A New Scalarization Technique to Approximate Pareto Fronts of Problems with Disconnected Feasible Sets R. S. Burachik∗
C. Y. Kaya†
M. M. Rizvi‡
May 23, 2013
Abstract We introduce and analyze a novel scalarization technique and an associated algorithm for generating an approximation of the Pareto front (i.e., the efficient set) of nonlinear multiobjective optimization problems. Our approach is applicable to nonconvex problems, in particular to those with disconnected Pareto fronts and disconnected domains (i.e., disconnected feasible sets). We establish the theoretical properties of our new scalarization technique and present an algorithm for its implementation. By means of test problems, we illustrate the strengths and advantages of our approach over existing scalarization techniques such as those derived from the Pascoletti-Serafini method, as well as the popular weighted-sum method.
Keywords Multiobjective optimization, Scalarization techniques, Pareto front, Efficient set, Numerical methods, Disconnected feasible set. AMS subject classifications. 90C26, 90C29, 90C30, 90C56.
1
Introduction
In many real-world problems, decisions depend on maximization or minimization of multiple competing objective functions. Since it is usually not possible to optimize the competing functions simultaneously, one can only hope to find a trade-off, or compromise, solution. The intent of a multiobjective optimization problem is to find out, in some sense, the best trade-off among these criteria. A range of solutions are found, while a compromise is made among the conflicting objective functions. Each of such compromise solutions is called a Pareto optimal solution, and the set, which contains the objective values of these solutions, is called the Pareto front. A common approach to solve a multiobjective optimization problem is to reformulate the problem as a scalar one, i.e., as a ∗
School of Mathematics and Statistics, University of South Australia, Mawson Lakes, SA 5095, Australia, email:
[email protected], † School of Mathematics and Statistics, University of South Australia, Mawson Lakes, SA 5095, Australia, email:
[email protected], ‡ Corresponding Author, School of Mathematics and Statistics, University of South Australia, Mawson Lakes, SA 5095, Australia,
[email protected].
1
single-objective problem by means of a set of parameters. This reformulation is referred to as a parameter-based scalarization. The scalarized problem can then be solved by using standard single-objective optimization techniques and associated (widely available) software. Some of the parameter-based scalarization approaches, that are widely employed in the literature, are the well-known weighted-sum method [1, 2], epsilon-constraint method [3], Pascoletti-Serafini method and its generalization [4, 5]. Further scalarization methods are discussed in [6-10, 26-27]. The Pareto front is in general an infinite set, which can in theory be generated by solving the scalarized problem over the whole range of values of the parameters (or, weights) of the scalarization. However, in practice, only an approximation of the Pareto front is obtained by solving the scalarized problem over a (finite) partition of the space of weights. It is often necessary to solve the scalarized problem many times to obtain as many points as one would need to get a reasonable approximation of the front. Computationally speaking, because solving each scalarized problem can in itself be very costly, one may be forced to obtain an approximation of the Pareto front by finding a minimal number of Pareto points. Moreover, the Pareto front and/or the feasible set (or domain) might be disconnected, to complicate the computational tasks further. In the present paper, we propose a scalarization technique, which is applicable, not only to problems with a disconnected Pareto front, but also to problems with a disconnected feasible set. We call this scalarization technique the kth-objective weighted-constraint problem, because it involves minimizing, for each fixed k, the weighted kth-objective function, while the rest of the weighted objective functions are incorporated as constraints. The new scalarization addresses the computational difficulties caused by the disconnected nature of the problem. We illustrate these difficulties and the remedy provided by our new scalarization method by means of the test problems we solve in Section 5. Algorithm 1 implements our new scalarization technique, and applies it to test problems to demonstrate its various features. We compare Algorithm 1 with Algorithms 2–4 implementing the Pascoletti-Serafini, modified Pascoletti-Serafini, and weighted-sum scalarization techniques. The paper is organized as follows. In Section 1, we describe the problem. The wellknown weighted-sum, Pascoletti-Serafini, and modified Pascoletti-Serafini methods, are reviewed in Section 2. In Section 3, we describe our new scalarization technique and its theoretical properties. Algorithm 1, which implements our approach, is given in Section 4, together with a brief description of Algorithms 2–4 implementing the other scalarization techniques of Section 2. Section 5 provides numerical examples that demonstrate and compare the performance of the methods. Finally, Section 6 contains our concluding remarks.
