Ann Oper Res (2008) 164: 115–126DOI 10.1007/s10479-008-0346-x

A graphical characterization of the efficient setfor convex multiobjective problems

Francisco Ruiz · Lourdes Rey · María del Mar Muñoz

Published online: 19 April 2008© Springer Science+Business Media, LLC 2008

Abstract In this paper, a graphical characterization, in the decision space, of the prop-erly efficient solutions of a convex multiobjective problem is derived. This characterizationtakes into account the relative position of the gradients of the objective functions and theactive constraints at the given feasible solution. The unconstrained case with two objectivefunctions and with any number of functions and the general constrained case are studiedseparately. In some cases, these results can provide a visualization of the efficient set, forproblems with two or three variables. Besides, a proper efficiency test for general convexmultiobjective problems is derived, which consists of solving a single linear optimizationproblem.

Keywords Multiobjective programming · Convex programming · Properly efficientsolutions · Efficiency test

1 Introduction

In many real situations, decisions must be made attending to several conflicting criteria,rather than by just optimizing a single objective. When such a problem is taken under con-sideration, the preferences of the decision maker must somehow be included in the solutionprocess. Many different methods of different nature have been developed for solving thesemultiobjective problems (see Steuer 1986 for an overview of methods). Although the typesof interaction between the decision maker and the method can differ significantly, a goodprevious knowledge of the structure of the efficient set is always extremely useful for theuser to successfully carry out the solution process.

The structure of the efficient set for convex multiobjective problems has been widelystudied in the scientific literature (see, for example, Sawaragi et al. 1985). The properties ofthe nondominated set in the objective space have been exploited in order to develop specific

F. Ruiz (�) · L. Rey · M.M. MuñozDepartment of Applied Economics (Mathematics), Faculty of Economy, University of Málaga,C/Ejido, 6, Málaga 29071, Spaine-mail: rua@uma.es

116 Ann Oper Res (2008) 164: 115–126

algorithms (see Miettinen 1999 and Steuer 1986), and also analytic properties of the efficientsets have been studied (see Lowe et al. 1984, Sawaragi et al. 1985 and Ward 1989). All thesestudies have allowed an in depth theoretical characterization of the efficient solutions of aconvex multiobjective problem, and the subsequent elaboration of solution methods.

In this paper, we develop a graphical characterization, in the decision space, of the prop-erly efficient solutions of convex multiobjective problems. This characterization has beenobtained based on the previously mentioned analytical results, and it takes into account therelative position of the gradients of the objective functions and the active constraints at thecorresponding point. This characterization allows checking the efficiency of the feasible so-lutions by means of drawing the corresponding gradients, which can be a helpful tool if weconsider problems with two or three variables (for example, for facility location problems).Besides, in certain cases it is possible to obtain an explicit expression of the efficient set orof its boundary.

Finally, based on this characterization, an analytical efficiency test for convex multiob-jective problems is proposed in this paper. Several such tests can already be found in theliterature. For example, in Charnes and Cooper (1961) and Ecker and Kouada (1975), ef-ficiency tests for linear multiobjective problems are obtained. The latter was generalizedafterwards for nonlinear problems in Wendell and Lee (1977). Some other efficiency testsfor nonlinear problems can be found in Benson (1978) and Brosowski and da Silva (1994).Nevertheless, in all these cases the tests are carried out by solving nonlinear problems, andseveral problems have to be solved in some of them. The test developed in this paper forconvex multiobjective problems checks the efficiency of a feasible solution by solving asingle linear optimization problem.

The remainder of this paper is organized as follows. In the next section, some basic con-cepts are given. Then, the graphical characterization is derived for unconstrained problemswith two objectives in Sect. 3, and the result is generalized for general unconstrained prob-lems in Sect. 4. The general constrained problem is considered in Sect. 5, and the efficiencytest is described in Sect. 6. The paper ends with the conclusions.

