) + q and ~ = Q~ + q that v = Qu. Because all ' and a~ in this case have the forms ( - e h, 0) and (0, eh), the realization of inf{Eh~ tt a~ul a~u [: u ~ U(p)} from Theorem 3 is

) I inf luj l lv j t ' (u ,v)~U(o) = i n f j = l I , j = l

>inf{ L j = l

as U(p)c_gu: I l u l 1 2 + IlQull2=p2}and II(u, Qu)ll 2= Ilull 2 tion A4, the crucial point is whether

inf lujl I(Qu)~l[: Ilul12+ IIQulI2=p 2 >0 . (13) k j = ]

Introduce here the concept of a P matrix. An n × n matrix A is called a P matrix if for all x ~ 0, there exists k such that Xk(AX) k > 0. See Fiedler and Ptak [3,4].

The set {u: II u II : + II Qu II z = p2} = {u: uT( I + QTQ)u = p 2} is the boundary of an ellipsoid centered around the origin, and is thus a nonempty compact set not containing the origin. So because the objective function in (13) is continuous, (13) is true subject to the following assumption.

ujl l(Qu)j I: (u, Qu) ~ U(p)

I(Qu)j]" ] l u l l 2 + HQull2=p2}, u j]

+ It Qu II 2 = p2. In order to employ assump-

A8. Q or - Q is a P matrix.

However, it is known, see for instance Murty [8], Theorem 3.13, that the feasible set (11) is a singleton when Q is a P matrix. So in this case we can just as well replace any original objective function by the zero function, and thus consider

rain{0: (x , y) ~ F } . (14)

This is the normal linear complementarity problem. Here it is known, see AI-Khayyal [1], that (14), when feasible, is equivalent to min{MxTy: (x, y) ~ C} for any M > 0. So if Q is a P matrix, the duality gap is always closed.

Therefore we replace assumption A8 by:

A8'. - Q is a P matrix.

So assumptions A7' and A8' imply assumption A4 via Theorem 3. When - Q is a P matrix the set C is automatically bounded as shown by:

I.emma 3. Under assumption A8', C = {(x, y) ~ ~2n: y = Qx + q} is bounded.

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Proof. By Rockafellar [10] it suffices to prove that the recession cone of C, O+C = {(x, y) ~ ~2,: y = Qx} consists only of the origin. Assume that (2, Y) ~ 0 ÷C and that (2, Y) ~ (0, 0). Let J = {j: 2~ > 0} ~ ¢. Because - Q is a P matrix there exists j ~ J with (Q2)j < 0. But this contradicts ~j > 0. []

For the complementari ty case the application of Theorem 1 is:

Proposition 2. Assume the set F is nonempty, f ( . ) is Lipschitz continuous on C, A7' and A8'. Then the duality gap between (10) and (12) is closed with a finite dual multiplier.

As a final remark on this section we comment on assumption A8'. Consider first the special case of (1) where the objective function and the constraint function both are concave and the set C is a polytope. This means that the constraint g(x) < 0 is a reverse convex constraint. Then the dual function w(.) in (3) is nondecreasing, polyhedral and bounded above by z. So clearly w(.) will eventually become horizontal. Note also that g(x) for an optimal x in (3) is a subgradient of w( ' ) . So when w(.) becomes horizontal an optimal x in (3) is also feasible in the primal problem. So in this special case it is easy to establish the closure of the duality gap by a finite dual multiplier.

Transforming this observation to the complementary case of this section, note that A8 ' absorbs the case where Q is negative definite. For that case the constraint

x T y < 0 ~ x T ( Q x + q ) < O

is reverse convex. So A8 ' is really a weakening of the requirement that xT(Qx + q) < 0 must be a reverse convex constraint.

5. Connections to general duality theory

We will now consider the general mathematical program

min f ( x )

s.t. g i ( x ) < O , i = 1 . . . . . m,

x ~ C .

We will assume that the Slater condition does not hold. That is,

V x ~ C : 3 i ~ {1 . . . . . m} such that gi (x ) >0.

Let g(x) = max{gl(x), . . . , gin(x)}. We then get the equivalent condition

Vx ~ C: g ( x ) >_ O.