2
2
Preliminaries
We recall now some basic notation and tools of multiobjective optimization problems. Let Rn be the n-dimensional Euclidean space. Denote by R+ the set of nonnegative real numbers and R++ the set of strictly positive numbers. For , m ∈ N, we consider the following multiobjective optimization problem (P): min f (x) := [f1 (x ), . . . , f (x )] (P) s.t. x ∈ X := {x ∈ Rn | gj (x) ≤ 0, j = 1, . . . , m} , (1) where the functions fi : Rn → R, i = 1, . . . , , and gj : Rn → R, j = 1, . . . , m, are continuous. We will assume that the functions fi are bounded below on the constraint set X, with a known lower bound. In this situation, there is no loss of generality in imposing that min {min fi (x)} > 0 . (2) i=1,..., x∈X
The solutions of (P) are called efficient points [11, 12], or Pareto points [13]. A less restrictive concept of solution of (P) is the one of a weak efficient point. To recall the definition of these types of solutions, we use the following standard notation. Given u, v ∈ R , we say that u v if and only if ui ≤ vi ∀ i = 1, . . . , and ∃ j such that uj < vj .
(3)
u < v if and only if ui < vi ∀ i = 1, . . . , . Definition 2.1 A point x¯ is said to be efficient for Problem (P) iff there is no x ∈ X such that f (x) f (¯ x). A point x¯ is said to be weak efficient for Problem (P) iff there is no x ∈ X such that f (x) < f (¯ x). We denote the set of weak efficient points of Problem (P) as WE(P). We mentioned before that one of the most common techniques for solving multiobjective optimization problems is the so-called parameter-based scalarization approach. Parameters appearing in the scalarization correspond to weights for each objective function. Define the set of non-negative weights, wi = 1 , W + := w ∈ R | wi ≥ 0 , i=1
and the set of positive weights, W ++ :=
w ∈ R | w i > 0 ,
wi = 1
.
i=1
A vector minimizing each of the objective functions is said to be an ideal vector. Obviously, if an ideal vector is feasible for Problem (P), then it is an efficient point of Problem (P). In general, an ideal vector does not exist because of the conflicting nature of the objectives. Therefore, it is usual to consider a reference vector called the utopia vector. 3
Definition 2.2 A utopia vector u ∈ R , associated with Problem (P), is defined as ui := fi∗ − εi , where εi > 0, for all i = 1, . . . , , and fi∗ is the optimal value of the optimization problem, min fi (x) .
(Pi )
x∈X
xi ), i = 1, . . . , . If x¯i is a minimizer of Problem (Pi ), then fi∗ := fi (¯ In our study, we focus on the following existing parameter-based scalarization approaches. The weighted-sum approach : The original Problem (P) is converted into a single objective optimization problem by associating a weight to each objective function and then minimizing the weighted-sum of the objective functions. The main advantage of this approach, introduced by Gass and Satty [1] and Zadeh [2], is that it is easy to implement. However, the weighted-sum approach cannot generate the points in nonconvex parts of the Pareto front. The weighted-sum problem for Problem (P), denoted by (Pw ), is defined, for a given weight w ∈ W + , as min wi fi (x) . (4) (Pw ) x∈X
i=1
Every solution of Problem (Pw ) is a weak efficient point (see [13]), and this fact is used for constructing an approximation of the Pareto front. The Pascoletti-Serafini approach : Another form of the parameter-based scalarization approach is presented by Pascoletti and Serafini [14], which is commonly used for generating points in the Pareto front. This method is also known as the goal-attainment method (see [13, 15, 16]). For a given w ∈ W ++ , the scalarization is stated as follows. (PS)
min
(α,x)∈R×X
α,
s.t. wi (fi (x) − ui ) ≤ α , ∀ i = 1, . . . , ,
(5)
where α ∈ R is a new variable and u is a utopia vector associated with Problem (P ). In [4], it is proved that every solution of Problem (PS) is a weak efficient point, which in turn is implemented for generating an approximation of the Pareto front. The modified Pascoletti-Serafini approach : An ordering cone and this parameterbased scalarization approach is presented in [4, 5]. For a given w ∈ W ++ , the method solves the following optimization problem. (MPS)
min
(α,x)∈R×X
α, s.t. wi (fi (x) − ui ) = α , ∀ i = 1, . . . , ,
(6)
where α ∈ R is a new variable and u is a utopia vector associated with Problem (P ). It turns out that a solution of Problem (MPS) may or may not be weak efficient (see [4, 5]). Therefore, the process of constructing a Pareto front using this approach is as follows. A set of optimal solutions is obtained by varying the weight vectors. Then, a weed-out process is applied to discard those points which are not weak efficient. However, there are cases where the process of weeding-out fails to eliminate non-Pareto points. We demonstrate this fact in Test Problems 5.1 and 5.2.