2 Basic definitions

The problem to be considered will be the following:

(MOP)

{min f(x) = (

f1(x), f2(x), . . . , fp(x))

s.t. gj (x) ≤ 0, j = 1, . . . ,m

where all functions fi (i = 1, . . . , p), gj (j = 1, . . . ,m), are convex and continuously dif-ferentiable. The feasible set of problem (MOP) will be denoted by X.

Now, let us give some basic definitions.

Definition 2.1 Given two vectors f, g ∈ Rp , the following notation will be used:

f < g ⇔ fi < gi (i = 1, . . . , p),

f ≤ g ⇔ fi ≤ gi (i = 1, . . . , p) and f �= g.

Definition 2.2 A feasible solution of (MOP), x∗, is said to satisfy the Kuhn-Tucker con-straint qualification if vectors ∇gj (x∗) = 0 (j ∈ J ) are linearly independent, where

J = {j ∈ {1, . . . ,m}/gj (x∗) = 0}.

Ann Oper Res (2008) 164: 115–126 117

Definition 2.3 A feasible solution of (MOP), x∗, is said to be Pareto optimal or efficient ifthere does not exist any other feasible solution y such that f(y) ≤ f(x∗).

Definition 2.4 A feasible solution of (MOP), x∗, is said to be weakly Pareto optimal orweakly efficient if there does not exist any other feasible solution y such that f(y) < f(x∗).

Definition 2.5 A feasible solution of problem (1), x∗, is said to be properly efficient (in thesense of Geoffrion) if it is efficient, and there exists a number M > 0, such that for eachi ∈ {1, . . . , p}, and each y ∈ X satisfying fi(y) < fi(x∗) there exists at least one j ∈{1, . . . , p} such that fj (x∗) < fj (y) and

(fi(x∗) − fi(y))/(fj (y) − fj (x∗)) ≤ M.

Given these definitions, it is obvious that every properly efficient solution is efficient, andevery efficient solution is in turn weakly efficient.

The next theorem characterizes the properly efficient solutions of (MOP), in terms of thewell known equivalent weighted problem:

Theorem 2.1 (See Sawaragi et al. 1985 for proof) Given problem (MOP), let us considera feasible solution x∗, such that the Kuhn-Tucker constraint qualification is satisfied at x∗.Then, x∗ is a properly efficient solution of (MOP) if and only if there exists a vector ofstrictly positive weights μ > 0, such that x∗ is an optimal solution of the following weightedproblem:

(Pμ)

{min

∑p

i=1 μifi(x)

s.t. gj (x) ≤ 0, j = 1, . . . ,m.

Given that the constraint qualification holds at x∗, and that (Pμ) is a convex problem,then x∗ is an optimal solution of (Pμ) if and only if the Karush-Kuhn-Tucker conditionshold at x∗, that is, if and only if there exist nonnegative multipliers λ1, . . . , λm, such that:

p∑i=1

μi∇fi(x∗) +m∑

j=1

λj∇gj (x∗) = 0,

λjgj (x∗) = 0 (j = 1, . . . ,m). (1)

If we denote by J the set of active constraints at x∗, that is,

J = {j ∈ {1, . . . ,m}/gj (x∗) = 0},then conditions (1) can be rewritten in the following way:

p∑i=1

μi∇fi(x∗) +∑j∈J

λj∇gj (x∗) = 0. (2)

Graphically, condition (2) means that there is a linear combination of the gradients ofthe objective functions, and the gradients of the active constraints at x∗, with nonnegativeweights, which is equal to the zero vector. Therefore, these gradients must be able to “com-pensate” for each other, so that the null linear combination exists. Intuitively, this will notbe possible if all the gradients lie in the same semi-space. This is the main point which willbe proved in this paper.

