Aggregate Objective Functions and Pareto Frontiers

 

Aggrregate Objecti O ive Fun nctions and Paareto Frrontierss: Requuired Relation R nships and a Praactical Implica I ations Achille Messac Cyriaqque Puemii-Sukam Emanuuel Melachhrinoudis

Corresp ponding Author h.D. Achille Messac, Ph D Chair Distinguuished Proffessor and Department Mechannical and Aeerospace Enngineering Syracusse Universitty, 263 Linkk Hall Syracusse, New York 13244, USA U Email: m [email protected] Tel: (3155) 443-2341 Fax: (3155) 443-3099 https://m messac.expressions.syr.edu/

nformation n Biblioggraphical In Messac, A., E Sukkam, C. P., and Melach hrinoudis, “Aggregate “ O Objective Functions and Pareto Frontierrs: Required Relationship R ps and Practiccal Implicatiions,” Optim mization and Engineering E Journal, Kluwer Publishers, Vol. V 1, No. 2, 2 June 2000,, pp. 171-1888.

Optimization and Engineering, 1, 171–188, 2000 c 2000 Kluwer Academic Publishers. Manufactured in The Netherlands. °

Aggregate Objective Functions and Pareto Frontiers: Required Relationships and Practical Implications ACHILLE MESSAC∗ AND CYRIAQUE PUEMI-SUKAM Rensselaer Polytechnic Institute, Multidisciplinary Design and Optimization Laboratory, Troy, NY 12190-3590, USA [email protected]; [email protected] EMANUEL MELACHRINOUDIS Northeastern University, Multidisciplinary Design Laboratory, Boston, MA 02115, USA [email protected] Received September 9, 1999; Revised June 15, 2000

Abstract. This paper addresses the problem of capturing Pareto optimal points on non-convex Pareto frontiers, which are encountered in nonlinear multiobjective optimization problems in computational engineering design optimization. The emphasis is on the choice of the aggregate objective function (AOF) of the objectives that is employed to capture Pareto optimal points. A fundamental property of the aggregate objective function, the admissibility property, is developed and its equivalence to the coordinatewise increasing property is established. Necessary and sufficient conditions for such an admissible aggregate objective function to capture Pareto optimal points are derived. Numerical examples illustrate these conditions in the biobjective case. This paper demonstrates in general terms the limitation of the popular weighted-sum AOF approach, which captures only convex Pareto frontiers, and helps us understand why some commonly used AOFs cannot capture desirable Pareto optimal points, and how to avoid this situation in practice. Since nearly all applications of optimization in engineering design involve the formation of AOFs, this paper is of direct theoretical and practical usefulness. Keywords: physical programming, Pareto optimality, multiobjective optimization

1.

Introduction

Various problems in practical applications of optimization, particularly in engineering design optimization, can be recast in the form of a nonlinear multiobjective Problem P:  µ1 (x)  µ (x)   2   minimize µ(x) =   ..  , x∈D  .  

(P)

µm (x) where D = {x ∈ R n | h(x) = 0, g(x) ≤ 0, α ≤ x ≤ β}, h : R n → R r , g : R n → R s , α ∈ (R ∪ {−∞})n , β ∈ (R ∪ {+∞})n , m is the number of objectives, or criteria, m ≥ 2, and ∗ Corresponding

author.

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r and s the numbers of equality and inequality constraints, respectively. For any decision vector x = (x1 , . . . , xn ), a criteria vector µ = (µ1 , . . . , µm ) is defined according to the function µ : R n → R m . Z = {z ∈ R m | z = µ(x), x ∈ D} is the set of images of all points in D. D is called the feasible region in decision space and Z the feasible region in objective space; (µ1 (x), . . . , µm (x)) are the coordinates of the image of x in objective space. For the multiobjective problem P, it is highly improbable to have a single x ∗ which minimizes every µi simultaneously; therefore, the solution is defined in terms of Pareto optimality in the following sense (see Cohon, 1978): a feasible solution to a multiobjective programming problem is Pareto optimal (noninferior, nondominated) if there exists no other feasible solution that will yield an improvement in one objective without causing a degradation in at least one other objective. So, x ∈ D is Pareto optimal if there does not exist y ∈ D, whose criteria vector, q = µ(y), dominates the criteria vector of x, p = µ(x), i.e. q ≤ p, p 6= q. (For any vectors v and w, v ≤ w implies that vi ≤ wi ∀i). A point x ∈ D is locally Pareto optimal if there is a neighborhood of that point in D where it is Pareto optimal. Thus, a point which is (globally) Pareto optimal is also locally Pareto optimal. The definition of Pareto optimality is extended to criteria vectors as well. So, p ∈ Z in the above definition is Pareto optimal (respectively, locally Pareto optimal) in the objective space. In engineering applications, the designer is often presented with several Pareto optimal points, representing alternative designs, from which s/he selects the one that offers the best trade-off among multiple objectives. Theoretically, the more Pareto optimal points are available (ideally the whole Pareto set), the better the final choice will be. Alternatively, and more commonly, Pareto optimal points are instead implicity generated in the natural course of engineering computational optimization. This optimization generally involves forming an Aggregate Objective Function (AOF) (or, some functional aggregation of the many conflicting criteria). Implicit in this process is the assumption that this AOF has the ability to indeed yield all the potentially useful/desirable design solutions. Unfortunately, the correctness of this assumption is more the exception than the norm. Since every Pareto point is of potential usefulness to the designer, any AOF should have the ability to yield all possible Pareto optimal points. An important set of related question follows: Can a given AOF yield a given set of Pareto optimal points? If not, why not and what can be done about it? It is these questions that the substance of this paper addresses. Unfortunately, it is not easy in practice to form the correct AOF, whether it is for the purpose of engineering design optimization or for that of generating Pareto optimal points for a representation of the Pareto frontier. The most common AOF structure is the weighted-sum approach, which involves forming a linear combination of objectives—minimized subject to the problem constraints. The aggregate objective function should have two characteristics. First, it should represent the designer’s preferences and objectives. Second, since every Pareto optimal point is of potential use to the designer, the AOF should therefore be able to capture every such point. The first characteristic is the subject of previous publications (Messac, 1996; Messac, 2000; Messac and Chen, 2000; Messac and Wilson, 1998; Messac, 1998). This article deals with issues related to the second characteristic; previous attempts can be found in publications (Athan and Papalambros, 1996; Chen et al., 1998; Das and Dennis, 1997; Das and Dennis, 1998; Koski, 1985; Messac et al., 2000; Messac et al., 1999; Osyczka, 1984; Rakowska et al., 1991). In these publications, the inability of the weighted-sum of the objectives to generate

