Approximating the Nondominated Set of an MOLP by Approximately Solving its Dual Problem

Transcription

1 Approximating the Nondominated Set of an MOLP by Approximately Solving its Dual Problem Lizhen Shao Department of Engineering Science The University of Auckland, New Zealand Matthias Ehrgott Department of Engineering Science The University of Auckland, New Zealand m.ehrgott and Laboratoire d Informatique de Nantes Atlantique Université de Nantes, France matthias.ehrgott@univ-nantes.fr August, 7 Abstract The geometric duality theory of Heyde and Löhne (6) defines a dual to a multiple objective linear programme (MOLP). In objective space, the primal problem can be solved by Benson s outer approximation method (Benson, 998a,b) while the dual problem can be solved by a dual variant of Benson s algorithm (Ehrgott et al., 7). Duality theory then assures that it is possible to find the nondominated set of the primal MOLP by solving its dual. In this paper, we propose an algorithm to solve the dual MOLP approximately but within specified tolerance. This approximate solution set can be used to calculate an approximation of the nondominated set of the primal. We show that this set is an ε-nondominated set of the original primal MOLP and provide numerical evidence that this approach can be faster than solving the primal MOLP approximately.

2 Introduction The goal of multiple objective optimization is to simultaneously minimize p noncomparable objectives. The objectives are usually conflicting in nature, a feasible solution optimizing all the objectives simultaneously does not exist. Therefore, the goal of multiobjective optimization is to obtain the nondominated set. For a multiple objective programme (MOP), a nondominated point in objective space corresponds to an efficient solution in decision space and an efficient solution is defined as a solution for which an improvement in one objective will always lead to a deterioration in at least one of the other objectives. The set of all nondominated points forms the nondominated set in objective space. It conveys the trade-off information to a decision maker (DM) we assume to prefer less to more in each objective. In this paper, we are concerned with multiobjective linear programming problems. Researchers have developed a variety of methods for generating the efficient set or the nondominated set such as multiobjective simplex methods, interior point methods, and objective space methods. Multiobjective simplex methods and interior point methods work in variable space to find the efficient set, see the references in Ehrgott and Wiecek (). Since the number of objectives of and MOLP is often much smaller than the number of variables and typically many efficient solutions in decision space are mapped to a single nondominated point in objective space, Benson (998a) argues that generating the nondominated set should require less computation than generating the efficient set. Moreover, it is reasonable to assume that a decision maker will choose a solution based on the objective values rather than variable values. Therefore, finding the nondominated set in objective space instead of the efficient set in decision space is more important for the DM. Benson has proposed an outer approximation method (Benson, 998a) to find the extended feasible set in objective space of an MOLP. Furthermore, for an MOLP, geometric duality theory (Heyde and Löhne, 6) specifies a dual MOLP and establishes that the extended primal and dual objective set are related to each other. Thus if the extended objective set of the dual MOLP is known, then the extended objective set of the primal MOLP can be obtained, and vice versa. Based on the geometric duality theory, Ehrgott et al. (7) develop a dual variant of Benson s algorithm to obtain the feasible set in objective space of the dual MOLP. Although it is theoretically possible to identify the complete nondominated set with the methods mentioned above, finding an exact description of this set often turns out to be practically impossible or at least computationally too expensive (see the examples in Shao and Ehrgott (6)). Therefore, many researchers focus on approximating the nondominated set, see Ruzika and Wiecek () for a survey. In the literature, the concept of ε-nondominated points has been suggested as a means to account for modeling limitations or computational inaccuracies. In Shao and Ehrgott (6), we have proposed an approximation version of Benson s algorithm to sandwich the extended feasible set of an MOLP with an outer approximation and an inner approximation. The nondominated set of the inner approximation is proved to be a set of ε-nondominated points. In this paper, we propose to solve the dual MOLP approximately, then to calculate a corresponding polyhedral set in objective space of the primal MOLP using a coupling function. The nondominated subset of this polyhedral set can be proved to be a set of ε- nondominated points of the original primal MOLP. Finally, we apply this approximate method to solve the beam intensity optimization problem of radiotherapy treatment planning. Three clinical cases were used and the results are compared with those obtained by the approximation version of Benson s algorithm which directly approximates the truncated extended feasible set of the primal MOLP, an algorithm developed in Shao and Ehrgott (6).