4
3
The Weighted-constraint Approach
Next, we describe our proposed approach. Fix k ∈ {1, . . . , } and w ∈ W ++ . Consider the problem (Pkw )
min wk fk (x), s.t. wi fi (x) ≤ wk fk (x), x∈X
i = 1, . . . , , i = k .
We refer to Problem (Pkw ) as the kth-objective weighted-constraint problem. Define the feasible set of Problem (Pkw ) (for fixed k and w) as Xwk := {x ∈ X | wi fi (x) ≤ wk fk (x), ∀i = k} ,
(7)
and the solution set of Problem (Pkw ) as Swk := {x ∈ X | x solves (Pkw )} . For each fixed w ∈ W ++ , we have X := ∪k=1 Xwk .
(8)
Indeed, for each x ∈ X take k verifying wk fk (x) := maxi=1,..., {wi fi (x)}. Hence x ∈ Xwk . We also use the following set. W (x) := {w ∈ W ++ | x ∈ Swk , ∀k = 1, . . . , } .
(9)
Note that W (x) may be empty for some x ∈ X. We will generate an approximation of the Pareto front by solving Problem (Pkw ) for all k ∈ {1, . . . , }, over a grid of values of w. Theorem 3.1 below implies that, for all w ∈ W ++ , we have ∩k=1 Swk ⊆ WE(P) ⊂ ∪w∈W ++ ∩k=1 Swk . In particular, if for some w ∈ W ++ we have that ∩k=1 Swk = ∅, then the left hand side of the inclusion above provides a way of computing weak efficient points. On the other k hand, if k=1 Sw = ∅, then Proposition 3.3 and Corollary 3.1 below induce a refinement among the solutions that generates new weak efficient points.
3.1
Theoretical Results
Theorem 3.1 x¯ ∈ X is a weak efficient solution of Problem (P), if and only if there exists some w ∈ W ++ such that x¯ solves (Pjw ) for all j ∈ {1, . . . , }. Proof. To prove the ‘if’ part, assume that w ∈ W ++ is such that x¯ solves (Pjw ) for all j. Suppose that x¯ ∈ X is not a weak efficient point of P. Then there exists x˜ ∈ X such that x) < fi (¯ x), i = 1, . . . , . fi (˜
(10)
From (8) there exists k such that x˜ ∈ Xwk . Then from (10) we have that wk fk (˜ x) < wk fk (¯ x), where wk > 0, which contradicts the fact that x¯ solves (Pkw ). To prove the ‘only if’ part, take x¯ ∈ X weak efficient for Problem (P). We will show that there exists w ∈ W ++ such that x¯ is a solution of (Pkw ) for all k. By the assumption in (2), fi (x) > 0 for all x ∈ X. Let 1/fi (¯ x) wi := . x) j=1 1/fj (¯ 5
Then wi > 0 and wi = 1. With this choice, w ∈ W ++ and x¯ satisfies all the constraints as equalities, that is, x) = wk fk (¯ x), i = 1, . . . , , i = k . wi fi (¯
(11)
If x¯ is not a solution of (Pkw ) for some k, then there exists x˜ ∈ X such that x) < wk fk (¯ x) , wk fk (˜
(12)
and wi fi (˜ x) ≤ wk fk (˜ x) i = k . Hence, x) ≤ wk fk (˜ x) < wk fk (¯ x), i = 1, . . . , and i = k . wi fi (˜ By (11), we can write x) < wk fk (¯ x) = wi fi (¯ x), i = 1, . . . , and i = k , wi fi (˜
(13)
since wi > 0, (12) and (13) yield, fi (˜ x) < fi (¯ x), i = 1, . . . , , which contradicts the weak efficiency of x¯.