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Pareto optimal points that lie on non-convex Pareto frontiers is well documented. What are then the properties or the mathematical conditions that an aggregate objective function should satisfy in order to capture Pareto optimal points on a convex or non-convex Pareto frontier? This article addresses this question directly, first, by developing a fundamental property (the admissibility property) that enables any AOF to capture points that are at least locally Pareto optimal. Second, by establishing necessary and sufficient conditions for the Pareto capturability of such admissible AOFs, on a given Pareto frontier. These conditions are then reduced to practical first and second order conditions, given reasonable differentiability requirements. An important practical implication is that an AOF whose curvature can be increased by using its free parameters (for example objective function exponents) can be made accordingly more effective at capturing Pareto optimal points, a property the weighted-sum of the objectives lacks. The paper is structured as follows: in Section 2, the admissible aggregate objective function is defined and its equivalence to the coordinatewise increasing function of the objectives is established; in Section 3, the conditions for the capturability of Pareto optimal points are presented; numerical examples are given in Section 4. The paper ends with a conclusion in Section 5. 2.

Admissible aggregate objective functions

Let f (µ) : R m → R, be a function that aggregates the objectives into a scalar—in a way that involves using designer-chosen parameters, hereby called free parameters. Such an aggregate objective function is generally constructed for the purpose of performing computational engineering design optimization, or of generating Pareto optimal points by varying the values of these parameters. For a prescribed set of parameter values, the AOF minimization Problem in Objective space (APO) is solved to provide a Pareto optimal point: min{ f (µ) | µ ∈ Z }.

(APO)

The ability of AOFs to generate Pareto optimal points emanates from a fundamental property of these functions. For utility functions, which are aggregate objective functions studied extensively in the literature, it is the coordinatewise increasing property. A coordinatewise increasing utility function implies that all objectives are in maximization form and that monotonicity holds for each one of them (regardless of the values at which the other objectives are held constant) (Steuer, 1989). In our multiobjective Problem P, where both the individual objectives and their aggregate objective function are to be minimized (less is better than more), the coordinatewise increasing property of f (µ) also implies monotonicity. In this section, the admissibility property is introduced as the fundamental property of aggregate objective functions. Unlike the coordinatewise increasing property, the admissibility property is based on the concept of Pareto optimality and thus it is more intuitive and general, whether the aggregate objective function is maximized or minimized. In addition, the admissibility property is more revealing of the shape of the aggregate objective contours, giving them a more insightful interpretation in generating Pareto optimal points. Nonetheless, an equivalence of the two properties is established, i.e., any admissible aggregate objective function is coordinatewise increasing in the objectives and vice-versa.

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Let c be a real constant. Using the aggregate objective function, define a set K (c) = {µ ∈ R m | f (µ) ≥ c} having boundary S(c) = {µ ∈ R m | f (µ) = c}. Having assembled the tools needed, the definition of admissible aggregate objective function and additional supporting definitions can be given. We assume in the rest of our work that the set S(c) is nonempty. Definition 1. f (µ) is an admissible aggregate objective function if S(c) is the Pareto optimal set of K (c), where c is any real constant. Definition 2. Let R m be the set of m-dimensional real numbers. T ⊆ R m , T 6= ∅, is called unconnected if there exist open sets O1 , O2 ⊆ R m with T ⊆ O1 ∪ O2 , O1 ∩ T 6= ∅, O2 ∩ T 6= ∅, and T ∩ O1 ∩ O2 = ∅. Hence, a set T is called connected if it is not unconnected. Definition 3. f (µ) is a locally admissible aggregate objective function if there exists a connected subset T of R m , different from a one-element set, where it is admissible. Definition 4. f (µ) is a coordinatewise increasing function of the criterion µi , i = 1, . . . , m, if it is a strict monotonically increasing function of each criterion µi , i = 1, . . . , m, regardless of the values at which the other criteria are held constant (Steuer, 1989). Figure 1 shows the contour of an admissible aggregate objective function in the simple case of two criteria. Given a criteria vector µ, its aggregate objective function value f (µ), represents the designer’s overall disutility, cost or penalty for such a solution. The lower the penalty, the more preferred the solution. A contour f (µ) = c contains criteria vectors having same preference. Another interpretation of the admissibility property is therefore that no point of the contour f (µ) = c is dominated by another point of the same contour. The reason being that a rational decision-maker would not ascribe the same penalty to a dominated criteria vector and to the criteria vector that dominates it. In figure 1, this means that for any given point of the contour f (µ) = c, the dominated rectangular cone C( p) should not contain any other point of the contour.

Figure 1.

The admissibility property in a simple biobjective case.