118 Ann Oper Res (2008) 164: 115–126

3 Unconstrained case. Two objective functions

First, the simplest case will be examined. Let us consider an unconstrained problem withjust two objective functions:

min f(x) = (f1(x), f2(x)) . (3)

Therefore, condition (2) holds if and only if there exist two strictly positive weights,μ1,μ2 such that:

μ1∇f1(x∗) + μ2∇f2(x∗) = 0. (4)

Obviously, condition (4) holds if and only if the gradients of both functions go in oppositedirections. In other words, the properly efficient set is formed by feasible solutions wherethe level curves of both objective functions are tangent, and grow in opposite directions.Relation (4), or more precisely, the following relation

∇f1(x∗) = −(μ2/μ1)∇f2(x∗) ⇔ ∇f1(x∗) = −α∇f2(x∗) (α > 0) (5)

constitutes, if the conditions of the implicit function theorem hold (see, for example, Man-gasarian 1969), an implicit parametrical description of the efficient set, where the decisionvariables are functions of the parameter α. Therefore, in these cases, the properly efficientset is a one-dimensional subset of the feasible set. Moreover, this expression may also leadin certain cases to an explicit formulation of the properly efficient set.

In order to illustrate these statements, let us consider the following simple example

min(50x4 + 10y4,30(x − 5)2 + 100(y − 3)4

). (6)

Making use of expression (5), the following parametrical definition of the efficient set isobtained:

x =5 3√

35 α

1 + 3√

35α

, y = 3 3√

10α

1 + 3√

10α, α > 0

which, in this case, can be turned into the following equation:

y = 3 3√

50x

(3√

50 − 3√

3)x + 5 3√

3, 0 < x < 5.

This properly efficient set can be seen in Fig. 1As it can be seen in Fig. 1, the properly efficient set is, in this case, a curve that joins

points (0, 0) and (5, 3), which are the global minima of f1 and f2, respectively (both pointsare not included in the properly efficient set). In all the points of this curve, the gradients ofboth functions go in opposite directions, and the level curves are tangent.

4 Unconstrained case. m objective functions

When a problem has more than two objectives, the characterization obtained in the previoussection can be extended in the following way. Given a properly efficient solution x∗, it must

Ann Oper Res (2008) 164: 115–126 119

Fig. 1 Properly efficient set of problem (6)

Fig. 2 Graphical characterization of the properly efficient solutions. Unconstrained case

be the optimal solution of a weighted problem with strictly positive weights, which in turnimplies that the Karush-Kuhn-Tucker conditions are satisfied at x∗, that is, the weighted sumof the gradients of the objective functions evaluated in x∗ is the zero vector. As previouslymentioned, it seems intuitive that this condition cannot be accomplished if all the gradientscan be included in the same open semi-space, because in this case they cannot compensateeach other. This idea is depicted in Fig. 2. On the left, the solution is properly efficientbecause no line can be found that leaves the three gradients in the same semi-space. Onthe right, the solution is not properly efficient because there are lines that leave the threegradients in the same semi-space.

Theorem 4.1 proves that this graphical characterization is correct. In order to prove thistheorem, it is necessary to use the following lemma (see Mangasarian 1969 for the proof):

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Lemma 4.1 (Stiemke’s Alternative Theorem) For each given matrix A, either

• Ax ≥ 0 has a solution x, or• AT y = 0,y > 0 has a solution y, but never both.

Theorem 4.1 Given the unconstrained problem

minx∈Rn

(f1(x), f2(x), . . . , fp(x)

)(7)

where all functions fi are convex, x∗ is a properly efficient solution of (7) if and only if, foreach a ∈ R

n, one of the two following conditions holds:

(i) ∀i = 1, . . . , p,aT ∇fi(x∗) = 0,(ii) ∃i, j ∈ {1,2, . . . , p}/aT ∇fi(x∗) > 0,aT ∇fj (x∗) < 0.

Proof Let us prove the first implication. Let

A =⎛⎜⎝

∇f1(x∗)T

...

∇fp(x∗)T

⎞⎟⎠ .