2
Remark 3.1 Note that the ‘only if’ part of Theorem 3.1 holds, in particular, for efficient points, since every efficient point is a weak efficient point. However, if a point solves (Pkw ) for all k ∈ {1, . . . , }, then this does not necessarily imply that the point is efficient, unless all objective functions are strictly convex. Example 3.1 and Proposition 3.1 illustrate this fact. Example 3.1 Let f : R2 → R2 given by f (x) := x, and consider the problem min f (x), s.t. x ∈ X = {x ∈ R2 | (x1 − 1)2 + (x2 − 1)2 ≤ 0.8, (x1 − 0.5) (x2 − 0.5) ≤ 0} . Here, x¯ = (0.5, 0.4) is a weak efficient point and wi are computed as in the proof of Theorem 3.1. Here, w1 = 4/9 and w2 = 5/9. For k = 1, 2, Problem (Pkw ) can be written as minx∈X w1 x1 s.t. w2 x2 − w1 x1 ≤ 0, and minx∈X w2 x2 s.t. w1 x1 − w2 x2 ≤ 0, and x¯ = (0.5, 0.4) is an optimum point of both problems. However, x¯ = (0.5, 0.4) is not efficient. It is easy to see from the definitions that every efficient point is weak efficient. In some cases, the converse is also true, as we state and prove next. Proposition 3.1 Suppose that X is convex and fi , i = 1, . . . , , are strictly convex functions defined on X. If x¯ ∈ X is a weak efficient solution of Problem (P), then x¯ is an efficient solution of Problem (P).
6
Proof. Let x¯ ∈ WE(P), then there exists no x ∈ X such that fi (x) < fi (¯ x), i = 1, . . . , .
(14)
Suppose that x¯ is not an efficient solution of Problem (P). Then there exists x˜ ∈ X such that x) ≤ fi (¯ x), i = 1, . . . , , (15) fi (˜ and, for at least one i, say i = k, fk (˜ x) < fk (¯ x).
(16)
Let xˆ = λ x˜ + (1 − λ) x ¯ ∈ X, where 0 < λ < 1. Since fi , i = 1, . . . , , are strictly convex, we have that the following statement holds for ∀i = 1, . . . , . x) = fi (λ x ˜ + (1 − λ) x¯) < λ fi (˜ x) + (1 − λ) fi (¯ x), i = 1, . . . , . fi (ˆ Combining the above expression with (15), we obtain fi (λ x ˜ + (1 − λ) x¯) < λ fi (¯ x) + (1 − λ) fi (¯ x) = fi (¯ x), i = 1, . . . , , which implies that x + (1 − λ) x¯) < fi (¯ x), fi (λ˜
i = 1, . . . , .
This contradicts (14). Hence x¯ is an efficient solution.
