Department of Social Systems and Management. Discussion Paper Series

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1 Department of Social Systems and Management Discussion Paper Series No Outer Approimation Method for the Minimum Maimal Flow Problem Yoshitsugu Yamamoto and Daisuke Zenke April 2005 UNIVERSITY OF TSUKUBA Tsukuba, Ibaraki JAPAN

2 OUTER APPROXIMATION METHOD FOR THE MINIMUM MAXIMAL FLOW PROBLEM Yoshitsugu Yamamoto University of Tsukuba Daisuke Zenke University of Tsukuba April 22, 2005 Abstract The minimum maimal flow problem is the problem of minimizing the flow value on the set of maimal flows of a given network. The optimal value indicates how inefficiently the network can be utilized in the presence of some uncontrollability. After etending the gap function characterizing the set of maimal flows, we reformulate the problem as a D.C. optimization problem, and then propose an outer approimation algorithm. The algorithm, based on the idea of ε-optimal solution and local search technique, terminates after finitely many iterations with the optimal value of the problem. Keywords: Network flow, minimum maimal flow, optimization over the efficient set, D.C. optimization, outer approimation, global optimization. 1. Introduction We are given a network (V, s, t, E, c), where V is the set of m + 2 nodes containing the source node s and the sink node t, E is the set of n arcs and c is the n-dimensional column vector whose hth element c h is the capacity of arc h. The set of feasible flows, denoted by X, is given by X = { R n A = 0, 0 c }, (1.1) where m n matri A is the incidence matri whose (v, h) element a vh is +1 if arc h leaves node v a vh = 1 if arc h enters node v 0 otherwise. The well-known conventional maimum flow problem is ma d s.t. X, where d is the n-dimensional row vector whose hth element is +1 if arc h leaves source s d h = 1 if arc h enters source s 0 otherwise. Definition 1.1 (minimum maimal flow problem) A vector X is said to be a maimal flow if there is no y X such that y and y. We use X M to denote the set of maimal flows, i.e., X M = { X there is no y X such that y and y }. (1.2) 1

3 A minimum maimal flow problem, abbreviated to (mmf ), is defined as min d (mmf ) s.t. X M. The purpose of this paper is to propose an algorithm for (mmf ), which is based on the outer approimation method (OA method for short) for a D.C. optimization problem. Note that D.C. stands for difference of two conve sets (or functions), which will be defined in Section 3. Below is our motivation to consider (mmf ). When we attempt to solve a maimum flow problem on condition that we should not be allowed to decrease arc flows, we often fail to obtain the maimum flow and are obliged to put up with a maimal flow. Under this restricted controllability, the minimum flow value attained by a maimal flow, i.e., the optimal value of (mmf ), indicates how inefficiently the network can be utilized. Figure 1 highlights the difference between maimum flow and minimum maimal flow. For network (a), both are 3. On the other hand, for network (b), the minimum maimal flow value reduces to 2 while the maimum flow value remains 3. The minimum maimal flow value does not monotonically increase as the capacities grow. 2 v v 1 1 s 1 t s 2 t 1 2 v 2 network (a) 1 2 v 2 network (b) Figure 1: Maimum flow vs. minimum maimal flow Shi-Yamamoto [24] first raised (mmf ) and proposed an algorithm. Several algorithms for (mmf ) combining local search and global optimization technique have been proposed in e.g., Gotoh-Thoai-Yamamoto [15] and Shigeno-Takahashi-Yamamoto [25]. An approach of D.C. optimization is found in Muu-Shi [18]. The difficulty of (mmf ) is mainly due to the nonconveity of X M. Indeed, (mmf ) embraces the minimum maimal matching problem, which is N P-hard (see e.g., Garay-Johnson [14]). It is readily seen that (mmf ) is a special case of the optimization problem over the efficient set of a multi objective optimization, which was first studied by Philip [20]. Applying a well-known result of multi objective optimization, X M is characterized as follows: The point is in X M if and only if there eists λ R p++ such that is an optimal solution of ma λ (SC(λ)) s.t. X. Therefore we can easily obtain a point X M by solving (SC(λ)) for an arbitrarily chosen λ R p++. Furthermore, for a sufficiently large M > 0 the following set Λ can substitute 2