3 Preliminaries In this paper we use the notation y to indicate y but y for y, R p whereas y < means yk <y k for all k =,...,p. The k-th unit vector in R p is denoted by e k and a vector of all ones is denoted by e. Given a mapping f : R n R p and a subset X R n we write f(x ):={f(x) :x X}. Let A R p. We denote the interior, and relative interior of A by int A, and ri A. Let C R p be a closed convex cone. An element y Ais called C-minimal if ({y} C\{}) A = and C-maximal if ({y} + C\{}) A =. A point y A is called weakly C-minimal (weakly C-maximal) if ({y} ric) A = (({y} +ric) A= ). We set wmin C A := {y A:({y} ri C) A= } and wmax C A := wmin ( C) A. In this paper we consider two special ordering cones, namely C = R p = {x Rp : x k,k =,...,p} and C = K := R e p = {y R p : y = = y p =,y p }. For the choice C = R p the set of weakly Rp -minimal elements of A (also called the set of weakly nondominated points of A) isgivenby wmin R p A := {y A:({y} int R p ) A= }. In case of C = K the set of K-maximal elements of A is given by max K A := {y A:({y} + K\{}) A= }. Note that ri K = K\{} so that weakly K-maximal and K-maximal elements of A coincide. Let us recall some facts concerning the facial structure of polyhedral sets (Webster, 99). Let A R p be a convex set. A convex subset F Ais called a face of A if for all y, Aand α (, ) such that αy +( α) F it holds that y, F. AfaceF of A is called proper if F A. Apointy Ais called an extreme point of A if {y} is a face of A. The extreme points are also called vertices. The set of all vertices of a polyhedron A is denoted by vert A. A polyhedral convex set A is defined by {y R p : By β}, whereb R m p and β R m. A polyhedral set A has a finite number of faces. A subset F of A is a face if and only if there are λ R p and γ R such that A { y R p : λ T y γ } and F = { y R p : λ T y = γ } A. Moreover, F is a proper face if and only if H := { y R p : λ T y = γ } is a supporting hyperplane to A with F = A H. We call hyperplane H = { y R p : λ T y = γ } supporting if λ T y γ for all y Aand there is some y Asuch that λ T y = γ. The proper (r )-dimensional faces of an r-dimensional polyhedral set A are called facets of A. A recession direction of A is a vector d R p such that y + αd Afor some y Aand all α. The recession cone (or asymptotic cone) A of A is the set of all recession directions A := {d R p : y + αd Afor some y Afor all α }. A recession direction d is called extreme if there are no recession directions d,d withd αd for all α> such that d = (d + d ).

4 A polyhedral convex set A can be represented by both a finite set of inequalities and the set of all extreme points and extreme directions of A (Rockafellar, 97, Theorem 8.). Let E = {y,...,y r,d,...,d t } be the set of all extreme points and extreme directions of A then r t r A = y Rp : y = α i y i + ν j d j with α i,ν j, and α i =. i= j= MOLP Duality Theory In this paper we consider multiple objective linear programming problems of the form min{px : x X}. () We assume that X in () is a nonempty feasible set X in decision space R n defined by X = {x R n : Ax b}. We have A R m n, b R m and P R p n. The feasible set Y in objective space R p is defined by i= Y = {Px: x X}. () It is well known that the image Y of a nonempty, polyhedron X under a linear map P is also a nonempty polyhedron of dimension dimy p. Definition. Afeasiblesolutionˆx X is an efficient solution of problem () if there exists no x X such that Px P ˆx. The set of all efficient solutions of problem () will be denoted by X E and called the efficient set in decision space. Correspondingly, ŷ = P ˆx is called a nondominated point and Y N = {Px : x X E } is the nondominated set in objective space of problem (). Definition. Afeasiblesolutionˆx X is called weakly efficient if there is no x X such that Px < Pˆx. The set of all weakly efficient solutions of problem () will be denoted by X WE and called the weakly efficient set in decision space. Correspondingly, the point ŷ = P ˆx is called weakly nondominated point and Y WN = {Px : x X WE } is the weakly nondominated set in objective space of problem (). The primal and dual pair of this MOLP formulated in Heyde and Löhne (6) are (P) wmin R p P (X ), X := {x R n : Ax b} (D) max K D(U), U := { (u, λ) R m R p :(u, λ),a T u = P T λ, e T λ = }, where K := {y R p : y = =... = y p =,y p } as defined before and D : R m+p R p is given by D(u, λ) := ( λ,..., λ p,b T u ) ( )( ) T Ip u = b T. λ The primal problem (P) consists in finding the weakly nondominated points of P (X ), the dual problem consists in finding the K-maximal elements of D(U). We use the extended polyhedral image sets P := P (X )+R p of problem (P) and D := D(U) Kof problem (D). It is known that the R p -minimal (nondominated) points of P and P (X )aswellasthek-maximal elements of D and D(U) coincide. Geometric duality theory (Heyde and Löhne, 6) states that there is an inclusion reversing one-to-one map Ψ between the set of all proper K-maximal faces