(17) 2
The following simple result states that, if there exists w ∈ W ++ such that a point solves (Pkw ) for all k, then w is unique. Using (9) this can be stated as follows. Proposition 3.2 If W (¯ x) = ∅, then W (¯ x) is a singleton. Consequently, x ∈ WE(P) if and only if x¯ ∈ ∩k=1 Swk¯ , where w¯i = (1/fi (¯ x))/( j=1 1/fj (¯ x)) . Proof. Assume that there are two elements w, ¯ wˆ ∈ W (¯ x). Since x¯ is feasible of all problems, for both w¯ and w, ˆ we have from (7) that w¯i fi (¯ x) = w¯k fk (¯ x), i = 1, . . . , and i = k,
(18)
x) = wˆk fk (¯ x), i = 1, . . . , and i = k. wˆi fi (¯
(19)
and From (18) and (19), we have wˆi x) fk (¯ w¯i = = , i = 1, . . . , and i = k. w¯k fi (¯ x) wˆk Since
i=1
w¯i =
i=1
(20)
wˆi = 1 and by (20), we can write
w ¯i fk (¯ x) wˆi 1 1 =1+ = 1+ =1+ = . w¯k w¯k fi (¯ x) wˆk wˆk i=1,i=k
i=1,i=k
i=1,i=k
Thus w¯k = wˆk holds for all k ∈ {1, . . . , }. Hence W (¯ x) is a singleton. The second statement follows from the first one and the proof of Theorem 3.1. 2 7
Remark 3.2 The ‘only if’ part of Theorem 3.1 demonstrates that if (Pkw ) has a common solution x¯ for all k ∈ {1, . . . , } then x¯ is weak efficient. However, if the solutions of (Pkw ) do not coincide and the corresponding optimum values are not all the same either, then our theory so far becomes inconclusive in determining a weak efficient point. On the other hand, one can make certain comparisons among the solutions of each Problem (Pkw ) and conclude whether any of these solutions is weak efficient or not, as outlined below in Proposition 3.3. Note that we may identify more than one weak efficient point as a result of this comparison procedure and we illustrate this latter situation in Test Problems 5.1 and 5.2. Proposition 3.3 Assume ∃ w ∈ W ++ such that Swj = ∅, ∀ j = 1, . . . , . Suppose that, for some k ∈ {1, . . . , }, ∃ x¯k ∈ Swk such that ∀r = k, ∃ x¯r ∈ Swr , which satisfies fr (¯ xr ) ≥ fr (¯ xk ) .
(21)
Then x¯k ∈ WE(P). Proof. Without any loss of generality, we may assume that k = 1. Suppose that x¯1 ∈ / WE(P). Then there exists xˆ ∈ X such that x) < fi (¯ x1 ), i = 1, . . . , . fi (ˆ
(22)
Note that xˆ ∈ X = ∪r=1 Xwr by (8). Case I. Suppose that xˆ ∈ Xw1 . Since w1 > 0, from (22) we can write w1 f1 (ˆ x) < w1 f1 (¯ x1 ) . The above situation contradicts the fact that x¯1 ∈ Sw1 . Hence, we are left only with Case II. Case II. Suppose that xˆ ∈ Xws , for some s = 1. From (22) we have that ws fs (ˆ x) < ws fs (¯ x1 ). x) < ws fs (¯ x1 ) ≤ ws fs (¯ xs ). This contradicts the fact that x¯s ∈ Sws . By (21) we have ws fs (ˆ 2 Since both cases lead to a contradiction, we conclude that x¯1 ∈ WE(P). Corollary 3.1 below is an immediate consequence of Proposition 3.3. Corollary 3.1 Let w ∈ W ++ . Suppose that (¯ x1 , . . . , x¯ ) ∈ Sw1 × . . . × Sw and that ∀r, k ∈ {1, . . . , }, xr ) ≥ fr (¯ xk ) . fr (¯ Then x¯k ∈ WE(P), for every k = 1, . . . , . Proof. In Proposition 3.3, we proved that, if ∃k such that (21) holds, then x¯k ∈ WE(P). In Corollary 3.1, it is assumed that ∀k, fr (¯ xr ) ≥ fr (¯ xk ) holds. Therefore, x¯k ∈ WE(P), for all k. 2
8
1
f2
0.5 (0.4, 0.37)
0 0
0.5
1
f1
Figure 1: Example 3.2 – The converse of Corollary 3.2 is not true.