4 for R p++ above: Λ = { λ R p++ λ e, λ1 = M }. (1.3) Shigeno-Takahashi-Yamamoto [25] showed that n 2 suffices for M defining Λ of (1.3) for (mmf ). It is also known and easily seen by applying the parametric optimization technique for (SC(λ)) that X M is a connected union of several faces of X. About the optimization problem over the efficient set, the reader should refer to e.g., White [31], Sawaragi- Nakayama-Tanino [22], Steuer [26] and Yamamoto [33]. For solution methods, see Benson [4 6], Bolintineanu [7], Ecker-Song [11], Fülöp [13], Dauer-Fosnaugh [10], Thach-Konno- Yokota [27], Sayin [23], Phong-Tuyen [21], Thoai [28], Muu-Luc [17], An-Tao-Thoai [3] and An-Tao-Muu [1, 2]. For simplicity we assume throughout this paper that the given network satisfies the following three assumptions. Assumption 1.2 (i) Each capacity takes a positive integer, i.e., c h Z and c h > 0 for each h E. (ii) There is some point X such that 0. (iii) There is no t-s-path. Note that Assumption 1.2 (i) ensures the integrality of vertices of X as well as the optimal value of (mmf ). Note also that 0 X M by Assumption 1.2 (ii), and min{ d X } = 0 by Assumption 1.2 (iii). In the net section we first introduce a gap function. We then etend the domain of the gap function to R n and reformulate (mmf ). Section 3 is devoted to a review of the OA method for D.C. optimization problems. Based on this method, we propose an algorithm for (mmf ) in Section 4, in which we introduce an ε-optimal solution and investigate the proper range of the parameter ε for the optimality condition. To make the algorithm more efficient, we incorporate a local search technique. Finally, we show that the algorithm with the local search technique terminates after finitely many iterations. Further works will be described in the last section. Throughout this paper we use the following notations: R n denotes the set of n-dimensional column vectors. Let R n + = { R n 0 } and R n ++ = { R n > 0 }. Let R n denote the set of n-dimensional row vectors, R n+ and R n++ are defined in the similar way. We use e to denote the row vector of ones, 1 to denote the column vector of ones, and e i to denote the ith unit row or column vector of an appropriate dimension. Let I denote the identity matri of an appropriate size. We use a and A to denote the transposed vector of a and the transposed matri of A, respectively. For a set S, we denote the interior of S by int S, the closure of S by cl S, and the relative boundary of S by S. We use P V to denote the set of vertices of a polyhedron P. For two vectors v and w, let [v, w] denote the line segment with endpoints v and w, and let (v, w] = [v, w]\{v}. Also [v, w) and (v, w) are defined in the similar way. 2. Reformulation of (mmf ) by the Etended Gap Function It is known that the gap function g : R n [, + ] given by defines the set of maimal flows X M as g() = ma{ ey y X, y } e, (2.1) X M = { X g() 0 }. 3

5 Then we can rewrite (mmf ) as min d (mmf ) s.t. X, g() 0. The function g has some nice properties such piecewise linearity and concavity; for more information, see e.g., Benson [4] and White [32]. The domain of g, denoted by dom g, is the set { R n g() > }. When we apply the OA method to (mmf ), we need to evaluate g at points outside of X. Unless there is a point y X satisfying y v, g(v) takes, and hence no information is available about how away the point v is from the domain of g. Then we etend the gap function g to R n in this section. The etended gap function ḡ : R n R is defined as ḡ() = ma{ ey βt y X, y + t, t 0 } e, (2.2) where the n-dimensional row vector β will be specified later. Clearly ḡ is also a piecewise linear concave function. The following theorem in Yamamoto-Zenke [34] shows that ḡ is an etension of g. Theorem 2.1 (i) The domain of ḡ is R n for any β 0. (ii) If β ne then ḡ = g on the domain of g. Proof: See Appendi for the proof. By Theorem 2.1, fiing β = ne, we can replace the constraint g() 0 in (mmf ) with ḡ() 0 to obtain an equivalent formulation of (mmf ): min d (mmf ) s.t. X, ḡ() 0, which is equivalent to min d (mmf ) s.t. X\int Ḡ, where Ḡ = { R n ḡ() 0 }. (2.3) Note that Ḡ is a conve set since ḡ is a concave function. By the definition of ḡ, it is clear that ḡ() 0 for all X, i.e., X Ḡ. Since Assumption 1.2 (ii) implies 0 X\X M, we see that ḡ(0) = g(0) > 0, i.e., 0 int Ḡ. Additionally we have the following lemma. Lemma 2.2 ḡ() > 0 for every point in the relative interior of X. Proof: Let be a point in the relative interior of X, i.e., A = 0 and 0 < < c. Letting = (1+ε) for a sufficiently small ε > 0, we see that A = 0 and 0 c, i.e., X and. Therefore ḡ() = g() e( ) = εe > Outer Approimation Method for D.C. Optimization Problems A set S is said to be a D.C. set if there are two conve sets Q and R such that S = Q\R. The optimization problem on a D.C. set is called the D.C. optimization problem, which is 4