If x∗ is a properly efficient solution of (7), then it is the optimal solution of a weightedproblem, for certain strictly positive weights, and thus, it satisfies the Karush-Kuhn-Tuckerconditions associated to that problem. Therefore, the following system:

AT μ = 0, μ ∈ Rp,μ > 0

has got some solution. Then, Lemma 4.1 implies that the system

Aa ≥ 0, a ∈ Rn

does not have any solution, that is, for each a ∈ Rn, either

∀i = 1, . . . , p, aT ∇fi(x∗) = 0,

or

∃i, j ∈ {1,2, . . . , p}/aT ∇fi(x∗) > 0, aT ∇fj (x∗) < 0,

and this completes the proof of the necessary condition. Conversely, let us suppose that x∗ isnot a properly efficient solution of (7). Then, x∗ is not the optimal solution of any weightedproblem with strictly positive weights. This implies that the system

AT μ = 0, μ ∈ Rp,μ > 0

does not have a solution. In this case, the alternative theorem of Stiemke assures that thesystem

Aa ≥ 0, a ∈ Rn

has got some solution, that is, there exists a vector a∗ ∈ Rn, such that

∀i = 1, . . . , p, aT ∇fi(x∗) ≥ 0,

∃j ∈ {1, . . . , p}/aT ∇fj (x∗) > 0.

Ann Oper Res (2008) 164: 115–126 121

Fig. 3 Properly efficient set of problem 8

But this contradicts the theses of the converse implication and thus, the proof is com-pleted.

Therefore, Theorem 4.1 assures that at a properly efficient solution of a general uncon-strained problem, no hyperplane exists that leaves the gradients of all the objective functionsin the same semi-space (except if the hyperplane itself contains all the gradients). As an ex-ample, let us consider the following problem:

min(50x4 + 10y4,30(x − 5)2 + 100(y − 3)4,70(x − 2)4 + 20(y − 4)4

), (8)

where a new function has been added to problem (6). Figure 3 shows the properly efficientset of this problem.

As it can be seen in Fig. 3, the point on the left is not properly efficient, because thethree gradients can be included in an open semi-space. The point on the right is properlyefficient because it is not possible to do so. The properly efficient set is the shadowed region(borders are not included). It can be observed that this region is delimited by the efficientsets of the functions considered two by two. As previously mentioned, in some cases it maybe even possible to obtain explicit expressions of these sets. If, on the other hand, thereare more than three functions, some of these sets formed considering subsets of functionsmay fall inside the properly efficient set. Anyway, the frontier of the properly efficient setis always formed by unions of such sets. This property can be extended to problems withany number of variables. In Lowe et al. (1984), Sawaragi et al. (1985) and Ward (1989), itis proved that the frontier of the properly efficient set is contained in the union of efficientsets corresponding to subproblems of the original problem with no more than n objectivesfunctions (where n is the number of decision variables). �

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Fig. 4 Graphical characterization of the properly efficient solutions. Constrained case

5 Constrained case

When the constraints are added to the problem, it is obvious that those solutions that areproperly efficient for the unconstrained problem and feasible for the constrained one, willalso be properly efficient for the constrained problem. And what about the properly efficientpoints for the unconstrained problem that are not feasible? It seems logical to think that, justas it happens with the single criteria problems, these solutions will somehow be projectedonto the frontier of the feasible set. That is, there will be active constraints at these newproperly efficient solutions, whose corresponding optimal Kuhn-Tucker multipliers for theweighted problem will be greater than zero. Therefore, the logical extension of the resultobtained in Sect. 4 to the constrained case is the following one: at a properly efficient so-lution x∗, the gradients of the objective functions and those of the active constraints at x∗cannot belong to the same open semi-space. In other words, if some hyperplane leaves thegradients of the objective functions in the same semi-space (that is, the point is not properlyefficient for the unconstrained problem), then there must exist an active constraint whosegradient lies in the other semi-space. This idea is shown in Fig. 4. The feasible set is theshadowed circle. The point on the left is not properly efficient because the gradients of theobjective functions and the gradient of the active constraint can be included in the same opensemi-space. On the other hand, the point on the right is properly efficient.