The general dual problem of (15) defined as in Tind and Wolsey [13] is

max v(0)

s.t. v ( g l ( x ) . . . . , g m ( X ) ) ~ f ( x ) V x ~ C ,

L 'EV,

where V is a set of nonincreasing functions, in the sense x < y We now convert (15) into a facial program

m i n { f ( x ) : x ~ C, g ( x ) < 0}.

v (x ) ~ v(y).

(15)

(16)

(17)

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Assuming the prerequisites for Theorem 1 are fulfilled. Then the dual of (17) is

max u

s.t. u - M m a x { g t ( x ) . . . . . g m ( X ) } < f ( x ) V x ~ C , (18)

M > O .

So in order to close the duality gap between (15) and (16), we need only consider the l, functions in V of form

L,(d, . . . . . din) = u - M max{d 1 . . . . . dm},

where u ~ [R and M ~ 1~+.

6. Concluding remarks

This paper has investigated under which conditions the duality gap between a facial program and its Lagrangean dual is closed. By introducing the concept of a perturbation function, we have generalized and simplified the work in Giannessi and Niccolucci [5]. Further we have introduced the concept of P matrices and nondegeneracy in order to close the duality gap in a linear complementarity context. Both concepts are well known in the theory of linear complementarity. We have also described a simple class of dual functions sufficient to close the duality gap for a general mathematical program not satisfying the Slater condition.

The results in Section 2 seem to be as general as possible, because the conditions which imply the duality gap to he closed, are not only sufficient but as shown in some sense also necessary. So further research should be oriented in finding particular instances of mathematical programs, which satisfy the assumptions A2 or A3 and A4. The basis for Sections 3 and 4 is the generation of the lower bound for the constraint function g ( x ) from (6), as given in Theorem 3. The feasible set F in (5) could also be represented as (1)via an alternative constraint function as noted by Sen and Sherali [11], Proposition 2.1. So a possible extension of our results could be to generate similar bounds for this function. Another possible extension could be to consider more than pairwise disjunctions for F in (4), which could lead to investigations of how to close the duality gap for other classes of facial disjunctive programs.

Acknowledgement

The authors wish to thank a referee for very careful comments, which lead to several improvements of the paper.

References

[1] F.A. AI-Khayyal, "Linear, quadratic, and bilinear programming approaches to the linear complementarity problem", European J. Oper. Res. 24 (1986) 216-227.

[2] E. Balas, "Disjunctive programming", Ann. Discrete Math. 5 (1979) 3-51. [3] M. Fiedler and P. Ptak, "On matrices with non-positive off-diagonal elements and positive principal minors", Czechoslot;ak.

Math. J. 12 (1962) 382-400. [4] M. Fiedler and P. Ptak, "Some generalizations of positive definiteness and monotonicity", Numer. Math. 9 (1966) 163-172. [5] F. Giannessi and F. Niccolucci, "Connection between nonlinear and integer programming problems", Symposia Mathematica

Vol. XIX, Inst. Nazionale Di Alta Math., Academic press, New York, 1976, 161-176. [6] B. Kalantari and J.B. Rosen, "Penalty for zero-one integer equivalent problems", Math. Programming 24 (1982) 229-232. [7] M. Minoux, Mathematical Programming, Theory and Algorithms, John Wiley, New York, 1986. [8] K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Heldermann-Verlag, Berlin, 1988. [9] M. Raghavachari, "On connections between integer programming and concave programming under linear constraints", Oper.

Res. 17 (1969) 680-684.

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[10] R.T. Rockafellar, Convex Analysis, Princeton University Press Princeton, NJ, 1970. [11] S. Sen and H.D. Sherali, "Nondifferentiable reverse convex programs and facetial convexity cuts via a disjunctive characteriza-

tion", Math. Programming 37 (1987) 169-183. [12] H.D. Sherali and C.M. Shetty, Optimization with Disjunctive Constraints, Springer-Verlag, Berlin, 1980. [13] J. Tind and L.A. Wolsey, "An elementary survey of general duality in mathematical programming", Math. Programming 21

(1981) 241-261.

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