5 of D and the set of all proper R p -minimal faces of P. The map Ψ is based on the two set-valued maps, H : R p R p, H(v) :={y R p : ϕ(y, v) =}, H : R p R p, H (y) :={v R p : ϕ(y, v) =}. where ϕ(y, v) = ( p i= y iv i + y p ) p i= v i v p is the coupling function. Let then λ(v) := ( ) p T v,...,v p, v i i= λ (y) := ( y y p,...,y p y p, ) T and H(v) = { y R p : λ(v) T } y = v p and H (y) = { v R p : λ (y) T } v = y p. Theorem. (Heyde and Löhne (6)) Let F be a proper weakly nondominated face of P and F be a proper K-maximal face of D, then Ψ(F ):= v F H(v) P. Ψ is an inclusion reversing one-to-one map between the set of all proper K-maximal faces of D and the set of all proper weakly nondominated faces of P and the inverse map is given by Ψ (F) = H (y) D. y F Moreover, for every proper K-maximal face F of D it holds dim F +dimψ(f )= p. Geometric duality theory gives us the following results. Corollar. (Heyde and Löhne (6)). If v is a K-maximal vertex of D, thenh(v) P is a weakly nondominated (p )-dimensional facet of P, and vice versa.. If F is a weakly nondominated (p )-dimensional facet of P, thereissome uniquely defined point v R p such that F = H(v) P.. If y is a weakly nondominated vertex of P, thenh (y) D is a K-maximal (p )-dimensional facet of D, and vice versa.. If F is a K-maximal (p )-dimensional facet of D, there is some uniquely defined point y R p such that F = H (y) D. The following pair of dual linear programming problems are useful for the proof of the geometric duality theory and they also play a key role in the algorithm of solving the dual problem to be introduced in the next section. (P (v)) min x X λ(v)t Px, X := {x R n : Ax b} (D (v)) max u T (v) bt u, T (v) := { u R m : u,a T u = P T λ(v) }

6 Solving the Primal and Dual MOLP Assuming X is bounded, Benson proposes an outer approximation algorithm to solve MOLP () in objective space (Benson, 998a,b). Benson defines Y = {y R p : Px y ŷ for some x X}, () where ŷ R p is chosen to satisfy ŷ>y AI. The vector y AI R p is called the anti ideal point for the problem () and is defined as y AI k =max{y k : y Y}. () According to the following theorem, instead of directly finding Y, Benson s outer approximation algorithm is dedicated to finding Y. Theorem. (Benson (998a,b)) The set Y R p is a nonempty, bounded polyhedron of dimension p and Y N = Y N. Benson s outer approximation algorithm can be illustrated as follows. First, a cover that contains Y is constructed and an interior point of Y is found. Then, for each vertex of the cover, the algorithm checks whether it is in Y or not. If not, the vertex and the interior point are connected to form a line segment and the unique boundary point of Y on the line segment is found. A cut (supporting hyperplane) of Y containing the boundary point is constructed and the cover is updated. The procedure repeats until all the vertices of the cover are in Y. As a result, when the algorithm terminates, the cover is actually equal to Y. In addition, to improve computation time, some improvements have been proposed in Shao and Ehrgott (6). In Ehrgott et al. (7), we have extended Benson s algorithm to MOLPs which are R p -bounded. The algorithm finds P instead of Y. Notice that P is an unbounded extension of Y as P = Y + R p and Y =(ŷ R p ) P. P, Y and Y have the same nondominated set. In the same paper we also propose a dual variant of Benson s algorithm to solve the dual MOLP. We show the dual variant of Benson s algorithm and how to obtain the nondominated facets of P from D. The algorithm is very similar to Benson s algorithm. Instead of Y, it finds D. In the course of the algorithm, supporting hyperplanes of D are constructed. The following proposition is the basis for finding supporting hyperplanes. Proposition. (Ehrgott et al. (7)) Let v max K D, then for every solution x of (P ( v)), H (P x) is a supporting hyperplane of D with v H (P x). The algorithm is shown in Algorithm. Algorithm (Dual variant of Benson s algorithm) Initialization: Choose some ˆd int D. Compute an optimal solution x of (P ( ˆd)). Set S = {v R p : λ(v),ϕ(px,v) } and k =. Iteration k. Step k. If vert S k D stop, otherwise choose a vertex s k of S k such that s k D. Step k. Compute α k (, ) such that v k := α k s k +( α k ) ˆd max K D. Step k. Compute an optimal solution x k of (P (v k )). Step k. Set S k := S k {v R p : ϕ(px k,v) }. Step k. Set k := k +andgoto(k). 6