Relationship Between Problems (Pw ) and (Pkw )
3.2
In this section, we prove that every point in the Pareto front, which can be generated by the weighted-sum scalarization, can also be generated by our kth-weighted objective approach. We also show that the converse of this fact is not true; namely, our approach can generate Pareto points which are not attainable by the weighted-sum method. The following is a direct consequence of Theorem 3.1. Corollary 3.2 If there exists w ∈ W + such that x¯ solves (Pw ), then there exists α ∈ W ++ such that x¯ solves (Pkα ) for all k. Proof. If there exists w ∈ W + such that x¯ solves (Pw ) implies the well-known fact [3] x¯ is weak efficient. By Theorem 3.1, the result follows. 2 The following example shows that the converse of Corollary 3.2 is not true. Example 3.2 Consider the problem min {x1 , x2 } , x∈X
where
X = {x ∈ R2 | (x1 − 1)2 + (x2 − 1)2 − 1 ≤ 0, 0.3 − x21 − x22 ≤ 0} .
Consider the weights w1 = 0.48 and w2 = 0.52. For k = 1, 2, Problems (P1w ) and (P2w ) can be written as minx∈X w1 x1 s.t. w2 x2 − w1 x1 ≤ 0, and minx∈X w2 x2 s.t. w1 x1 − w2 x2 ≤ 0, respectively. Here, x¯ = (0.4, 0.37) is located in the concave part of the front (see Figure 1) and it is well known that x¯ cannot obtained as a solution of (Pw ) for w ∈ W + . However, x¯ ∈ Sw1 ∩ Sw2 for w = (w1 , w2 ). Thus the converse of Corollary 3.2 is not true.
4
Algorithms
In this section, we introduce Algorithm 1, which implements our proposed scalarization technique, the weighted-constraint scalarization, for generating an approximation of the 9
Pareto front of bi-objective optimization problems. We also provide Algorithms 2–4, which implement the Pascoletti-Serafini, modified Pascoletti-Serafini, and weighted-sum, scalarization techniques, respectively, for comparison purposes. In Step 2 of Algorithm 1, each objective function, subject to the original constraints of the problem, is minimized. This individual minimization yields the “outer” end points of the Pareto front, as defined in this step of the algorithm. Note that, in the bi-objective case, the Pareto front is a (connected or disconnected) curve. Next, we determine the interval of weights that will be used to generate the rest of the points in the Pareto front. In Step 3 of Algorithm 1, a regular partition of the interval of weights associated with the Pareto front is created. The incremental weights in the partition correspond to rays emanating from the utopia point. If the utopia point is chosen sufficiently away from the Pareto front, these rays become almost parallel to one another with a more-or-less equal spacing between each consecutive ray. Parallel rays and regular weight partition result in an almost uniformly spaced points in the Pareto front. Note that the scheme of rays we implement in Algorithm 1 can also be found in [15]. In [6, 17, 18], slightly different approaches are taken for generating rays and in turn generating equally spaced points in the Pareto point. In Step 4 of Algorithm 1, for each weight value in the partition, first Problem (P1w ) and then Problem (P2w ) is solved. If the solutions of Problems (P1w ) and (P2w ) are the same, then the point in the function value space is accepted as a Pareto point, in line with Theorem 3.1. If Problems (P1w ) and (P2w ) yield different solutions, then Proposition 3.3 is evoked to accept either or both of the solutions as constituting a Pareto point. It should be noted that the results given in Theorem 3.1 and Proposition 3.3 are applicable to problems with two or more objectives. The steps of the algorithm described for two objectives here can be generalized to the cases with three of more objectives by employing weight grid generation techniques similar to those