6 studied in e.g., Tuy [29, 30] and Horst-Tuy [16]. In this section we eplain the OA method for the canonical form D.C. optimization problem, abbreviated to (CDC), which is defined as min p (CDC) s.t. D, h() 0, where p R n is the cost vector, D R n is a nonempty compact conve set and h : R n R {+ } is a conve function. We assume that int { R n h() 0 } = { R n h() < 0 }. Defining the conve set H = { R n h() 0 }, (CDC) can be written as min p (CDC) s.t. D\int H, and hence (CDC) is a D.C. optimization problem. To make this problem simple we further assume that 0 D int H, and min{ p D } = 0. (3.1) Note that (CDC) reduces to (mmf ) when D = X, H = Ḡ and p = d Regularity and optimality condition Problem (CDC) is said to be regular when D\int H = cl (D\H). (3.2) Figure 2 shows an eample of (CDC) that is not regular, where cl (D\H). D\int H, while H D p Figure 2: The case where (CDC) is not regular The regularity assumption yields the optimality condition Theorem 3.1, which was given by Horst-Tuy [16]. To make this paper self-contained, we give the proof in Appendi. In the followings we denote D(η) = { D p η }, (3.3) for η R. Theorem 3.1 Let be a feasible solution of (CDC). D(p ) H. Proof: See Appendi for the proof. Then is an optimal solution if 5

7 3.2. OA method for (CDC) Let be an optimal solution of (CDC) and k D\int H be the incumbent at iteration k. In the OA method, we construct polytopes P 0, P 1,, P k, such that P 0 P 1 P k D(p ). If p k = 0, we have done by (3.1). In the case where p k > 0, we check the optimality condition D(p k ) H by evaluating h(v) at each verte v of P k. Namely, if h(v) 0 for each verte v of P k, meaning P k H, then k solves (CDC). Otherwise we construct P k+1 by adding some linear inequality to P k. Here we describe the OA method for (CDC). /** OA method for (CDC) **/ 0 (initialization) Find an initial feasible solution 0 of (CDC) and construct an initial polytope P 0 such that P 0 D(p 0 ). Compute the verte set PV 0 of P 0. Set k := 0. k (iteration k) Solve ma{ h(v) v PV k } to obtain vk. k1 (termination) If either p k = 0 or h(v k ) 0, meaning P k H, then stop. (The current incumbent k is an optimal solution of (CDC)). Otherwise, obtain the point k [0, v k ) H. k2 (cutting the polytope) If k D, set k+1 := k and P k+1 := P k { R n l() 0 } for some function l : R n R such that l(v k ) > 0 and l() 0 for all D(p ). If k D, set k+1 := k and P k+1 := P k { R n p p k+1 }. k3 Compute the verte set P k+1 V of P k+1. Set k := k + 1 and go to k. Remark 3.2 Note that adding a linear inequality to P k makes P k+1 and the verte set PV k of P k is at hand. Subroutines for computing the verte set P k+1 V from the knowledge of PV k are provided in e.g., Chen-Hansen-Jaumard [8], Subsection 7.4 of Padberg [19] and Chapter 18 of Chvátal [9]. Due to the possible degeneracy of P k, a sophisticated implementation should be needed e.g., Fukuda-Prodon [12]. 4. Outer Approimation Method for (mmf ) By Assumption 1.2 (ii)-(iii), we have 0 X int Ḡ, and min{ d X } = 0, (4.1) which correspond to (3.1). Then we can apply the OA method to (mmf ) if the regularity condition is met Regularity and optimality condition Unfortunately, the problem (mmf ) is not regular, hence we introduce a positive tolerance ε and consider, instead of (mmf ), min d (mmf ε ) s.t. X\int Ḡε, where Ḡ ε = { R n ḡ() ε }. (4.2) We call an optimal solution of (mmf ε ) an ε-optimal solution of (mmf ). First we show that any positive ε ensures the regularity of (mmf ε ). 6

