An interactive interior point algorithm for multiobjective linear programming problems

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An interactive interior point algorithm for multiobjective linear programming problems

B. Aghezzaf

^∗

, T. Ouaderhman

Departement de Mathematiques et d’Informatique, Faculte des Sciences Ain chock, B.P. 5366 Marif, Casablanca, Morocco Received 19 July 1999; accepted 10 January 2001

Abstract

We propose an interactive interior point method for .nding the best compromise solution to a multiple objective linear programming problem. We construct a sequenceX⁰; X1; : : : ; X^k; : : :of smaller and smaller polytopes which shrink towards the compromise solution. During the kth iteration, we move from the center of polytope X^k−1 to the center of polytope X^k by performing a local optimization which consists of maximizing a linear function over an ellipsoid. c2001 Elsevier Science B.V. All rights reserved.

Keywords:Multiobjective linear programming; Interior point methods; Interactive methods; Compromise solution

1. Introduction

The area of multicriteria decision making (MCDM) in general, and that of multiobjective mathematical programming (MOMP) have been very active in recent years. As a result, a number of methods for addressing these problems have become available (see, [7,13]). The purpose of these solution methods is to optimize the decision maker’s utility (preference) function. Depending on the assumption made on the utility function, the methods can be classi.ed as utility theory, goal programming, and progressive articulation and posterior articulation approaches. The progressive articulation approach, called the interactive method, is becoming increasingly popular and is considered a promising approach. The interactive methods are designed to .nd one “most preferred”

∗Corresponding author.

E-mail address: aghezzaf@facsc-achok.ac.ma (B. Aghezzaf ).

solution for problem (MOMP), that is dictated by an explicit or implicit utility function that belongs to the DM. Every algorithm in this class approaches has to develop two basic capabilities: (1) trade-oB mecha- nism, and (2) an optimization algorithm. We discuss each of these aspects brieCy below.

Although researchers have developed numerous interactive solution methods, there are only a few multiobjective interior point methods in the literature [1–6,14]. The solution approaches that are currently available use the aEne scaling primal algorithm, or the primal–dual algorithm with various schemes for de.ning a single interior search direction [2,5,6]. Two other approaches developed by Trafalis et al. [14] use the, so-called method for centers to both linear and nonlinear multiple objective programming problems, where the decision space is a convex polyhedron de- .ned by linear inequalities. The .rst algorithm is a generalization of Renegar’s algorithm [10] to multiple objective setting, that .nds an eEcient point

PII: S 0167-6377(01)00089-X

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of the polytope. The second, is based on techniques of algebraic geometry related to the parametrization of algebraic varieties inn-dimensional spaces. It ap- proximates a portion of the set of eEcient faces by an algebraic surface using the nonnegative orthant as the ordering cone to reduce the decision space.

Since the cone depends on the inscribed ellipsoid, the best compromise solution may not be retained in the reduced space.

To remedy this situation, we present an interactive interior point algorithm for determining a best compromise solution to a multiobjective linear programming problem (MOLP) in situations with an implicitly de.ned utility function, which is illustrated by deriving the local trade-oBs (the marginal rates of substitution) by interacting with the decision maker (DM). The algorithm is based, in principle, on the same idea used by Vaidya [15] in a single objective algorithm. By employing this approach, we construct a sequenceX⁰; X¹; : : : ; X^k; : : :of smaller and smaller polytopes which shrink towards the compromise solution, the algorithm generates a sequence of points x⁰; x¹; : : : ; x^k; : : : ;such thatx^kis in the interior of poly- topeX^k, by maximizing a linear function over an ellipsoid.

The paper is organized as follows. In Section 2, the problem statement and some fundamental de.nitions are stated. Section 3 describes the algorithmic procedure of the method and the results of the computational studies with the method are included in Section 4.

2. Problem denition and preliminary discussions Multiobjective linear programming (MOLP) problem is concerned with determining scalar values of ncontinuous decision variables in order to maximize l(l¿2) objectives simultaneously. Amathematical formulation of the problem is

(P) max Cx={z1(x); : : : ; zl(x)}

s:t: x∈X={x∈Rⁿ|Ax¿b};

whereC andAarel×nandm×nmatrices respec- tively, andb∈R^m. Here, the row c^T_i;16i6lofC are the coeEcients of 1 linear objective functionsz_i. In order to make the problem nontrivial, it will be assumed that the objectives are in conCict. Due to the conCicting nature of the objectives, an optimal solu-

tion that simultaneously maximizes all the objectives is usually not obtainable. Instead, there are several solutions, called eEcient solutions (nondominated or pareto optimum), that have the property that no improvement in any one objective is possible without sacri.cing one or more of the other objectives. The set of all eEcient solutions (in continuous case) is known as the eEcient frontier.