7 In Step k, we use the on-line vertex enumeration algorithm of Chen and Hansen (99) to calculate the vertices of S k = S k {v R p : ϕ(px k,v) }. This algorithm needs to remember the adjacency list of each vertex of S k. The adjacency list of a vertex v includes its adjacent vertices and its adjacent cutting planes. We say that a supporting hyperplane H (Px k )={v R p : ϕ(px k,v)=} is adjacent toavertexs k if the vertex is on the hyperplane, i.e., ϕ(px k,s k )=. Wesaya cut is degenerate if the supporting hyperplane does not support D in a facet. The principle of the on-line vertex enumeration algorithm is to find the vertex sets of S k on both sides of the supporting hyperplane H (Px k )={v R p : ϕ(px k,v)=} and then to use adjacency lists of vertices to identify all edges of S k intersecting H (Px k ). The corresponding intersection points are computed and the adjacency lists updated. When the algorithm terminates, we have the following Proposition (see Ehrgott et al. (7) for a proof). Proposition. () The set of K-maximal vertices of D is vert S k. () The set { y R p : ϕ(y, v) for all v vert S k } is a nondegenerate inequality representation of P. () All R p -minimal (nondominated) vertices of P are contained in the set W := {Px,Px,...,Px k }. () The set {v R p : λ(v),ϕ(y, v) for all y W}is a (possibly degenerate) inequality representation of D. Possibly degenerate means that this inequality representation may include redundant inequalities (a redundant inequality is produced during the iteration if a supporting hyperplane supports a face but not a facet of D). () and () in Proposition. give the vertex set and the inequality representation of D while () gives the inequality representation of P and () gives a set of nondominated points of P which includes all the vertices of P. Proposition. suggests that we can get both P and D when the algorithm terminates. At each iteration k, the hyperplane given by H (Px k )={v R p : ϕ(px k,v)= } is constructed so that it cuts off a portion of S k containing s k,thuss S S... S k S k = D. Geometric duality theory establishes a relationship between P and D. P has the property that P = P + R p while D has the property that D = D Kand the projection to its first p components is the polytope {t R p,t, p i= t i }. Now we are going to extend the geometric duality theorem to two special polyhedral convex sets which have the same property as P and D, respectively. Property. In this paper, we consider special convex polyhedral sets S R p with the property that S = S Kand the projection to its first p components is the polytope {t R p,t, p i= t i }. Lemma. For S with Property., S = K. Proof. The proof is part of the proof of Proposition. in Ehrgott et al. (7). Definition.6 For a polyhedral convex set S R p with Property., we define D(S) ={y R p : ϕ(y, v), for all v vert S}, whereϕ(y, v) = p ( i= y iv i + y p ) p i= v i v p. 7

8 Proposition.7 Let S R p with Property.. Then D(S) =D(S)+R p. Proof. It is obvious that D(S) D(S) +R p, therefore we only need to show D(S) D(S)+R p. Let w + d D(S) +R p with w D(S) andd Rp. Since w D(S), we have ϕ(w, v) for all v vert S. ϕ(w + d, v) =ϕ(w, v)+ d, λ(v) where λ(v) = (v,...,v p, p i= v i) T.Sinceϕ(w, v) and it is obvious that d, λ(v) because both d and λ(v) R p,wehaveϕ(w + d, v) andw + d D(S). Corollary.8 For S R p with Property., Theorem. holds for D = S and P = D(S). Proposition.9 Let S and S be polyhedral convex sets with Property. and S S,thenD(S ) D(S ). Proof. By Lemma. we have S = S = K. This means that S and S have only one extreme direction d = e p =(,...,, ). Suppose S has r vertices, v,...,v r. We need to show that for y D(S ), i.e., ϕ(y, v),v = v,...,v r, it holds that y D(S ). Let v be a vertex of S,thenv S as S S. Therefore, v can be expressed as v = r i= α iv i + νd with α i,ν, r i= α i =, and d = e p. We calculate ϕ(y, v ) as follows. ϕ(y, v ) = = = = = p y k vk + y p p( vk ) v p k= k= i= k= p r p y k ( α i vk i + νd k)+y p ( ( p y k k= r i= p α i vk i + y p ( r α i ϕ(y, v i )+ +ν i= r α i ϕ(y, v i )+ν i= k= i= k= r α i vk i + νd k)) i= r α i vk) i i= k= r α i vp i νd p i= r p p α i vp i + y k νd k y p νd k νd p k= Since ϕ(y, v i ), α andν, we have ϕ(y, v ). This means that any y D(S ) is also contained in D(S ). This proves D(S ) D(S ). For the dual variant of Benson s algorithm, Proposition.9 indicates that D(S k ) enlarges with the iteration and when the algorithm terminates at iteration k, D(S k )= D(D) = P. We give an example to illustrate the dual variant of Benson s algorithm and we show how S k and D(S k ) change with iteration k. Example. Consider the MOLP min{px : Ax b}, where ( ) P =,A=,b=. 8