Theorem 5.1 shows that this idea is correct. In order to prove this theorem, it is necessaryto use the following lemma (see Mangasarian 1969 for the proof):

Lemma 5.1 (Tucker’s Alternative Theorem) Let B,C and D be given matrices, with B

being nonvacuous. Then either

(i) Bx ≥ 0,Cx ≥ 0,Dx = 0 has a solution x, or(ii) BT y2 + CT y3 + DT y4 = 0,y2 > 0, y3 ≥ 0 has a solution y2, y3, y4 but never both.

Theorem 5.1 Let us consider problem (MOP), and a feasible solution x∗, such that theconstraint qualification holds. Then, x∗ is a properly efficient solution of (MOP) if and onlyif, for each vector a ∈ R

n such that

∀i = 1, . . . , p, at∇fi(x∗) ≤ 0,

∃j ∈ {1, . . . , p}/at∇fj (x∗) < 0

Ann Oper Res (2008) 164: 115–126 123

Fig. 5 Properly efficient set of problem (9)

there exists at least one k ∈ {1, . . . ,m} such that:

gk(x∗) = 0, at∇gk(x∗) > 0.

Proof The proof of this theorem is analogous to that of Theorem 4.1, using Lemma 5.1. �

As an example, let us consider the following problem:

⎧⎨⎩

min(50x4 + 10y4,30(x − 5)2 + 100(y − 3)4

)s.t. (x − 2)2 + (y − 2)2 ≤ 1,

x + y ≥ 4(9)

which is problem (6) with two constraints. Figure 5 shows the properly efficient set for thisproblem.

In Fig. 5, the feasible set is the shadowed region, and the properly efficient set is formedby the arcs AB , BC and CD (without including the extreme points A and D). The curvejoining points (0, 0) and (5, 3) is the properly efficient set of problem (6), that is, of theunconstrained problem. As it can be seen, the arc BC is formed by the feasible points thatare properly efficient for the unconstrained set, while arcs AB and CD are formed by theprojections onto the frontier of the feasible set of the rest of the properly efficient solutionsof the unconstrained problem. The figure also shows the gradients of the objective functionsand of the active constraints in two points. The point lying on the arc AB is properly effi-cient, while the point lying on the upper top of the circumference is not. For the case withmore objective functions and/or variables, the comments made in Sect. 4 about the form ofthe properly efficient set also hold here.

124 Ann Oper Res (2008) 164: 115–126

6 A proper efficiency test

Based on Theorem 5.1, proved in Sect. 5, lets now propose a proper efficiency test for prob-lem (MOP). Given a feasible solution x∗, let us denote by J the set of indices correspondingto the active constraints in x∗:

J = {j ∈ {1, . . . ,m}/gj (x∗) = 0

}.

Next, let ε be any strictly positive number and let us consider the following problem:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

mina

p∑i=1

∇fi(x∗)ta

s.t. ∇fi(x∗)ta ≤ 0 (i = 1, . . . , p),

∇gj (x∗)ta ≤ 0 (j ∈ J ),

p∑i=1

∇fi(x∗)ta ≥ −ε.

(10)

Theorem 5.1 assures that, if x∗ is a properly efficient solution of (MOP), then there isno feasible vector for problem (10), a, such that ∇fk(x∗)ta < 0, for some k ∈ {1, . . . , p}.Therefore, in this case the optimal value of problem (10) is 0. Conversely, if x∗ is not aproperly efficient solution of (MOP), then there exists at least one feasible vector for (10),a, such that ∇fk(x∗)ta < 0, for some k ∈ {1, . . . , p}. Given that if a vector a satisfies thiscondition, any other multiple of a also does, in this case the optimal value of problem (10)is—ε. In fact, the last constraint is included in the problem so as to assure that the optimalsolution is not unbounded if x∗ is not properly efficient.