Denition 2.1. Asolution Lx∈X is said to be eEcient if there exists no pointx∈X such that

c^T_ix¿c^T_ixL ∀i;

c^T_ix ¿ c^T_ixL for at least onei:

With the above de.nition, the solution to a MOMP problems reduces to the problem of .nding all eE- cient solutions. Often, there are an in.nite number and they are not comparable. Hence, it is assumed that the decision maker has a real valued utility function (value function or preference function)U, de.ned on the values of objectives, which has not been accessed and is not explicitly known, with this assumption, the MOMP problem can be rede.ned as follows:

(Pu) max U(z₁(x); : : : ; z_l(x)) s:t: x∈X:

Within this mathematical framework, the primary objective of the interactive approaches is to .nd a best compromise solution through interaction with the DM.

Denition 2.2. Abest compromise solution (or optimal solution) to MOLP is an eEcient solution that maximizes the decision maker’s utility function. That is, a feasible solution Lxis the best compromise solution if and only if

U(Cx)L ¿U(Cx) for allx∈X:

Many other technical rules were developed to address the MOLP problem (P). One class of approaches is developed in [14] by considering an ordered cone.

With the Trafalis algorithm, an approximation of a portion of the eEcient frontier is given by construct- ing a sequence of algebraic surfaces {S_k}. The general surfaceS_kcan be considered the loci of analytical centers of intersection of an ordering coneand the polytopeX, where the apexofis moving onSk−1

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Fig. 1.

by taking the initializing manifoldS0 to be a part of the inscribed ellipsoid ofX. But the question which arises is how to construct the conethat illustrates the DM preferences.

In using the cone of increasing direction C^¿= {x∈Rⁿ=Cx¿0} as the ordered cone, the portion of the eEcient frontier obtained by Trafalis algorithm also depends on the geometry of the polytope X as it depends on in the general case, since the algorithm is dependent on the inscribed ellipsoid of X. Consequently, the .nal surface S^∗ generated by the algorithm might possibly be a portion of the ef- .cient frontier that is of no further interest, as one can never be completely sure about the existence of one^k which allows the desired solution in the re- stricted polytope and how to select it if that exists (see Fig. 1).

3. The proposed algorithm

In this paper, with (Pu) as the problem under con- sideration, an interactive interior point algorithm that can be used to solve MOLP will be developed and tested. For the development of the method, it is assumed thatX is bounded, andU is strictly increasing in each of its arguments, concave and diBerentiable on R^l. As the polytopeX is bounded, we can assume that m¿n, and the columns ofAare linearly independent.

To solve the multiobjective optimization problem, we have to .nd a way to approximate the utility function

at the current iterate and take a step in a direction that results in an improvement of the value of the objective function. The partial information on the utility function of interest is the local trade-oBs (marginal rates of substitution) between the objective functions which are obtained by interacting with the decision maker.

This concept is .rst proposed in multiple criteria decision making by GeoBrion et al. [8].

We assume thatz1(x) is a reference criterion.

Denition 3.1. The tradeoB ratio between the objec- tivez₁(x) andz_i(x) at pointx^kis

!_1i=(@U=@z_i)(x^k) (@U=@z1)(x^k):

Consider that the property of the utility function, the partial derivative ofU at a pointx^kis

_x^kU= @U

@x^k₁; : : : ; @U

@x^k_n

:

= @U

@z1

l i=1

!^k_1i @zi

@x₁^k; : : : ; l

i=1

!^k_1i @zi

@x^k_n

= @U

@z1

˙_x^kU= @U

@z1

!^kC;

where!^k= [!^k₁₁; : : : ; !^k_1l] and ˙_x^kU=!^kC;˙_x^kU is termed a positively scaled local gradient ofU atx^k.

In developing our approach, the motivation has been to overcome the diEculties encountered with the Trafalis algorithm mentioned before, and generating a single interior direction at current iterate that improves the utility function without combin- ing it with other directions resulting from anchoring points as in [2,6]. This algorithm also follows the Renegar approach [10], but there are three criti- cal diBerences which enable us to solve the above diEculties.

First, the reduced space is obtained by using an updating formula to update the lower bounds of objective functions instead of the ordered cone.

Second, the potential function is formulated in a manner to orient the iterates to the desired solution. Third, the generate iterate is computed along a direction that could be diBerent from the direction in Newton method. On the other hand, the problem is how much the lower bounds values

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Fig. 2.

of the objective functions can be changed in order that the reduced space contains the compromise solution.

In the following, we propose an appropriate way to update them.