8 Theorem 4.1 The problem (mmf ε ) is regular for any ε > 0. Proof: We show that X\int Ḡε = cl(x\ḡε), (4.3) holds for any ε > 0. ( ) Since X\int Ḡε is closed and X\int Ḡε X\Ḡε, we have X\int Ḡε = cl(x\int Ḡε) cl(x\ḡε). ( ) Let be an arbitrary point of X\int Ḡε and let N δ () denote its δ-neighborhood, i.e., N δ () = { R n < δ }. We show that there is always a point, say in N δ () (X\Ḡε). If ḡ() > ε then there eists γ > 0 such that ḡ( ) > ε for any point N γ () by the continuity of ḡ. This implies N γ () Ḡε, and hence int Ḡε. Therefore the assumption X\int Ḡε implies that X and ḡ() ε. By Theorem 2.1, we have ḡ() = g(). When ḡ() < ε, take as. Clearly = Ḡε and = N δ (), and we have done. When g() = ḡ() = ε, there is an optimal solution y of ma{ ey y X, y } such that e(y ) = ε, and hence y. Take λ such that 0 < λ < min{ 1, δ/ y } and let = λy + (1 λ). Since = λ y < δ, we see N δ (). Also we see that X by the conveity of X, and hence g( ) = ḡ( ) by applying Theorem 2.1 again. Since and, we have ḡ( ) = g( ) = ma{ ey y X, y } e < ma{ ey y X, y } e = e(y ) = ε. Therefore we see that Ḡε. This completes the proof. Net we discuss the upper bound of ε, which will be crucial for the convergence of the algorithm. Lemma 4.2 If ε (0, 1) then 0 int Ḡε, and (0, v) Ḡε for any point v such that ḡ(v) 0. Proof: We have ḡ(0) > 0 since 0 int Ḡ. Note that ḡ(0), which coincides with g(0), takes an integer value by the integrality property of X, and hence ḡ(0) 1. Then we have ḡ(0) > ε, i.e., 0 int Ḡε for any ε (0, 1). The continuity of ḡ ensures the last assertion. For the following lemma, we use δ s to denote the number of arcs leaving node s, i.e., δ s = { i d i = +1 }. (4.4) Lemma 4.3 Let and ε be an optimal solution and an ε-optimal solution of (mmf ), respectively. Then 0 d d ε εδ s. Proof: Since X and ḡ( ) 0, is a feasible solution of (mmf ε ), and hence d ε d. Let y ε be an optimal solution of ma{ ey y X, y ε }. Clearly y ε X M, i.e., y ε is a feasible solution of (mmf ), and hence d dy ε. We see that yεi εi ε for each i = 1,..., n, since y ε ε 0 and e(y ε ε) ε. That implies d(y ε ε) ε { i d i = +1 } = εδ s, hence d ε d dy ε d ε + εδ s. 7

9 Then d ε coin- Theorem 4.4 Let ε be an ε-optimal solution for some ε (0, 1/δ s ). cides with the optimal value of (mmf ). Proof: From Lemma 4.3 we see that 0 d d ε < 1. This inequality and the integrality of d gives the assertion. In the sequel we choose ε from the open interval (0, 1/δ s ). Note that ḡ( ε) ε holds for an ε-optimal solution ε of (mmf ). Therefore ḡ() 0 for any accumulation point of { ε} ε 0+. This observation leads to the following corollary. Corollary 4.5 Let { ε} ε 0+ be a sequence of ε-optimal solutions of (mmf ) for ε converging to 0 from above. Then any accumulation point of { ε} ε 0+ is an optimal solution of (mmf ). As seen in Theorem 3.1, letting X(η) = { X d η }, (4.5) for η R, the optimality condition of the OA method is X(d ε ) Ḡε for some ε X\int Ḡε. We can further rela this condition. Theorem 4.6 Let ε X\int Ḡε for some ε (0, 1/δ s ). If X( d ε 1 ) Ḡε ε > 0 then d ε coincides with the optimal value of (mmf ). for some Proof: Let and ε be an optimal solution and an ε-optimal solution of (mmf ), respectively. Since ε is a feasible solution of (mmf ε ), we have d ε d ε. It is also clear that d ε d. If d < d ε then we have X( d ε 1 ) Ḡε since d is integer, and hence ḡ( ) ε > 0, which contradicts that ḡ( ) = 0. Then we have d ε d. Hence by Lemma 4.3 we obtain d ε d ε d d ε + εδ s < d ε + 1. This completes the proof. We construct a polytope P satisfying X( d ε 1 ) P for some ε X\int Ḡε. Let v be a verte minimizing ḡ(v) over P V and ε = ḡ(v ). For any P we have ḡ() ḡ(v ), i.e., 0 ḡ() ḡ(v ) = ḡ() ε, and hence P Ḡε. This implies that X( d ε 1 ) Ḡε. Therefore if ε > 0 then the optimal value of (mmf ) is obtained by Theorem Local search For v X M X V, we define the set of efficient vertices linked to v by an edge as N M (v) = { v X M X V [v, v ] is an edge of X } (4.6) = { v X V [v, v ] is an edge of X and g(v ) 0 }. Whenever we find a feasible solution w X M, we apply the Local Search procedure starting with w (LS(w) for short) for further improvement. The procedure is described as follows. /** LS(w) procedure **/ 0 (initialization) If w X V then find the face F of X containing w in its relative interior and solve min{ d F } to obtain a verte v 0 X M X V, otherwise set v 0 := w. Set k := 0. 8