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In the next theorem, the equivalence between admissibility and the coordinatewise increasing property is established. Subsequent theorems therefore are proved using either one of the two properties, whichever is more convenient to use. Theorem 1. An aggregate objective function f (µ) is admissible if and only if it is a coordinatewise increasing function of the criterion µi , i = 1, . . . , m. Proof: (1-1) Assume that f (µ) is a coordinatewise increasing function of the criterion µi , i = 1, . . . , m; we wish to prove that S(c) = {µ ∈ R m | f (µ) = c} is the Pareto optimal set of K (c) = {µ ∈ R m | f (µ) ≥ c} for each real c. By contradiction, consider a real value c, and assume that S(c) is not the Pareto optimal set of K (c); then there exists p ∈ S(c) which is dominated by a point q, q ∈ K (c), i.e. f (q) ≥ c. For such a point, q, we have q ≤ p, q 6= p. Since f (µ) is coordinatewise increasing, f (q) < f ( p). Or, p ∈ S(c), so f ( p) = c; we have f (q) ≥ c and f (q) < c, which is impossible. Therefore, S(c) is Pareto optimal for each real constant c and by Definition 1, f (µ) is an admissible aggregate objective function. (1-2) Assume that f (µ) is admissible; we wish to prove that it is coordinatewise increasing. By contradiction, suppose that there exist two points a, b ∈ R m , b ≤ a, b 6= a and f (b) ≥ f (a). Let f (a) = c; since f (µ) is admissible, S(c) is the Pareto optimal set of K (c) and therefore a is Pareto optimal; we have f (b) ≥ f (a), i.e. f (b) ≥ c; so, b ∈ K (c), b ≤ a, b 6= a and a is Pareto optimal in K (c), which is impossible. Thus, f (µ) is coordinatewise increasing. 2 Since the admissibility property can also hold only locally, Corollary 1 follows directly from the above theorem. Before giving Corollary 1, we have the following definition of local optimum. Definition 5. A point µ∗ ∈ Z is a local optimum of APO if there exists a neighborhood V of µ∗ in Z where it is an optimum of APO. The relationship between local optimum and locally admissible aggregate objective function is that if a point µ∗ ∈ Z is a local optimum of APO in a connected neighborhood V of µ∗ , V ⊆ Z , and the function f (µ) is admissible in V , then f (µ) is a locally admissible aggregate objective function. Corollary 1. An aggregate objective function f (µ) is locally admissible if and only if it is a locally coordinatewise increasing function of the criterion µi , i = 1, . . . , m. The aggregate objective function is constructed for the purpose of generating Pareto optimal points, either through computational optimization, or through explicit generation. So, its quality is measured by its ability to capture such points. For the sake of the following development, a formal definition of the term “capturable” is provided. Definition 6. A point µ∗ ∈ Z is capturable by an aggregate objective function f (µ) if it is a local optimum of Problem APO.

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Suppose an admissible aggregate objective function captured a point. What is the assurance that it is Pareto optimal? The following important theorem asserts that an admissible aggregate objective function captures points that are at least locally Pareto optimal. Theorem 2. Given an admissible aggregate objective function f (µ), if µ∗ ∈ Z is captured by f (µ), then µ∗ is locally Pareto optimal. Proof: Since µ∗ is capturable, it is an optimum of APO in a neighborhood V of µ∗ , V ⊆ Z . Let f (µ∗ ) = c; therefore f (µ) ≥ c for every µ ∈ V , which implies that V ⊆ K (c). Since f (µ) is an admissible aggregate objective function, S(c) is the Pareto optimal set of K (c) and we have µ∗ ∈ S(c). Therefore, µ∗ is Pareto optimal in K (c). It follows that µ∗ is Pareto optimal in V . 2 The next step is to prove the converse of Theorem 2, which is done in Theorem 3. The latter provides the assurance that any Pareto optimal point can be captured if we have the appropriate aggregate objective function. Given a Pareto optimal point in a multicriteria maximization problem, Soland (1979) proved that there exists a coordinatewise increasing utility function, which can capture this point when the utility function is being maximized subject to the problem constraints. For a multicriteria minimization problem, the same result can be proved, but the same proof cannot be used. This is because the negative of the admissible aggregate objective function used for the maximization problem cannot be taken since it will not be coordinatewise increasing. The following theorem deals with minimization of the aggregate objective function. So, without any restriction on convexity and continuity we have: Theorem 3. If µ∗ ∈ Z is locally Pareto optimal, then there exists a locally admissible aggregate objective function that can capture µ∗ . Proof: Since µ∗ ∈ Z is locally Pareto optimal, there exists a neighborhood V of µ∗ , V ⊆ Z where it is Pareto optimal. Without loss of generality, we assume that V is connected and open; therefore, there exists a positive real constant ε such that for each µ ∈ V , having coordinates (µ1 , . . . , µm ), we have |µk − µ∗k | < ε for k = 1, . . . , m, or equivalently ε for each k = 1, . . . , m. Consider the aggregate objective function −ε < µk − µ∗k < P defined as f (µ) = m k=1 f k (µk ), where  ∗  µk − µk −ε if µk ≤ µ∗k f k (µk ) = m  1 − exp(µ∗k − µk ) if µk > µ∗k We wish to prove that f (µ) is coordinatewise increasing in V . Let µ1 and µ2 ∈ V , have coordinates (µ1 , . . . , µk−1 , µ1k , µk+1 , . . . , µm ) and (µ1 , . . . , µk−1 , µ2k , µk+1 , . . . , µm ), respectively, such that µ1k < µ2k ; we have f (µ1 ) − f (µ2 ) = f k (µ1k ) − f k (µ2k ). Further let 0 = f k (µ1k ) − f k (µ2k ). We wish to prove that 0 ≤ 0. The five cases to examine are: µ1k < µ2k < µ∗k , µ1k < µ∗k < µ2k , µ∗k < µ1k < µ2k , µ1k = µ∗k and µ2k = µ∗k

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1. µ1k < µ2k < µ∗k ; 0 =

µ1k −µ2k m µ1k −µ∗k m

2. µ1k < µ∗k < µ2k ; 0 = − ε − (1 − exp(µ∗k − µ2k )) ∗ 1 2 ∗ 3. µk < µk < µk ; 0 = exp(µk − kk2 ) − exp(µ∗k − µ1k ) 4. µ1k = µ∗k ; 0 = −ε − (1 − exp(µ∗k − µ2k )) 5. µ2k = µ∗k ; 0 =

µ1k −µ∗k m

So in all the cases the condition 0 ≤ 0 is verified, and f (µ) is coordinatewise increasing in V and by Corollary 1 locally admissible. We will now prove that µ∗ is an optimum of Problem APO in V . Without loss of generality, we assume that: µk < µ∗k µk = µ∗k µk > µ∗k

for k ∈ {1, . . . , r }, for k ∈ {r + 1, . . . , s}, for k ∈ {s + 1, . . . , m}.