9 P and D are shown in Fig.. The vertices of D are (, ), (, ), (, ), (, ), (, ). Their corresponding facets (supporting hyperplanes) of D are =,y + =,y + =,y + =,y =. The four vertices of P, (, ), (, ), (, ) and (, ) correspond to the facets (supporting hyperplanes) of D v + v =,v + v =, v + v =and v + v =, respectively. v P D y Figure : P and D for Example.. v Fig. shows the change of S k with each iteration k. As can be seen, with iteration k, S k becomes smaller and smaller until at termination it is the same as D. The vertices of the initial cover S are (, )and(, ). The first hyperplane cuts off vertex (, ), the vertices of S are (, ), (, ), and ( 8, ). The second hyperplane cuts off vertex (, ), thus the vertices of S are (, ), ( 8, ), ( 8, ) and (, ). The third hyperplane cuts off vertex ( 8, ), thus the vertices of S are (, ), (, ), (,,( 8, )and(, ). The fourth hyperplane cuts off vertex ( 8, ) and the vertices of S are (, ), (, ), (,,(, )and(, ). After the fourth cut, we have S = D. Therefore, the vertices of S are also the vertices of D. v v v v v S S S S S ˆd v ˆd v ˆd v ˆd v ˆd v Figure : The reduction of S k with iteration k. The change of D(S k ) after each iteration k can be seen in Fig.. The calculation of D(S k ) is according to the definition D(S k )={y R p : ϕ(y, v) for all v vert S k }. For example, D(S )={y R p : ϕ(y, v) forv =(, ), (, )}, i.e. D(S )={y } { }. In contrast to the reduction of Sk, D(S k ) enlarges with iteration k. When the dual variant of Benson s algorithm terminates, S = D and D(S )=D(D) =P. With Proposition.9 the process of outer approximation of D can be interpreted as a process of inner approximation of P.. Obtaining the Nondominated Facets of P from D As seen in Proposition. (), vertex v =(v,v,...,v p )ofd corresponds to a supporting hyperplane of P which supports P in a weakly nondominated facet. The 9

10 D(S ) D(S ) D(S ) y y y D(S ) D(S ) y y Figure : The enlarging of D(S k ) with iteration k. hyperplane is λ(v) T y = v p,whereλ(v) =(v,...,v p, p i= v i) T and λ(v). If λ(v) >, then the supporting hyperplane supports P in a nondominated facet instead of a weakly nondominated facet. We call v an inner vertex of D if λ(v) >, otherwise, we call it a boundary vertex of D. To calculate the nondominated facets of P, we only need to consider inner vertices of D. Now we are going to show how to calculate the nondominated facet F which corresponds to an inner vertex v of D. When the algorithm terminates, Proposition. () gives us the vertex set of D and () gives us the inequality representation of D. The inequality representation of D is D = {v R p : λ(v),ϕ(y, v) for all y W}where W = {Px,Px,...,Px k }. As we mentioned before, at Step k, we use the on-line vertex enumeration algorithm of Chen and Hansen (99). Therefore, we have the adjacency list for each vertex of D. The adjacency list of a vertex includes the adjacent vertices and the adjacent supporting hyperplanes of D. The set of adjacent supporting hyperplanes of an inner vertex is a subset of {v R p : ϕ(y, v) = for all y W}. ForaninnervertexofD, if all its adjacent supporting hyperplanes (cuts) are non degenerate, i.e., each cut supports D in a facet, then we can find all the points in P which correspond to the cuts of D by geometric duality theory. These points are the vertices of F. If not all of its adjacent cuts support D in facets, i.e., some cuts are degenerate, then we need to use the following result. Proposition. Let inner vertex v of D correspond to nondominated facet F of P. Suppose v has a degenerate adjacent cut (supporting hyperplane), then the cut corresponds to a nondominated point p F,butp is not a vertex of F. Proof. Suppose vertex v has k non degenerate cuts (supporting hyperplanes). The k non degenerate cuts correspond to the k vertices of F. The degenerate cut can be expressed by a linear combination of the k non degenerate cuts. Correspondingly, the point that the degenerate cut corresponds to can also be expressed by the same linear combination of the k vertices. Therefore, we have p F.