10 k (iteration k) Solve min{ dv v N M (v k ) } to obtain a solution v. If dv dv k then stop, v k is the local optimal verte of (mmf ). Otherwise set v k+1 := v, k := k+1 and go to k. Remark 4.7 If w X M, the face F of X containing w in its relative interior is contained in X M since X M is a connected union of several faces of X OA method for (mmf ) We describe the OA method for (mmf ) as follows. /** OA method for (mmf ) **/ 0 (initialization) Find an initial feasible verte w 0 X M X V of (mmf ). If N M (w 0 ) = then stop. (w 0 is a unique feasible solution of (mmf )). Otherwise, apply the LS(w 0 ) procedure to obtain a local optimal verte 0 X M X V. Solve ζ := ma{ e X, d d 0 1 } and construct an initial polytope P 0 X(d 0 1) by setting P 0 := { R n e ζ, d d 0 1, 0 }. Compute the verte set PV 0 of P 0. Set k := 0. k (iteration k) Solve min{ ḡ(v) v PV k } to obtain a verte vk. k1 (termination) If either d k = 0 or ḡ(v k ) > 0 then stop. (The optimal value of (mmf ) is d k ). Otherwise, obtain the point k ε (0, v k ) Ḡε. (Note that Lemma 4.2 ensures that (0, v k ) Ḡε ). k2 (update) If k ε X, obtain the point k (0, v k ] Ḡ. k2.1 If k X, meaning k X M, then obtain the local optimal verte z k X M X V by applying the LS( k ) procedure, and further obtain the point z k ε (0, z k ) Ḡε. Set k+1 := argmin{ dz k ε, d k ε } and P k+1 := P k { R n d d k+1 1 }. k2.2 If k X, meaning k X M, then set k+1 := k ε and P k+1 := P k { R n d d k+1 1, l() 0 }. k3 If k ε X then set k+1 := k and P k+1 := P k { R n l() 0 }. k4 Compute the verte set P k+1 V of P k+1. Set k := k + 1 and go to k. Remark 4.8 The function l : R n R in Step k2.2 and Step k3 is given by one of the inequalities ±A 0 and c not satisfied by the point v k, i.e., (i) l() = e j c j for some j {1,..., n} such that vj k > c j, or (ii) l() = sgn(a i v k )a i for some i {1,..., m} such that a i v k 0, where a i is the ith row of A, and { +1 when α > 0 sgn(α) = 1 when α < 0. Lemma 4.9 Let z k be the local optimal verte obtained by applying the LS( k ) procedure starting with k in Step k2.1 at iteration k, and suppose dz k > 0. Then dz k < dz k for iteration k such that k > k. Proof: It suffices to show that dz k+1 < dz k. By the construction of P k+1 we have P k+1 { d d k+1 1 }. Since k+1 (0, v k+1 ] P k+1 and z k+1 is obtained by LS( k+1 ), we have dz k+1 d k+1 d k