3.1. Generating a polytopes

Let x^k be an interior point of X and consider in the following the updating formula used to update the objectives lower levels, that gives in iterationk, the added setC^k of constraints:

C^k={x∈Rⁿ|C_x¿(x^k)};

where

(x^k) = [₁^k+1; : : : ; ^k+1_l ]^T; _i^k+1=^k_i +^k_i(c^T_ix^k−^k_i);

16i6l and 06^k_i ¡1;

_i⁰6min

x∈X c^T_ix; 16i6l; (1)

where the coeEcients{^k_i} will be changed, in general, from one iteration to another. By choosing con- veniently the values of the step-sizes^k_i (06^k_i¡1) at each iteration in the updating formulas, any point in the eEcient frontier can be obtained, which is not available in the algorithm of Trafalis et al. (see Fig. 2).

And at each iteration, also we add another constraint given by the partial information. That constraint, which is used to eliminate a certain portion of

the decision space, is considered in such a way that the reduced decision spaceX^k contains the best compromise solution andx^k in his interior. The proposed constraint is given by

˙_x^kUx¿^k; (2)

where

^k=!^k(x^k) +( ˙_x^kUx^k−!^k(x^k)) and

0¡ ¡1: (3)

Thus, the original decision space is further reduced by adding the above constraints to the existing set of constraintsX. LetX^k denote the reduced decision space at thekth iteration, then

X^k={x∈Rⁿ|A^kx¿b^k} ∩C^k

={x∈Rⁿ|Ax¿b; ˙_xkUx¿^k;

K= 0;1; : : : ; k} ∩C^k: (4) By using the following equality:

i=l i=1

(^k+1_i −^k_i) =^i=l

i=1

^k_i(c^T_ix^k−^k_i)

=^i=l

i=1

^k_i(c^T_ix^k−z_i)+^i=l

i=1

^k_i(z_i−^k_i);

(5) where z_i=c^T_ix; x∈X, and if the lower bounds satisfy the aspirations of the decision maker, he will prefer not to update them, hence:_i=l

i=1(^k+1_i −^k_i) = 0.

The coeEcients{^k_i}that yieldsz=(x^k) will satisfy _i=l

i=1 ^k_i(c^T_ix^k−zi) = 0, which could be guaranteed by ^k_i =!^k_1iorc^T_ix^k=zi.

So, we propose that the values of the step-sizes^k_i depend on the DM preferences and given by

^k+1_i =^k_i +!^k_1i √ mk

i=l i=1

!^k_1i

(c^T_ix^k−_i^k);

16i6l and mk=m+l+k:

3.2. Locating an improving direction of U in the reduced space

The purpose of this step is to locate a direction of increase for (MOLP) problem in the reduced decision

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space. The scaled local gradient ofU is calculated at the current interior pointx^k, that is completely speci- .ed once the local tradeoBs are obtained from the DM, and used to locate an improving direction ofUin the reduced spaceX^k.

Consider the following problem:

(P^k) max ˙_xkUx s:t: x∈X^k:

The associated potential function would be F^k(x) =

mk

i=1

ln(A^k_ix−bi) + l

s=1

ln(c^T_sx−_s^k+1)

+(m_k) ln( ˙_x^kUx−^k):

Since the columns ofA(andm¿n), soA^k, are linearly independent,F^k(x) is a strictly concave function over the interior ofX^k, and the analytic center ofX^kgiven by maximizingF^kis indeed a unique point. Hence, the problem is converted into a new form by maximizing F^k(x) over the interior ofX^k.

One more interesting concept is to introduce the inscribed ellipsoid ofX^k, denoted byE^k(1), and maxi- mizingF^k over it. The ellipsoidE^k(1) is given by E^k(1) ={x∈Rⁿ|(x−x^k)^TH^k(x−x^k)61};

where H^k=^m^k

i=1

A^k_i(A^k_i)^T

(A^k_ix−b^k_i)² +^l

s=1

c_sc^T_s (c^T_sx−^k+1_s )²

+(m_k)( ˙_x^kU)^T( ˙_x^kU) ( ˙_x^kUx−^k)² :

At iterationk, the next iterate is obtained by maximizing the linear approximation ofF^k(x) overE^k(1) and we have to solve

max F(x^k)x;

s:t: x∈E^k(1):

Letz^k be the solution point of this problem. We shall now describe how to .nd the next iteratex^k+1 from x^k. From the theory of convex optimization, (z^k−x^k) satis.es the system of linear equations, given by the (KKT) conditions:

H^k(z^k−x^k) =tF(x^k): (6)

For some scalar t ¿0, we .rst compute"^k, a vector in the direction of (z^k−x^k) by solving the system of linear equations

H^k"^k=F(x^k):

Next, we .nd a scalart^k¿0, such that (t^k)²("^k)^TH^k"^k61:

Once the direction"^kandt^kare available, we take the next interior step according to the updating formula given by

x^k+1=x^k+t^k"^k; 0¡ 61:

Line search: the line search is executed along the directiont^k"^k by varying thevalue.

3.3. Algorithm

In this section, the major steps of the algorithm are presented

Step1:k= 0.