11 v D P v y Figure : Vertex (, ) and its degenerate cut. Figure : P and the point corresponding to a degenerate cut Proposition. shows that the points of P which correspond to the adjacent cuts of a vertex of D are on the same facet of P. Moreover, all non degenerate cuts of D correspond to the vertices of the facet of P and all degenerate cuts correspond to points which are not vertices of the facet. Therefore, we can use the convex hull of the points to find the facet. We show an example with a degenerate cut. Example. For Example., vertex (, )ofd is adjacent to three cuts, they are v + v =, v + v =andv =.Amongthem,v = is a degenerate cut because it only supports D in vertex (, ) instead of a facet. This can be seen in Fig.. Vertex (, )ofd corresponds to the line segment between (, ) and (, ), a facet F of P, as shown in Fig.. Cut v + v = corresponds to (, ) and cut v + v = corresponds to (, ); while v = corresponds to (, ), a nondominated point of P. The two points (, ) and (, ) are the vertices of F, while point (, )isinf, but it is not a vertex of F. The degenerate cut v = can be obtained by simply adding the two adjacent non degenerate cuts together with equal weights, i.e., (v + v )+ ( v + v )= + orv =. On the other hand, (, ) can also be obtained by simply adding the two vertices (, ) and (, ) together with the same equal weights, i.e., (, ) + (, ) = (, ). Solving the Dual MOLP Approximately If the nondominated set of an MOLP is curved, which means there are very many facets and very many vertices of P, then there will be very many vertices and facets of D according to Corollar.. Therefore, whether we are solving the primal problem with Benson s algorithm or whether we are solving the dual problem with dual variant of Benson s algorithm, computation time may be a problem. To improve the computation time, an approximation version of Benson s algorithm has been proposed in Shao and Ehrgott (6). For a vertex of S k not in Y, but with Euclidean distance to the boundary point of Y on the line segment between the vertex and the interior point less than tolerance ɛ (ɛ R, ɛ>), we omit constructing the cut and remember both the vertex and the boundary point as a point of the outer approximation and a point of the inner approximation, respectively. Finally, we have an outer approximation Y o of Y and an inner approximation Y i of Y to sandwich Y, i.e., Y o Y Y i. Moreover, each vertex of Y o has a corresponding point of Y i, and the Euclidean distance between the two points is less than tolerance ɛ. As a result of approximation, all the points of Y i calculated are on the boundary of Y and the weakly nondominated set of Y i has been proved to be a set of weakly ε-nondominated points of Y (ε = ɛe).

12 In this section, we propose approximately solving the dual MOLP but controlling the approximation error to get an approximate extended feasible set in objective space D o, and then finding the polyhedral set D(D o ). D(D o ) is an inner approximation of P, the original extended primal feasible set. Finally we will show that the nondominated set of D(D o ) is actually an ε-nondominated set of the original MOLP. Our approximate dual variant of Benson s algorithm is identical to Algorithm except for Step k. Let ɛ R,ɛ> be a tolerance, then the changes are as follows. Step k. If, for each v vert S k, v D+ ɛe p is satisfied, then the outer approximation of D denoted by D o is equal to S k. Stop. Otherwise, choose any v k vert S k such that v k / D+ ɛe p and continue. To check v D+ ɛe p, we need solve D (v) and get its objective value f. If v p f ɛ, thenv D+ ɛe p. Since D o D, Proposition.9 implies D(D o ) D(D) =P. This means that D(D o ) is an inner approximation of P. WeuseP i to represent D(D o ). We illustrate the algorithm by using the same example as Example.. Example. In Example., let us set ɛ =. After two cuts there are two vertices of S (v =( 8, )andv =( 8, )) outside D. The boundary points of D which have the same v value as v and v are bd =( 8, 8 )andbd =( 8, 8 ), respectively. Both the Euclidean distance between v and bd and the Euclidean distance between v and bd are equal to 8. We accept these two infeasible points v, v for the outer approximation of D due to the distances to their corresponding boundary points being less than ɛ, i.e., v, v D+ ɛe q. When the algorithm terminates, the total number of iterations k is equal to and S = D o. Figs. 6 and 7 show D o and its corresponding polyhedral set P i = D(D o ). v D(D o )=P i D o Figure 6: D o v y Figure 7: P i We are going to evaluate the approximation quality of P i as inner approximation of P. Letusfirstrecallthedefinitionofε-efficient solutions. Definition. (Loridan (98)) Consider the MOLP () and let ε R p.. A feasible solution ˆx X is called an ε-efficient solution of () if there does not exist x X such that Px P ˆx ε. Correspondingly, ŷ = P ˆx is called an ε-nondominated point in objective space;. A feasible solution ˆx X is called a weakly ε-efficient solution if there does not exist x X such that Px < Pˆx ε. Correspondingly, ŷ = P ˆx is called a weakly ε-nondominated point in objective space.