It must be noted that problem (10) is a linear problem, and 0 is always a feasible solution.Thus, this proper efficiency test is simple to implement and to carry out for any convexproblem, provided that the gradients of the functions are known.

As an example, let us consider the following problem:

⎧⎨⎩

min((x − 5)2 + (y − 5)2 + z2, (x − 6)2 + (y − 6)2 + (z − 6)2

)s.t. x + y + z ≤ 5,

x, y, z ≥ 0.

(11)

As seen in Fig. 6, the feasible set is the tetrahedron delimited by the axes and the planex + y + z = 5. The properly efficient set (indicated by a thick line) is the segment joiningpoints (5/2, 5/2, 0) and (5/3, 5/3, 5/3), which lies on the plane x + y + z = 5. Let us carryout the proper efficiency test on points A = (2,2,1), and B = (1,1,3). In both cases, theconstraint x + y + z ≤ 5 is active. Taking ε = 10, the test for A takes the following form:

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

mina

−14a1 − 14a2 − 8a3

s.t. − 6a1 − 6a2 + 2a3 ≤ 0,

− 8a1 − 8a2 − 10a3 ≤ 0,

a1 + a2 + a3 ≤ 0,

− 14a1 − 14a2 − 8a3 ≥ −10

whose optimal solution is a1 = a2 = a3 = 0, and the optimal value of the objective functionis 0. Therefore, (2, 2, 1) is properly efficient. On the other hand, the test for B takes the

Ann Oper Res (2008) 164: 115–126 125

Fig. 6 Feasible and properlyefficient sets of problem (11)

Fig. 7 Projections of the gradients at points A and B onto the plane x = y

form:

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

mina

−18a1 − 18a2

s.t. − 8a1 − 8a2 + 6a3 ≤ 0,

− 10a1 − 10a2 − 6a3 ≤ 0,

a1 + a2 + a3 ≤ 0,

− 18a1 − 18a2 ≥ −10

whose optimal solution is a1 = 1;a2 = −4/9;a3 = −5/9, and the optimal value of the ob-jective function is—10. Therefore, (3, 3, 1) is not a properly efficient solution of prob-lem (11).

As the gradients of the objective functions and the active constraint at both points areall contained in the plane x = y, Fig. 7 shows the projections of these gradients onto theaforementioned plane. For point A (on the left), it can be seen that the three gradients cannotbe included in the same open semi-plane. Therefore, no open semi-space in R

3 contains themall. For point B (on the right), an open semi-space determined by any plane that contains thedotted line (except the plane x = y itself) contains the three gradients.

126 Ann Oper Res (2008) 164: 115–126

7 Conclusions

In this paper, we have obtained a graphical characterization, in the decision space, of theproperly efficient solutions of a convex multiobjective problem. Namely, the proper effi-ciency of a feasible solution is characterized in terms of the relative position of the gradientsof the objective functions and the active constraints at the given point. This result allowsus to visualize the efficiency of a solution in the cases when a graphical representation ofthe feasible set is possible. In particular, this can be helpful for location problems. Severalexamples and figures have been used to illustrate the characterization.

On the other hand, based on this property, we have proposed a proper efficiency test forgeneral convex multiobjective problems. Given a feasible solution, its proper efficiency ischecked by solving a single linear optimization problem, which has always the vector 0as a feasible solution. Therefore, the test is easy to implement and carry out. This simpleefficiency test for convex problems can be an important support tool for certain interactiveor goal programming algorithms where the efficiency of the solution is not guaranteed.

Acknowledgements This research has been partially supported by the Andalusian Regional Ministry ofInnovation, Science and Enterprise (PAI group SEJ-445), and by the Spanish Ministry of Education andScience (Research Project MTM2006-01921). F. Ruiz’s research has also been supported by the SpanishMinistry of Education and Science (mobility program, PR2005-0212).

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