11 Since we assume that dz k > 0, we have 0 < dz k ε < dz k by the choice of z k ε. Therefore in Step k2.1 d k+1 1 < d k+1 = min{d k ε, dz k ε} < dz k. Combining the two inequalities yields the desired result. Theorem 4.10 The OA method for (mmf ) works correctly and terminates after finitely many iterations. Proof: (correctness) If N M (w 0 ) = at the initialization step, we can conclude from the connectedness of X M that w 0 is a unique feasible solution of (mmf ) and hence solves the problem. When the algorithm terminates in Step k1, the optimal value of (mmf ) is equal either to zero by Assumption 1.2 (iii), or to d k by Theorem 4.6. So the optimal value is obtained whenever the algorithm terminates. We suppose that the algorithm has not yet terminated at iteration k, i.e., d k > 0 and ḡ(v k ) 0, and show that each step of the algorithm can be done. Lemma 4.2 ensures that there are points k ε (0, v k ) Ḡε and z k ε (0, z k ) Ḡε, in Step k1 and Step k2.1, respectively. Since 0 int Ḡ and vk int Ḡ, there also eists a point k (0, v k ] Ḡ. When k ε X, clearly v k X, and hence the function l : R n R of Remark 4.8 can be found in Step k3. To show that the function l : R n R is found in Step k2.2 we have only to show that v k X. Suppose the contrary, i.e., v k X. By the assumption that ḡ(v k ) 0 and the fact that ḡ() 0 for all X, we have ḡ(v k ) = 0, i.e., v k Ḡ, and hence v k X\int Ḡ = X M. This implies k = v k X M by the choice of k, which contradicts that we are currently at iteration k2.2. Therefore we have seen that v k X in Step k2.2. (finiteness) Suppose that the polytope P ν at iteration ν meets the condition P ν X and P ν X M =, (4.7) after updated either in Step k2 or in Step k3, and consider the net iteration. Since v ν is chosen from P ν, we have v ν X\X M and consequently ḡ(v ν ) > 0. Then the algorithm stops at Step k1. Therefore we have only to prove that (4.7) holds within a finite number of iterations. Note first that both Step k2.2 and Step k3 are done at most a finite number of times. Indeed, the polytope, say P k, when 2m + n cuts l() 0 have been added to the initial polytope P 0, is contained in X. Therefore v k as well as k ε lies in X, and hence we obtain that k = v k X M in the same way as in the former part of this proof. Therefore we come to neither Step k2.2 nor Step k3 after iteration k. Namely, Step k2.1 followed by Step k4 repeats itself after iteration k. For iteration k with k k + 1, we have k X M. We then locate z k X M X V by applying the LS( k ) procedure and obtain a point z k ε (0, z k ) Ḡε. If dz k = 0 for some k k + 1 then we set k+1 := z k ε since dz k ε = dz k = 0 d k ε. Then the incumbent value d k+1 becomes zero, and hence the algorithm stops in Step k1 at the net iteration. If dz k > 0 for all k with k k + 1, we see that dz k+1 < dz k for all k k + 1 by Lemma 4.9. Since X M X V is finite, we eventually obtain a point z ν 1 X M X V such that dz ν 1 dz for all z X M X V. Also we have dz ν 1 ε < dz ν 1 by the choice of z ν 1 ε. The polytope P ν is then defined as P ν := P ν 1 { d d ν 1 }, where ν satisfies that d ν = min{ d ν 1 ε, dz ν 1 ε } < dz ν 1. This means that P ν (X M X V ) =. Since X M is a connected union of several faces of X, we see that dz ν 1 d for all X M. Therefore we conclude that P ν X M =. 10

12 We illustrate the OA method for (mmf ) in Figure 3, in which we use a two-dimensional general polyhedron X = { R 2 B b, 0 } for B R m 2 and b R m because the set of feasible flows X = { R n A = 0, 0 c } is unsuitable for illustration. We obtain a local optimal verte 0 X M X V and set up an initial polytope P 0 (See (a)). It is easy to enumerate all vertices of P 0 because this polytope is simply given by P 0 := { R n e ζ, d d 0 1, 0 }. We obtain a point v 0 minimizing ḡ(v) over PV 0, and a point 0 ε (0, v 0 ) Ḡε (See (b)). We see that 0 ε X, and hence set 1 := 0 and cut off v 0 from P 0 (See (c)). Using PV 0, we compute P V 1. In the net iteration, we obtain points v 1, 1 ε and 1. Since 1 X M, we apply the LS( 1 ) procedure to obtain a point z 1, and obtain a point z 1 ε (0, z 1 ) Ḡε (See (d)). We find a point z 1 ε X\int Ḡε such that dz 1 ε < d 1 ε. We then set 2 := z 1 ε and construct P 2 by adding the cut d d 2 1 to P 1 (See (e)). Because ḡ(v) > 0 for all vertices v of P 2 (See (f)), we terminate at the net iteration with the optimal value d Further Works The OA method provides the optimal value but may fail to provide an optimal solution of (mmf ). Finding an optimal solution is still a hard task even when its value is known, however, the following lemma affords a clue to the way of finding an optimal solution. Lemma 5.1 Let ε (0, 1), ε be an ε-optimal solution of (mmf ) and ε = { ξ R n Aξ = 0, ξ 0, eξ ε }. (5.1) If ε + ξ is an integer vector for some ξ ε then ε + ξ is an optimal solution of (mmf ). Proof: (feasibility) Let = ε + ξ and y be an optimal solution of ma{ ey y X, y }. Note that and also since X { y y } inherits the integrality property of X. Suppose we have the inequality e is integer, (5.2) e ε e ey, (5.3) ey is integer, (5.4) ey < e ε + 1. (5.5) Then by (5.3) and (5.5) together with the integrality of e and ey we see that e = ey. Hence ḡ( ) = ey e = 0, meaning that X M. The inequality (5.5) is seen as follows. Let y ε be an optimal solution of ma{ ey y X, y ε }, and let ξ = y ε ε. We see that Aξ = Ay ε A ε = 0, ξ 0 and eξ = e(y ε ε) = g( ε) ε, and hence ξ ε. Then ey ε = e( ε + ξ ) e ε + ε < e ε +1. The point y is a feasible solution of ma{ ey y X, y ε }, since y X and y = ε + ξ ε. Then we see that e(y ε y ) 0, and hence ey ey ε < e ε + 1. (optimality) We show that solves (mmf ). Clearly, d ξ e ξ since d e and ξ 0. For any v X M X V, we see that g(v) ε, and v is an integer vector by the integrality property of X. Since ε = ξ is an optimal solution of (mmf ε ), we have d ε d for 11