13 Now we proceed to show that the weakly nondominated set of P i is actually a set of weakly ε-nondominated points of P. First let us move D along e p by ɛ, then we get D u = D + ɛe p. When the approximate algorithm terminates, the set of K-maximal points of D o will lie in between the set of K-maximal points of D and the set of K-maximal points of D u, i.e., D D o D u. We can calculate P = D(D) andp u = D(D u ). According to Proposition.9, we have P P i P u. Therefore, the set of weakly nondominated points (R p - minimal points) of P i = D(D o ) is in between the set of weakly nondominated points (R p -minimal points) of P and the set of weakly nondominated points of Pu. Theorem. Suppose the dual approximation error is ɛ. Let ε = ɛe, then the nondominated set of P i is a set of ε-nondominated points of P. Proof. First we show that the nondominated set of P u is actually a set of ε- nondominated points of P. According to Corollary.8, Theorem. applies to D = D u and P = P u,thus we can use duality theory to find P u. Let v =(v,v,...,v p )beavertexofd. Then there is a corresponding vertex v u of D u and v u = v+ɛe p. Suppose vertex v of D corresponds to facet λ(v) T y = v p of P, where λ(v) =(v,...,v p, p i= v i) T and suppose vertex v u =(v u,vu,...,vu p ) of D u corresponds to facet λ(v u ) T y = vp u of Pu,whereλ(v u )=(v u,...,vu p, p i= vu i )T.Asv +ɛe p = v u,wehaveλ(v u )=λ(v). Therefore, the facet λ(v)y = v p of P is parallel to the facet λ(v u )y = vp u of Pu. Let F be a facet of D, then there is a corresponding facet F u of D u and F u is parallel to F. Suppose F corresponds to vertex y =(y,,...,y p )ofp, then the equation of the hyperplane spanned by F is (y p y )v +(y p )v +...+(y p y p )v p +v p = y p. Suppose F u corresponds to vertex y u =(y u,y u,...,yp u )ofp u, then the equation of the hyperplane spanned by F u is (yp u yu )v +(yp u yu )v +...+(yp u yp )v u p + v p = yp u. F u and F are parallel and F u is obtained by moving F along e p by ɛ, therefore, we have yp u = y p + ɛ, and(yp u yu )=(y p y ),..., (yp u yu p )=(y p y p ), i.e., y u = y + ɛe. Above all, P u can be obtained by moving every point y of P to y+ɛe. Therefore, the nondominated set of P u is a set of ε-nondominated points of P. Moreover, as P u P i P, so the nondominated set of P i is also a set of ε-nondominated points of P. Example. As in Example., ɛ =. Fig. 8 and Fig. 9 show D, Du, P and P u. The vertices of P are (, ), (, ), (, ) and (, ). The corresponding vertices of P u are (, ), (, ), (, )and(, ). The nondominated set of P u is a set of ε-nondominated points of P where ε = e. 6 Application: Radiotherapy Treatment Planning The aim of radiotherapy is to kill tumor cells while at the same time protecting the surrounding tissue and organs from the damaging effect of radiation. To achieve these goals computerized inverse planning systems are used. Given the number of beams and beam directions, beam intensity profiles that yield the best dose distribution under consideration of clinical and physical constraints are calculated. This is called beam intensity optimization problem.

14 v D D u v Figure 8: D and D u P P u y Figure 9: P and P u We have formulated the beam intensity optimization problem as an MOLP in Shao and Ehrgott (6). The objectives of the MOLP are to minimize the maximum deviation y,, of delivered dose from tumor lower bounds, from critical organ upper bounds and from the normal tissue upper bounds, respectively. We used both Benson s algorithm and an approximation version of Benson s algorithm to solve the MOLP of an acoustic neuroma (AC), a prostate (PR) and a pancreatic lesion (PL) case, respectively. In this paper, we solve the dual problems of the same three clinical cases as above, i.e., the acoustic neuroma (AC), the prostate (PR) and the pancreatic lesion (PL) both by the dual variant of Benson s algorithm and the approximate dual variant of Benson s algorithm. Both the acoustic neuroma case and the prostate case can be solvedexactly with the dual variant of Benson s algorithm. We show the results of solving the dual problem exactly with the dual variant of Benson s algorithm and approximately with the approximate dual variant of Benson s algorithm. The results for the acoustic neuroma case are shown in Fig.. The set max K D and the union of the nondominated facets of P obtained by the dual variant of Benson s algorithm are on the top left and right, respectively. The other pictures show the results of solving the dual problem approximately with the approximate dual variant of Benson s algorithm. Pictures on the left are max K D o while pictures on the right are the union of the nondominated facets of P i. From top to bottom, they are for approximation error ɛ =., and ɛ =., respectively. The results for the prostate case are in Fig.. The pictures on the top left and right are max K D and the union of the nondominated facets of P, theywere obtained by the dual variant of Benson s algorithm. The rest of the pictures show the results of solving the dual problem approximately with the approximate dual variant of Benson s algorithm. Pictures on the left show max K D o while pictures on the right show the union of the nondominated facets of P i. From top to bottom, they are for approximation error ɛ =., ɛ =., and ɛ =., respectively.