Initialization: x⁰; Ax⁰¿ b and select a .nite ⁰_i6min_x∈Xc^T_ix; 16i6l

Step2: Iterationk:

(a) Determine the tradeoBs vectors!^katx^k. (b) Evaluate

_i^k+1=^k_i + !^k_1i (√mk)_i=l

i=1 !^k_1i(c^T_ix^k−^k_i);

16i6l;

^k=!^k(x^k) +( ˙_x^kUx^k−!^k(x^k));

06 ¡1:

(c) Solve the linear system:

H^k"^k=F(x^k):

(d) Determinet^k: (t^k)²("^k)^TH^k"^k61:

(e) Find the new iteratex^k+1 from:

x^k+1=x^k+t^k"^k; 0¡ 61:

Step3: If the DM acceptx^k+1 as the compromise solution, stop.

Otherwise increment the iteration counter:k:=k+1, go to step 2.

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4. Illustrative example

The algorithm proposed in this paper will be illustrated using the following numerical example [4]. We have chosen this example because the approach will yield a .nal interior point better than that found in A.

Arbel approach max x1

max x₂

s:t: x₁+ 5x₂664;

x1+x2620;

4x₁+x₂656;

2x1−x2622;

x₁−x₂610;

x1+ 4x2¿8;

6x1+x2¿6;

−3x₁+ 2x₂616;

−x1+ 2x2620;

x₁¿0;

x2¿0:

For this example, an initial point and the initialization of^k_i are available through

x⁰= [2;5]; ⁰₁=⁰₂= 0:

Table 1

Solution results with= 0:1;(= 1)

k x1 x2 !^k (normalized) u(x)

0 0.714286 0.285714 2., 5. 10.

1 0.687195 0.312805 2.46878, 5.4236 13.3896 2 0.662035 0.337965 2.97453, 5.82676 17.3319 3 0.638485 0.361515 3.51675, 6.21107 21.8428 4 0.616244 0.383756 4.09585, 6.57722 26.9394 5 0.595118 0.404882 4.71206, 6.92603 32.6359 6 0.575029 0.424971 5.36462, 7.25887 38.9411 7 0.556005 0.443995 6.05124, 7.57783 45.8552 8 0.538168 0.461832 6.7672, 7.88575 53.3644 9 0.521732 0.478268 7.50415, 8.18611 61.4298 10 0.507032 0.492968 8.2476, 8.48289 69.9635 11 0.494588 0.505412 8.97196, 8.7798 78.772 12 0.485236 0.514764 9.62986, 9.07746 87.4147 13 0.480261 0.519739 10.1315, 9.36195 94.8506 14 0.482106 0.517894 10.317, 9.6041 99.0859

Fig. 3.

Table 2

Solution results when varying

kmax x1 x2 u(x)

0.1 14 10.317, 9.6041 99.0859

0.2 16 10.297, 9.59045 99.7528

0.3 20 10.1945, 9.76225 99.5215

0.4 27 10.1327, 9.82124 99.5156

0.5 48 10.0269, 9.97138 99.9821

Fig. 4.

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Fig. 5.

Table 3

Solution results by other testing example (= 0:1)

Example kmax x1 x2 u(x)

Arbel et al. [2] 8 7.112 6.217 80.207

Stueur [13] (p. 377) 21 5.00279 4.99688 1295.18

Arbel [3] 6 4.87026 5.06697 24.68

Sherali [11] 10 3.56098 4.50153 −12:82

To simulate the DM’s choice behavior, we will assume the following implicit preference function in order to test our algorithm:

U(x) =x₁x₂:

The vector optimization problem has a unique solution given through

x^?= [10;10] and U^?= 100:

Following the steps outlined above, we generate a sequence of steps summarized in Table 1 and illustrated by Fig. 3.

We mentioned earlier that the value of the scalar has some eBect on the procedure, this eBect on the progress of the iterative process when varying, , is illustrated in Table 2. The utility values at the current iterate are shown in Fig. 4 below, and in Fig. 5 we il- lustrate how a sequenceX⁰; X¹; : : : ; X^k; : : :is smaller and smaller polytopes which shrink towards the compromise solution.

In addition to the illustrative example, the problems presented by Arbel [2,3], Stueur [13, p. 377] and Sher- ali [11] were solved for comparison purposes. The results are given in Table 3.

5. Conclusion

In this paper, we have presented an interactive interior point algorithm for multiobjective linear programming problems. The algorithm is based on the modi- .cation of the method of centers to MOLP problems, this modi.cation is based on .nding an approximation to the gradient of the implicitly-known utility function by interacting with the DM. The marginal rates of substitution are obtained and used in deriving the approximation gradient and to update the lower bounds of the objectives.

6. For further reading See Refs. [9,12]

References

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