13 v 0 P 0 P 0 0 ε Ḡ ε Ḡ ε X 0 d X 0 d (a) polytope P 0 (b) obtaining v 0 and 0 ε v 0 { l() = 0 } LS( 1 ) 0 ε P 0 z 1 v 1 1 Ḡ ε Ḡ ε z 1 ε 1 ε X 1 d X P 1 1 d (c) cutting off v 0 from P 0 (d) obtaining v 1, 1 ε and z 1 ε { d = d 2 1 } v 1 v 3 ḡ(v 3 ) > 0 Ḡ ε 2 Ḡ ε P 2 2 X P 1 d X (e) cutting off v 1 from P 1 (f) termination Figure 3: An eample of the OA method for (mmf ) 12

14 all X such that g() ε, and hence d ε dv for all v X M X V. Then we see that d = d ε + d ξ dv + e ξ < dv + 1. Since both and v are integer vectors, we have d dv for all v X M X V. Since the dimension of X is n m, it would be desirable to reduce the number of variables that we have to handle in the algorithm. Yamamoto-Zenke eplains an idea in [34], however, with the proviso that it does not work generally. Computational eperiment should be carried out to improve the efficiency of the algorithm in this paper. Appendi Proof of Theorem 2.1 Proof: (i) The etended gap function ḡ() of (2.2) is given by the optimal value of ma ey e βt (P G ()) y,t s.t. Ay = 0, 0 y c, y + t, t 0, whose dual problem is min (D G ()) π,α,β s.t. αc β e (π, α, β) Ω, where Ω = { (π, α, β) R m+2n πa + α β e, α 0, 0 β β }. For any R n, (D G ()) is feasible, e.g., take π = β = 0 and α e, and has the finite optimal value. By the duality theorem of linear programming, for any R n, (P G ()) also has the finite optimal value, and hence ḡ() is finite for any R n. (ii) Let be a point in the domain of g. By the similar observation in (i), the gap function g() of (2.1) is given by the optimal value of min αc β e (D G ()) π,α,β s.t. (π, α, β) Ω, where Ω = { (π, α, β) R m+2n πa + α β e, α, β 0 }. If β is so large that every verte v of Ω satisfies v β then we have ḡ() = g() by the theory of linear programming. Replacing π by π 1 π 2 with π 1, π 2 0 and introducing a slack variable vector γ 0, Ω is rewritten as (π 1 ) (π 1 ) (π 1 ) (π 2 ) ( Ω = α A A I I I ) (π 2 ) α β β = 1, (π 2 ) α β 0. γ γ γ 13

15 Let v be a verte of Ω. Then it is a basic solution of the system defining Ω, i.e., v = (w B, w N ) = (B 1 1, 0) for some nonsingular n n submatri B of (A A I I I). Since the incidence matri A is totally unimodular, so is (A A I I I). Therefore the matri B 1 is composed of 1, 0 and +1, and hence B 1 1 n1. This completes the proof. Proof of Theorem 3.1 Proof: Suppose that D\intH is not an optimal solution of (CDC), i.e., there eists y D\intH such that py < p. Clearly, y D(p ) and h(y) 0. If h(y) > 0 then y is not contained in H, and hence y D(p )\H. By the regularity assumption, if h(y) = 0, i.e., y H then we can take y N δ (y) D such that py < p and h(y ) > 0 for a sufficiently small δ > 0, where N δ (y) = { y R n y y < δ }, and hence we see that y D(p )\H. Acknowledgement This research is supported in part by the Grant-in-Aid for Scientific Research (B)(2) of the Ministry of Education, Culture, Sports, Science and Technology of Japan. References [1] An, L. T. H., Tao, P. D. and Muu, L. D.: Numerical Solution for Optimization over the Efficient Set by D.C. Optimization Algorithms, Oper. Res. Lett., Vol. 19 (1996), [2] An, L. T. H., Tao, P. D. and Muu, L. D.: Simplicially-constrained DC optimization over the efficient and weakly sets, J. Optim. Theory Appl., Vol. 117 (2003), [3] An, L. T. H., Tao, P. D. and Thoai, N. V.: Combination between global and local methods for solving an optimization problem over the efficient set, European J. Oper. Res., Vol. 142 (2002), [4] Benson, H. P.: Optimization over the Efficient Set, J. Math. Anal. Appl., Vol. 98 (1984), [5] Benson, H. P.: An All-Linear Programming Relaation Algorithm for Optimizing over the Efficient Set, J. Global Optim., Vol. 1 (1991), [6] Benson, H. P.: A Finite, Nonadjacent Etreme-Point Search Algorithm for Optimization over the Efficient Set, J. Optim. Theory Appl., Vol. 73 (1992), [7] Bolintineanu, S.: Minimization of a Quasi-Concave Function over an Efficient Set, Math. Program., Vol. 61 (1993), [8] Chen, P. C., Hansen, P. and Jaumard, B.: On-Line and Off-Line Verte Enumeration by Adjacency Lists, Oper. Res. Lett., Vol. 10 (1991), [9] Chvátal, V.: Linear programming, W.H. Freeman and Campany, New York, [10] Dauer, J. P. and Fosnaugh, T. A.: Optimization over the Efficient Set, J. Global Optim., Vol. 7 (1995), [11] Ecker, J. G. and Song, J. H.: Optimizing a Linear Function over an Efficient Set, J. Optim. Theory Appl., Vol. 83 (1994), [12] Fukuda, K. and Prodon, A.: Double Description Method Revisited, in Combinatorics and Computer Science,