15 8 6 9 v 8 7. v v 6 y v 8 7. v v 6 y v 8 7. v v 6 y 6 8 Figure : The results for the AC case.

16 v. v v. y v. v v. y v. v v. y v. v v. y Figure : The results for the PR case. 6

17 The dual variant of Benson s algorithm cannot solve the problem of pancreatic lesion case within hours of computation. Therefore, we show the results solved by the approximate dual variant of Benson s algorithm in Fig.. Pictures on the left are max K D o and pictures on the right are the union of the nondominated facets of P i. Pictures on the top are for ɛ =. while pictures on the bottom are for ɛ =.. v 8 6. v v. y v 8 6. v v. y Figure : The results for the PL case. Summarizing information comparing the number of vertices and number of cuts of the dual variant of Benson s algorithm and the approximate dual variant of Benson s algorithm with various values of ɛ is given in Table. The dual variant of Benson s algorithm can solve the dual problem of the first two cases exactly in one hour, but not the problem of the pancreatic lesion case. The approximate dual variant of Benson s algorithm can solve all three problems within minutes with approximation error ɛ =.. Table and Figs., and clearly show the effect of the choice of ɛ. The smaller the approximation error, the more vertices and the more cuts are generated and the longer the computation time. To make comparisons with the results obtained by Benson s algorithm and the approximation version of Benson s algorithm, we show Y and Y o with different values of approximation error ɛ for the above three cases. Benson s algorithm can exactly solve the acoustic case and the prostate case. Fig. shows Y for the acoustic case and Fig. shows Y for the prostate case. Y o with ɛ =. for the acoustic case obtained by the approximation version of Benson s algorithm is shown in Fig., while Y o with ɛ =. for the prostate case obtained by the approximation version of Benson s algorithm is shown in Fig. 6. Benson s algorithm cannot solve the pancreatic lesion case within hours of computation, 7

18 we show Y o with approximation error ɛ =. in Fig. 7 and ɛ =. in Fig y y 8 Figure : AC: Y solved by Benson s algorithm. Figure : AC: Y o solved by approximation version of Benson s algorithm with ɛ =.. y y Figure : PR: Y solved by Benson s algorithm. Figure 6: PR: Y o with ɛ = γ 6 6 α β y Figure 7: PL: Y o with ɛ =.. Figure 8: PL: Y o with ɛ =.. The number of vertices and number of cuts of Y o with various values of ɛ are also given in Table. Comparing the computation time of exactly solving the primal and exactly solving the dual problem for both the acoustic case and the prostate case, solving the primal problem exactly using Benson s algorithm needs more computation time than solving the dual problem exactly using the dual variant of Benson s algorithm. If the approximation error ɛ in the approximation version of Benson s algorithm and the approximation error ɛ in the approximate dual variant of Benson s algorithm are the same, then both algorithms guarantee finding ε-nondominated set (ε = ɛe). Thus we can compare the computation time for solving Y o and P i with the same 8

19 approximation error ɛ. In Table, we observe that for the three clinical cases solving the dual approximately is always faster than solving the primal approximately with the same approximation error ɛ. Table : Running time and number of vertices and cutting planes to solve the dual problem and the primal problem for three cases with different approximation error ɛ (ɛ = means it is solved exactly). Case ɛ Solving the dual Solving the primal Time Vertices Cuts Time Nondominated Cuts (seconds) of D o of D o (seconds) vertices of Y o of Y o AN PR PL Solving the primal MOLP gives us Y. If we only need the nondominated facets, we need to get rid of the weakly nondominated facets. Solving the dual problem allows us to directly calculate the nondominated facets. This can be taken as an advantage of solving the dual problem. 7 Conclusion In this paper, we have developed an approximate dual variant of Benson s algorithm to solve MOLPs in objective space. We have shown the algorithm guarantees to find ε-nondominated points with a specified accuracy ɛ. This algorithm was applied to the beam intensity optimization problem of radiation therapy treatment planning. Three clinical cases were used and the results were compared with those obtained by approximately solving the primal with the approximation version of Benson s algorithm. When both algorithms use the same approximation error ɛ, both of them guarantee producing ε-nondominated set (ε = ɛe), we found that approximately solving the dual with the approximate dual variant of Benson s algorithm is faster than approximately solving the primal with the approximation version of Benson s algorithm for all the three clinical cases. References Benson, H. P. (998a). Hybrid approach for solving multiple-objective linear programs in outcome space. Journal of Optimization Theory and Applications, 98, 7. Benson, H. P. (998b). An outer approximation algorithm for generating all efficient 9

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