If r = m or s = m, then µ is feasible and dominates µ∗ , which contradicts the Pareto optimality of µ∗ . So m > s, i.e. m − 1 ≥ s; thus m − s ≥ 1. We have m ≥ r , so 1 ≥ mr and it follows that m − s ≥ mr , i.e. m − (s + mr ) ≥ 0. Therefore: f (µ) − f (µ∗ ) ¶ r µ s m X X X µk − µ∗k −ε + (−ε) + (1 − exp(µ∗k − µk )) − (−εm). = m k=1 k=r +1 k=s+1 Considering −ε < µk − µ∗k < ε, k = 1, 2, . . . , m, we have: µ

f (µ) − f (µ∗ ) ≥ −

rε m

¶ − r ε − (s − r )ε +

m X

(1 − exp(µ∗k − µk

¢¢

+ (εm),

k=s+1

or equivalently, µ µ ¶¶ m X r (1 − exp(µ∗k − µk )). ε+ f (µ) − f (µ∗ ) ≥ m − s + m k=s+1 Since m − (s + mr ) ≥ 0, and for each k ∈ {s + 1, . . . , m}, (1 − exp(µ∗k − µk )) ≥ 0, we have f (µ) − f (µ∗ ) ≥ 0 and therefore f (µ) ≥ f (µ∗ ). Note that the case µ ≥ µ∗ was not included above because f (µ) is coordinatewise increasing. Thus µ∗ is an optimum of Problem APO in V and by Definition 6 capturable by f (µ). Hence, there exists a locally 2 admissible aggregate objective function that can capture µ∗ .

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Pareto capturability of admissible aggregate objective functions

In the previous section, the concept of admissibility property of aggregate objective functions was introduced and its equivalence to the coordinatewise increasing property established. The proof that such functions capture points that are at least locally Pareto optimal has also been provided. Although admissibility is a necessary property of the aggregate objective function, it is not sufficient for capturing Pareto optimal points. In this section, necessary and sufficient conditions of an admissible aggregate objective function to capture a given Pareto optimal point are derived. These conditions provide the tools for designing appropriate aggregate objective functions, which play a pivotal role in the process of computational engineering design optimization. Considering the optimization problem P, the first interest is in the Pareto optimal decision vectors. So, in the rest of this paper, it is assumed that each µi , i = 1, . . . , m, is defined from D, the feasible region in the decision space. Let us introduce next the notion of an open transformation (see Marlow, 1978), which is used in the remainder of the paper. Definition 7. Let X and Y be two spaces; an open transformation h from X to Y maps open sets of X into open sets of Y . An open transformation is equivalent to mapping each neighborhood of a point e of X into a neighborhood of its image h(e) in Y . It is assumed that each µi , i = 1, . . . , m, is an open and continuous transformation. Let g : D → R be the function defined by g(x) = f (µ1 (x), . . . , µm (x)) = f (µ(x)), for each x ∈ R n , and APD be the Problem that minimizes g(x) in Decision space: min{g(x) | x ∈ D},

(APD).

Definition 8. a ∈ D is decision-space capturable by an aggregate objective function g(x) if it is a local optimum of Problem APD. The following proposition establishes the equivalence of decision-space capturability, enabled by g(x), and (objective-space) capturability, enabled by f (µ). Proposition 1. Given an aggregate objective function f (µ), a ∈ D is decision-space capturable by g(x), if and only if p = µ(a) is capturable by f (µ). Proof: (1-1) Assume that a ∈ D is decision-space capturable by g(x); we wish to prove that p = µ(a) is capturable by f (µ). Since a is decision-space capturable by g(x), it is an optimum of Problem APD in a neighborhood V of a, V ⊆ D; so µ(V ) = {µ(x) | x ∈ V } is a neighborhood of µ(a) in the objective space. We have V ⊆ D, then µ(V ) ⊆ Z . For each x ∈ V , g(a) ≤ g(x); therefore f (µ(a)) ≤ f (µ(x)). So f ( p) ≤ f (q) for each q ∈ µ(V ); thus p is an optimum of [APO] in µ(V ), and by Definition 6 capturable by f (µ). (1-2) Assume that p = µ(a) is capturable by f (µ); we wish to prove that a is decisionspace capturable by g(x). Since p = µ(a) is capturable by f (µ), it is an optimum of [APO] in a neighborhood W of p, W ⊆ Z . Since µ is continuous, µ−1 (W ) = {x ∈ D | µ(x) ∈ W } is a neighborhood of a in D; we wish to prove that a = arg min{g(x) | x ∈ µ−1 (W )}, i.e.