16 [13] Fülöp, J.: A Cutting Plane Algorithm for Linear Optimization over the Efficient Set, Lecture Notes 405, Economics and Mathematical Systems, Springer-Verlag, Heidelberg, [14] Garay, M. R. and Johnson, D. S.: Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, [15] Gotoh, J., Thoai, N. V. and Yamamoto, Y.: Global Optimization Method for Solving the Minimum Maimal Flow Problem, Optim. Methods Softw., Vol. 18 (2003), [16] Horst, R. and Tuy, H.: Global Optimization, Springer-Verlag, Berlin, third and revised and enlarged edition, [17] Muu, L. D. and Luc, L. T.: Global optimization approach to optimization over the efficient set, Lecture Notes 452, Economics and Mathematical System, Springer-Verlag, Heidelberg, [18] Muu, L. D. and Shi, J.: D.C. Optimization Methods for Solving Minimum Maimal Network Flow Problem, Lecture notes 1349, captivation of conveity ; fascination of nonconveity, Research Institute for Mathematical Sciences, Kyoto University, [19] Padberg, M.: Linear programming, Algorithms and Combinatorics 12, Springer-Verlag, Heidelberg, [20] Philip, J.: Algorithms for the Vector Maimization Problem, Math. Program., Vol. 2 (1972), [21] Phong, T. Q. and Tuyen, H. Q.: Bisection Search Algorithm for Optimizing over the Efficient Set, Vietnam J. Math., Vol. 28 (2000), [22] Sawaragi, Y., Nakayama, H. and Tanino, T.: Theory of Multiobjective Optimization, Monographs and Tetbooks 176, Academic Press, Orlando, [23] Sayin, S.: Optimizing over the Efficient Set Using a Top-Down Search of Faces, Oper. Res., Vol. 48 (2000), [24] Shi, J. and Yamamoto, Y.: A global optimization method for minimum maimal flow problem, Acta Math. Vietnam., Vol. 22 (1997), [25] Shigeno, M., Takahashi, I. and Yamamoto, Y.: Minimum Maimal Flow Problem - An Optimization over the Efficient Set -, J. Global Optim., Vol. 25 (2003), [26] Steuer, R. E.: Multiple Criteria Optimization: Theory, Computation, and Application, John Wiley & Sons, New York, [27] Thach, P. T., Konno, H. and Yokota, D.: Dual Approach to Minimization on the Set of Pareto-Optimal Solutions, J. Optim. Theory Appl., Vol. 88 (1996), [28] Thoai, N. V.: Conical Algorithm in Global Optimization for Optimizing over Efficient Sets, J. Global Optim., Vol. 18 (2000), [29] Tuy, H.: D.C. Optimization: Theory, Methods and Algorithms, in Horst, R. and Pardalos, P. M. eds., Handbook of Global Optimization, Kluwer Academic Publishers, Netherlands, [30] Tuy, H.: Conve Analysis and Global Optimization, Kluwer, Dordrecht, [31] White, D. J.: Optimality and Efficiency, John Wiley & Sons, Chichester, [32] White, D. J.: The Maimization of a Function over the Efficient Set via a Penalty Function Approach, European J. Oper. Res., Vol. 94 (1996), [33] Yamamoto, Y.: Optimization over the Efficient Set: Overview, J. Global Optim., Vol. 22 (2002), [34] Yamamoto, Y. and Zenke, D.: Cut and Split Method for the Minimum Maimal Flow Problem, to appear in Pacific Journal of Optimization. 15