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g(a) = min{g(x) | x ∈ µ−1 (W )}. By contradiction, suppose that there exists b ∈ µ−1 (W ), with g(b) < g(a). Since b ∈ µ−1 (W ), then µ(b) ∈ W ; and since g(b) < g(a), f (µ(b)) < f (µ(a)), i.e., f (µ(b)) < f ( p), which contradicts the fact that p = arg min{ f (µ), µ ∈ W }. i.e. f ( p) = min{ f (µ), µ ∈ W }. So, for each x ∈ µ−1 (W ), we have g(a) ≤ g(x); and therefore a is a minimum of Problem APD in µ−1 (W ), and by Definition 8 decision-space capturable by g(x). 2 Let’s assume that the Pareto hypersurface and the AOF hypersurface verify the theorem of implicit functions (see Greenberg, 1998; Marlow, 1978). That is, these hypersurfaces satisfy the hypothesis of the theorem of implicit functions, which can therefore be applied. Since the Pareto hypersurface can in general be represented by φ(µ1 , . . . , µm ) = 0, it can be rewritten in the form µi ≡ µif (µi− ), where µi− = (µ1 , . . . , µi−1 , µi+1 , . . . , µm ). Let a ∈ D and p = µ(a) be Pareto optimal; the AOF hypersurface at point p is represented by f (µ) = f ( p) and can be rewritten in the form µi ≡ µi0 (µi− ). So, at p the Pareto hypersurface and the AOF hypersurface have the same value in terms of their ith coordinates, i.e. µi0 (µi− ) = µif (µi− ). Let 0(µi− ) = µif (µi− ) − µi0 (µi− ). We have 0( p− ) = 0. The following theorem gives the necessary and sufficient condition for a point in the decision space to be captured by a coordinatewise increasing function. Theorem 4. Assume that f (µ) is a coordinatewise increasing function of µ; a is decisionspace capturable by g(x) if and only if there exists a neighborhood of p = µ(a) in the objective space Z where the function 0(µi− ) is nonnegative. Proof: (4-1) Assume that a is decision-space capturable by g(x); therefore it is an optimum of Problem APD in a neighborhood V of a, V ⊆ D; µ(V ) is a neighborhood of p = µ(a) in the objective space and according to Proposition 1, p is an optimum of Problem APO in µ(V ). We wish to prove that 0(µi− ) ≥ 0 in µ(V ), i.e. µif ≥ µi0 in µ(V ). By contradiction, assume that there exists a point q such that µif (q− ) < µi0 (q− ) with q ∈ µ(V ), where q and q− have coordinates (q1 , . . . , qi−1 , qi , qi+1 , . . . , qm ) and (q1 , . . . , qi−1 , qi+1 , . . . , qm ), respectively. Let us consider the points q1 and q2 whose coordinates are (q1 , . . . , qi−1 , µif (q− ), qi+1 , . . . , qm ) and (q1 , . . . , qi−1 , µi0 (q− ), qi+1 , . . . , qm ), respectively. The point q2 is on the aggregate objective hypersurface whose equation is f (µ) = f ( p), so we have f (q2 ) = f ( p); and the point q1 is on the Pareto hypersurface. Since q is in µ(V ), q1 is in µ(V ), and since p = arg min{ f (µ), µ ∈ µ(V )}, i.e. f ( p) = min{ f (µ), µ ∈ µ(V )}, we have f (q1 ) ≥ f ( p). So, f (q1 ) ≥ f (q2 ). But µif (q− ) < µi0 (q− ) implies that q1 ≤ q2 , q1 6= q2 . Since f (µ) is coordinatewise increasing, so f (q1 ) < f (q2 ). We have f (q1 ) ≥ f (q2 ) and f (q1 ) < f (q2 ), which is impossible. So, for each q ∈ µ(V ), µif (q− ) ≥ µi0 (q− ) and it follows that µif ≥ µi0 in µ(V ). (4-2) Assume that there exists a neighborhood W of p in objective space where µif ≥ µi0 . We note that µ−1 (W ) = {x ∈ D | µ(x) ∈ W } is a neighborhood of a in D. We wish to prove that a is an optimum of Problem APD in µ−1 (W ). By contradiction, suppose that there exists b ∈ µ−1 (W ), with g(b) < g(a), (or, (i)). Since b ∈ µ−1 (W ), we have µ(b) ∈ W . Let s = µ(b), therefore s and s− have coordinates (µ1 (b), . . . , µi−1 (b), µi (b), µi+1 (b), . . . , µm (b)) and (µ1 (b), . . . , µi−1 (b), µi+1 (b), . . . , µm (b)), respectively. Consider the points s1 and s2 whose coordinates are (µ1 (b), . . . , µi−1 (b), µif (s− ), µi+1 (b), . . . , µm (b)) and (µ1 (b), . . . ,

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µi−1 (b), µi0 (s− ), µi+1 (b), . . . , µm (b)), respectively. Then s1 lies on the Pareto hypersurface and s2 lies on the AOF hypersurface whose equation is f (µ) = f ( p); so, f (s2 ) = f ( p). By hypothesis, µif ≥ µi0 in W ; hence, µif (s− ) ≥ µi0 (s− ), and it follows that s1 ≥ s2 . Since f (µ) is coordinatewise increasing, we have f (s1 ) ≥ f (s2 ), i.e. f (s1 ) ≥ f ( p), (or, (ii)). Since b is a feasible point in decision space, s = µ(b) is feasible in objective space. We note that s and s1 have coordinates (µ1 (b), . . . , µi−1 (b), µi (b), µi+1 (b), . . . , µm (b)) and (µ1 (b), . . . , µi−1 (b), µif (s− ), µi+1 (b), . . . , µm (b)), respectively. If we assume that µi (b) < µif (s− ), then s ≤ s1 , s 6= s1 , which implies that s dominates s1 , which is impossible because s1 is Pareto optimal. So µi (b) ≥ µif (s− ) and s ≥ s1 . Since f (µ) is coordinatewise increasing, therefore f (s) ≥ f (s1 ), (or, (iii)). Statements (ii) and (iii) imply f (s) ≥ f (s1 ) ≥ f ( p). But from (i), we have g(a) > g(b), thus f (µ(a)) > f (µ(b)), i.e. f ( p) > f (s). Hence, we have f ( p) > f (s) ≥ f (s1 ) ≥ f ( p), which is impossible. Therefore, for each point b ∈ µ−1 (W ), we have g(b) ≥ g(a). It follows that a is an optimum of Problem APD in µ−1 (W ), and by Definition 8 decision-space capturable by g(x). 2 Proposition 1 and Theorem 4 link the decision space and the objective space. Geometrically, in the case when m = 2, it follows from Theorem 4 that at a capturable point, the curves representing the Pareto frontier and the aggregate objective function are tangent and there is a neighborhood of the point where the Pareto frontier is above the aggregate contour. The following remarks and corollaries can be derived from Theorem 4. Remark 1. Applying Proposition 1, “a is decision-space capturable by g(x)” in Theorem 4 can be replaced by “ p = µ(a) is capturable by f (µ)”; it follows from Theorem 1 that “ f (µ) is a coordinatewise increasing function of µ” in Theorem 4 can be replaced by “ f (µ) is an admissible aggregate objective function”. Corollary 2. Assume that f (µ) is a locally coordinatewise increasing function of µ in O and p = µ(a) is locally Pareto in a neighborhood E, with E ⊆ O. The point p is capturable by f (µ) if and only if there exists T1 , a neighborhood of p, with T1 ⊆ E and 0(µi− ) is nonnegative in T1 . Proof:

In Theorem 4, consider only E in place of the objective space Z .

2

Corollary 2 gives us necessary and sufficient conditions for a locally Pareto point to be captured. However, these conditions are not readily used to establish Pareto point capturability. Assuming that the aggregate objective function and the Pareto optimal frontier satisfy certain differentiability requirements, these general conditions can be converted into more useful conditions in the following Corollary 3. These are first and second order capturability conditions, which provide clear insights for designing suitable aggregate objective functions. Corollary 3. Assume that f (µ) is locally coordinatewise increasing for the criteria in O, that p = µ(a) is locally Pareto optimal in a neighborhood E, E ⊆ O and that the functions corresponding to the Pareto hypersurface and the aggregate hypersurface are twice differentiable. Under these assumptions: (a) If p is capturable, then ∇ 2 0(µi− )| p is positive semidefinite. (b) If ∇0(µi− )| p = 0 and ∇ 2 0(µi− )| p is positive semidefinite, then p is capturable.

AGGREGATE OBJECTIVE FUNCTIONS

181

Proof: (a) Assume that p is capturable; we wish to prove that ∇ 2 0(µi− )| p is positive semidefinite. We have 0( p− ) = 0, and since p is capturable, by Corollary 2, there exists T1 , a neighborhood of p, with T1 ⊆ E and 0(µi− ) ≥ 0 in T1 . The expansion of 0 in T1 about p yields: ¯T 1 0(µi− ) ≈ 0( p− ) + ∇0(µi− ) ¯ p (µi− − p− ) + (µi− − p− )T 2 × ∇ 2 0(µi− ) | p (µi− − p− ). We can prescribe the free parameters of the objective function such that ∇0(µi− )| p = 0. Therefore: 0(µi− ) ≈

1 (µi− − p− )T ∇ 2 0(µi− ) | p (µi− − p− ). 2

Since 0(µi− ) ≥ 0 in T1 , it follows that ∇ 2 0(µi− )| p is positive semidefinite. (b) Assume that ∇0(µi− )| p = 0 and ∇ 2 0(µi− )| p is positive semidefinite. We wish to prove that p is capturable. The expansion of 0 in E about p yields: 0(µi− ) ≈ 0( p− ) + ∇0(µi− ) |Tp (µi− − p− ) 1 + (µi− − p− )T ∇ 2 0(µi− ) | p (µi− − p− ). 2 Since 0( p− ) = 0 and ∇0(µi− )| p = 0, we have: 0(µi− ) ≈

1 (µi− − p− )T ∇ 2 0(µi− ) | p (µi− − p− ). 2

Because ∇ 2 0(µi− )| p is positive semidefinite, it follows that 0(µi− ) ≥ 0 in E, and by Corollary 2, p is capturable. 2 Remark 2. If ∇0(µi− )| p = 0 and ∇ 2 0(µi− )| p = 0, then in the neighborhood of p considered for the expansion, we have 0(µi− ) = 0; and therefore all the points of that neighborhood which lie on the Pareto hypersurface are capturable. Our attention is now turned to the most Pmof aggregate objective functions, Pm popular form wi µi with i=1 wi = 1. This linear aggregate the weighted-sum form, e.g. f (µ) = i=1 objective function is clearly globally admissible. Its simplicity makes it very effective in generating Pareto optimal solutions in linear multiobjective problems. However, as shown in the following corollary, it has significant limitations in capturing Pareto optimal solutions in nonlinear multiobjective problems, which present a significant deficiency in the case of engineering design optimization. Corollary 4. Assuming that the aggregate objective function is the weighted-sum of the objectives, p is capturable if and only if it lies on a convex part of the Pareto hypersurface.

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Proof: (4-1) Assume that p is capturable, we wish to prove that it lies on a convex part of the Pareto hypersurface. We have ∇ 2 0(µi− )| p = ∇ 2 µif (µi− )| p − ∇ 2 µi0 (µi− )| p ; since the aggregate objective function is of the weighted-sum kind, ∇ 2 µi0 (µi− ) = 0. By Corollary 3(a), ∇ 2 0(µi− )| p is positive semidefinite, i.e. ∇ 2 µif (µi− )| p is positive semidefinite, and it follows that p lies on a convex part of the Pareto hypersurface. (4-2) Assume that p lies on a convex part of the Pareto hypersurface; we wish to prove that p is capturable. Since the aggregate objective function is of the weighted-sum kind, ∇ 2 µi0 (µi− ) = 0. In addition we can prescribe the weights such that ∇0(µi− )| p = 0. Therefore we have ∇0(µi− )| p = 0 and ∇ 2 0(µi− )| p is positive semidefinite. It follows from Corollary 3(b) that p is capturable. 2 Corollary 3 states that the Hessian of the function defined as the difference between the Pareto hypersurface and the AOF hypersurface is positive semidefinite at capturable points. This means that, in a neighborhood of a capturable point, that function is convex. Corollary 4 defines limitations of the weighted-sum approach as capturing only points on convex parts of the Pareto frontier. An important practical implication is that an objective function whose curvature can be increased by adjusting its free parameters (e.g. exponents) can be made accordingly more effective at capturing Pareto optimal points, which is not the case with the weighted-sum AOF approach. Also, by continually increasing the curvature of an AOF, we are able to capture more and more points—and in the limit, all Pareto points. This is illustrated in the following section.

4.

Numerical examples

In the previous section, the conditions for capturing a Pareto optimal point in objective space were given. In practical applications, the designer may wish to obtain a Pareto optimal point in decision space. Proposition 1 establishes the equivalence between decision-space capturability and (objective-space) capturability. For the purpose of illustration, the following three examples deal with the simple case of two criteria µ1 and µ2 in objective space. Example 1. ·

Minimize Subject to

µ1 µ= µ2

¸

µ22 ≥1 9 µ41 + µ42 ≥ 16 µ21 +

µ31 + µ32 ≥ 1 27 0 ≤ µ1 ≤ 2.9 0 ≤ µ2 ≤ 2.9

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Here we have µi− = µ1 ; the gradient and the Hessian become the first and the second derivatives, 0 0 (µ1 ) and 0 00 (µ1 ), respectively. As shown in figure 2(a), the feasible region in objective space is bounded by the arcs AB, BC, CD and the line segments AE, ED, where A(0.256, 2.9), B(0.743, 1.990), C(1.980, 0.893); D(2.9, 0.459) and E(2.9, 2.9) are the corner points. As an aggregate objective function, the weighted-compromise function is used, J = µs1 + bµs2 , with parameters b > 0 and s, a positive integer. This aggregate objective function is chosen because it offers the required flexibility to change its curvature and orientation by adjusting the free parameters for capturing Pareto optimal points; in addition, it is locally admissible in the first quadrant and globally admissible for s odd. The parameters s and b are varied and the software Matlab (1984) is used to capture Pareto optimal points by solving the problem that minimizes the aggregate objective function, subject to the above constraints. Captured points are depicted by asterisks in figure 2(b)–(e). For s = 1, the aggregate objective function becomes the weighted-sum of the objectives, and only the Pareto optimal corner points A, B, C, D are captured (see figure 2(b)). Other Pareto optimal corner points lie on the concave part of the Pareto frontier and could not be captured, according to Corollary 4. For s = 2, the corner points A, B, C, D, are captured as in the previous case. In addition, all the Pareto optimal points between A and B are captured; this is done with the value of 1/9 for the parameter b, a degenerate case (see figure 2(c)). For the arc AB, 0 0 (µ1 ) = ; 0 0 (µ1 ) = 0 yields b = 1/9. For this value of b, 0 00 (µ1 ) = 0 and therefore −µ1 −1+9b bµ2 Corollary 3(b) is satisfied. 0 00 (µ1 ) = 0 also indicates that the curvature of the constantAOF contour is the same as that of the Pareto hypersurface, thus all points of the Pareto hypersurface are captured, according to Remark 2. No point of the arc BC is captured; this −µ22 +µ21 b is because (for2that part of the Pareto frontier) we have 0 0 (µ1 ) = −µ1 bµ ; 0 0 (µ1 ) = 0 3 2 µ2 2 4 4 2 00 implies b = µ2 ; therefore 0 (µ1 ) = − µ7 (µ1 + µ2 )µ1 , which is negative for all points of 1 2 the arc BC. So, even though the gradient condition is satisfied, the Hessian is not positive semidefinite, thus violating Corollary 3(b). The same holds true for Pareto optimal points of −27µ2 1 µ2 bµ1bµ ; 0 0 (µ1 ) = 0 implies b = µ271 µ2 , and it follows the arc CD, where 0 0 (µ1 ) = − 27 2 2 3 3 µ +27µ 1 µ1 1 µs 2 , thus 0 00 (µ1 ) < 0. that 0 00 (µ1 ) = − 729 2 For s = 3, corner Pareto optimal points A, B, C, D are captured. All points on the arcs AB and CD are captured. The points captured for different ranges of the parameter b are depicted in Table 1, and provided approximately (see figure 2(d)). 1 −µ2 ; No interior point of arc BC is captured; this is because, 4for 4this arc, 0 0 (µ1 ) = −µ21 bµbµ 3 2 µ2 2 µ1 +µ2 0 00 00 0 (µ1 ) = 0 implies b = µ1 and hence 0 (µ1 ) = −µ1 7 ; 0 (µ1 ) < 0 and Corollary 3 (b) is not satisfied. For s = 4, all the Pareto optimal points are captured and we have the Table 2: The following useful observations shall be understood in terms of Corollary 3(b). (1) For µ2 9µ2 +µ2 µ2 −9bµ2 arc AB, 0 0 (µ1 ) = µ1 1 bµ3 2 ; 0 0 (µ1 ) = 0 yields b = 9µ12 , and therefore 0 00 (µ1 ) = 18 1µ3 2 , 2

2

2

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Figure 2. Manipulating the aggregate objective function to capture Pareto points: (a) feasible region; (b) partial capture of Pareto points for s = 1; (c) partial capture of Pareto points for s = 2; (d) partial capture of Pareto points for s = 3; (e) full capture of Pareto points for s = 4.

185

AGGREGATE OBJECTIVE FUNCTIONS Table 1.

Points captured for s = 3 in figure 2(d). b

Table 2.

Points/range captured

(0, 1]

A, B and arc AB

[1, 3]

B and C

[3, 26]

C

[26, 28]

C, D and arc CD

[28, ∞)

D

Points captured for s = 4 in figure 2(e). b

Points/range captured

(0, 0.9]

A, B and arc AB

[0.9, 1.1]

B, C and arc BC

[1.1, 59]

C

[59, 171]

C, D and arc CD

[171, ∞)

D

thus 0 00 (µ1 ) ≥ 0. (2) For arc BC, 0 0 (µ1 ) = −µ31 b−1 ; 0 0 (µ1 ) = 0 implies b = 1 and thus bµ3 00

1 µ 3645 1

µ31 +27µ32

2

1 2 −bµ2 +27µ1 µ ; 0 0 (µ1 ) 27 1 bµ32 hence 0 00 (µ1 ) ≥ 0.

0 00 (µ1 ) = 0; for arc CD, 0 0 (µ1 ) =

= 0 implies b =

27µ1 , µ2

and

, therefore 0 (µ1 ) = µ2 For s ≥ 5, all Pareto optimal points are captured. So, by increasing the value of s, the curvature of the aggregate objective function is increased, which enables it to capture different Pareto optimal arcs. For a fixed value of s, by varying b, the orientation of the aggregate objective contour is changed to capture different Pareto optimal points on a given arc. 0 00 (µ1 ) = 0 is a special case where a single value of b can be used to capture all Pareto optimal points of an arc. Dominated points are not captured since the chosen aggregate objective function is admissible. This is illustrated in figure 2 where no point of the arcs AE, ED and no interior points are captured. In the weighted-sum approach, by varying the criteria weights, only the orientation of the aggregate objective contour is changed, but not its curvature. It is for this reason that such an aggregate objective function cannot capture points on concave parts of the Pareto frontier. The second example illustrates this fact; that is, the curvature of the aggregate objective function must be increased in order to capture Pareto optimal points. Example 2. ·

µ1 µ2

¸

Minimize

µ=

Subject to

(µ1 − 1)4 + (µ2 − 3)4 ≥ 16 1.6 ≤ µ1 ≤ 2.9798 3.8933 ≤ µ2 ≤ 6

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Figure 3. Increasing the curvature of the aggregate objective function to capture Pareto points: (a) Pareto points captured for s = 1, . . . , 6; (b) Pareto points captured for s = 7.

The feasible region in the objective space is shown in figure 3(a). The Pareto frontier is the feasible part of the boundary of the first constraint. The same aggregate objective function, J = µs1 + bµs2 , is used in this example. For s = 1, . . . , 6, only corner points that are Pareto optimal are captured. This is because the Hessian ∇ 2 0(µi− ) is negative definite for these cases, thus violating Corollary 3(b). For s = 7, only points around the center of the arc are captured (see figure 3(b)). For the points on the left part of the curve, numerical conditioning issues become important, as the curve is nearly horizontal; in these cases, the form of the AOF may help or exacerbate the difficulties. For the point on the right part of the curve, the curvature of the aggregate objective function is not large enough to capture the Pareto optimal points. As s increases, the curvature of the aggregate objective function increases, which enables it to capture more and more points on the Pareto frontier. For s ≥ 13 the entire Pareto frontier is captured. The next example deals with a truss optimization problem; a more complete definition of this truss problem can be obtained in Koski (1985). Example 3. The dynamic behavior of the two-bar truss shown in figure 4(a) is considered in this example. Its structural behavior is simple to analyze and only two criteria are included to enable a graphic presentation in the objective space; the first criterion is the square of the fundamental frequency and the second criterion is the volume of the structure. The only constraints are on member areas. The optimization problem has the form: · ¸ µ1 Minimize µ = µ2 ¯ i = 1, 2 Subject to A ≤ Ai ≤ A, The structure is idealized by concentrating half of the mass of both√the members on the free node, as shown in figure 4(a). Note that we have m = ρ L2 (A1 + 2A2 ), L = 100 cm, ρ = 7850 kg/m3 , E = 200 Gpa, A = 1.0 cm2 , A¯ = 2.0 cm2 .

AGGREGATE OBJECTIVE FUNCTIONS

Figure 4.

187

Two-bar truss biobjective optimization: (a) structure; (b) objective space.

The feasible region in the objective space is presented in figure 4(b); the same aggregate objective function, J = µs1 + bµs2 , is used for this example. The Pareto frontier is the arc BCA, which is non-convex. The line AB was drawn purposely in figure 4(b) to highlight the concavity of the arc BCA. Thus, only the end-points A and B of this frontier are obtained by the weighted-sum method. For s = 2, in addition to the points A and B, the arc BC is captured; as s increases, points on arc CA are captured and the entire Pareto frontier is obtained with s ≥ 11. It is important to note that this nearly trivial case requires an aggregate objective function (with s ≥ 11) that is almost never used in practice. The failure to recognize this requirement has serious practical consequences. Namely, a severely suboptimal design may result from the optimization process.

5.

Conclusion

In this paper, the problem of capturing Pareto optimal points in nonlinear optimization problems is addressed by studying the properties of the aggregate objective functions. A fundamental property of an aggregate objective function, the admissibility property, is developed, and its equivalence to the coordinatewise increasing property is proved. First and second order conditions for admissible aggregate objective functions to capture Pareto optimal points are presented. These conditions are obtained by assuming that the objective functions are open transformations, the Pareto and aggregate objective function hypersurfaces are implicit functions at least twice differentiable, conditions which generally occur in practice (at least in the computational sense, and locally). This paper provides us important insights in developing suitable aggregate objective functions for the capture of Pareto optimal points in the process of design optimization. Aggregate Objective Functions must allow the flexibility of changing their curvature, in order to be effective in practice. Although this paper makes an important contribution in the construction of appropriate aggregate objective functions (i.e., to capture Pareto optimal points in nonlinear optimization problems), research work should continue to investigate the case of Pareto optimal corner points where the differentiability conditions are not satisfied.

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Acknowledgments This research was partially supported by the National Science Foundation CAREER Grant number DMI 9702248 for Dr. Achille Messac. References T. W. Athan and P. Y. Papalambros, “A note on weighted criteria methods for compromise solutions on multiobjective optimization,” Engineering Optimization vol. 27, no. 2, pp. 155–176, 1996. W. Chen, M. M. Wiecek, and J. Zhang, “Quality utility—A compromise programming approach to robust design,” in Proceedings of the DETC’ 98, ASME Design Engineering Technical Conference, Atlanta, Georgia, Sept. 13–16, 1998. Paper # DAC5601. J. L. Cohon, Multiobjective Programming and Planning, Mathematics in Science and Engineering vol. 140, Academic Press: San Diego, 1978. I. Das and J. E. Dennis, “A closer look at drawbacks of minimizing weighted-sums of objectives for Pareto set generation in multicriteria optimization problems,” Structural Optimization vol. 14, pp. 63–69, 1997. I. Das and J. E. Dennis, “Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems,” SIAM Journal on Optimization vol. 8, pp. 631–657, 1998. M. D. Greenberg, Advanced Engineering Mathematics, 2nd edition., Prentice Hall: New Jersey, 1998. J. Koski, “Defectiveness of weighted method in multicriterion optimization of structures,” Communications in Applied Numerical Methods vol. 1, no. 6, pp. 333–337, 1985. W. H. Marlow, Mathematics for Operations Research, John Wiley & Sons: New York, N.Y., 1978. Matlab, The Mathworks, Inc. Natick, MA, 1984. A. Messac, “Physical programming: effective optimization for computational design,” AIAA Journal vol. 34, no. 1, pp. 149–158, 1996. A. Messac, “Control-structure integrated design with closed-form design metrics using physical programming,” AIAA Journal vol. 36, no. 5, pp. 855–864, 1998. A. Messac, “From the dubious construction of objective functions to the application of physical programming,” AIAA Journal vol. 38, no. 1, pp. 155–163, 2000. A. Messac and X. Chen, “Visualizing the optimization process in real-time using physical programming,” Engineering Optimization Journal vol. 32, no. 5, 2000. A. Messac, G. J. Sundararaj, R. V. Tappeta, and J. E. Renaud, “Interactive physical programming: tradeoff analysis and decision making in multicriteria optimization,” AIAA Journal, vol. 38, no. 5, pp. 917–926, 2000. A. Messac, G. J. Sundararaj, R. V. Tappeta, and J. E. Renaud, “The ability of objective functions to generate points on non-convex Pareto frontiers,” AIAA Journal vol. 38, no. 3, 2000. A. Messac and B. Wilson, “Physical programming for computational control,” AIAA Journal vol. 36, no. 2, pp. 219–226, 1998. A. Osyczka, Multicriterion Optimization in Engineering with Fortran Programs, Ellis Horwood Series in Engineering, Halsted Press: New York, 1984. V. Pareto, Manuel d’Economie Politique, Paris, Giard, 1909. J. Rakowska, R. T. Haftka, and L. T. Watson, “Tracing the efficient curve for multi-objective control-structure optimization,” Computer Systems Engineering vol. 2, pp. 461–471, 1991. R. M. Soland, “Multicriteria optimization: A general characterization of efficient solutions,” Decisions Sciences vol. 10, pp. 26–38, 1979. E. R. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Krieger: Melbourne, Florida, 1989.