described in [18]. Algorithm 1 (Weighted-constraint scalarization for = 2) Step 1 (Input) Choose a utopia point u = (u1 , u2). Set the number of partition points (N + 1) in the interval of weights. Set s = 0. Step 2 (Determine the outer end points of the Pareto front) (a) Find x¯0 that solves problem (P2 ). x0 ) and f2∗ := f2 (¯ x0 ). Let f¯1 := f1 (¯ ∗ ¯ Let w0 := (f2 − u2 )/[(f1 − u1 ) + (f2∗ − u2 )]. x0 ), f2 (¯ x0 )]. Set F (s) := [f1 (¯ (b) Find x¯f that solves problem (P1 ). Let f1∗ := f1 (¯ xf ) and f¯2 := f2 (¯ xf ). Let wf = (f¯2 − u2 )/[(f1∗ − u1 ) + (f¯2 − u2 )]. xf ), f2 (¯ xf )]. Set F¯ := [f1 (¯ Step 3 (Generate a weight partition) Set w¯t := w0 + t(wf − w0 )/N, t = 0, 1, . . . , N. Set t = 0. Step 4 (Solve the scalarized problems) Let w = (w1 , w2 ) := (w¯t , 1 − w¯t ). (a) (i) Find x¯1 that solves Problem (P1w ). 10
(ii) Find x¯2 that solves Problem (P2w ). (b) Determine the weak efficient points: Let s := s + 1. If x¯1 = x¯2 = x¯, then set F (s) := [f1 (¯ x), f2 (¯ x)], and go to Step 5. Otherwise, one of the following three cases is executed. x1 ) ≤ f2 (¯ x2 )] and [f1 (¯ x2 ) ≤ f1 (¯ x1 )] then, set F (s) := [f1 (¯ x1 ), f2 (¯ x1 )], (i) If [f2 (¯ let s := s + 1, and set F (s) := [f1 (¯ x2 ), f2 (¯ x2 )]. x1 ) ≤ f2 (¯ x2 )], then set F (s) := [f1 (¯ x1 ), f2 (¯ x1 )]. (ii) If [f2 (¯ x2 ) ≤ f1 (¯ x1 )], then set F (s) := [f1 (¯ x2 ), f2 (¯ x2 )]. (iii) [f1 (¯ Step 5 (Stopping criterion) If t := N, then stop. Otherwise, set t := t + 1 and go to step 3. Step 6 (Output) Set F (s + 1) := F¯ . The array of Pareto points, F , is an approximation of the Pareto front, with (s + 1) elements. The number of distinct Pareto minima generated by Algorithm 1 might turn out to be more than (N + 1); i.e., it is possible to have the final value of s to be greater than N. In Steps 4(a)(i)-(ii) of Algorithm 1, having to solve two problems in order to find one Pareto point may at first look like a disadvantage. However, one has to note that the problem in Step 4(a)(ii) should normally use the solution from Step 4(a)(i) as an initial guess. In the case when the solutions found in Steps 4(a)(i)-(ii) are the same, Step 4(a)(ii) would consume a negligible amount of computational time in finding the same solution. If, on the other hand, the solution in Step 4(a)(ii) is different from that in Step 4(a)(i), then the solution in Step 4(a)(ii) might also be a Pareto solution, which is in fact checked in Step 4(b). The latter situation would be encountered when the Pareto front and/or the domain is disconnected. The prospect of finding two Pareto points for a single choice of weights is an advantage over the existing scalarization techniques, which are troubled especially with a disconnected domain. In the case when only one of the different solutions found in Steps 4(a)(i)-(ii) is Pareto, then this necessarily points to either a disconnected Pareto front or the pathological case of a disconnected domain, which again indicates an advantage of the new approach. Algorithm 2 (Pascoletti-Serafini scalarization for = 2) Steps 1-3 Do the same as in Steps 1-3 of Algorithm 1. Step 4 (Solve the scalarized problem) Let w = (w1 , w2 ) := (w¯t , 1 − w¯t ). Find x¯ and α that solves Problem (PS). Let s := s + 1. x), f2 (¯ x)]. Set F (s) := [f1 (¯ Steps 5-6 Do the same as in Steps 5-6 of Algorithm 1.
11
Algorithm 3 (Modified Pascoletti-Serafini scalarization for = 2) Steps 1-6 Do the same as in Steps 1-6 of Algorithm 2, except that, in Step 4 of Algorithm 2, solve Problem (MPS), instead of Problem (PS).
Algorithm 4 (Weighted-sum scalarization for = 2) Steps 1-6 Do the same as in Steps 1-6 of Algorithm 2, except that: (i) In Step 2 of Algorithm 2, set w0 = 0 and wf = 1. (ii) In Step 4 of Algorithm 2, solve Problem (Pw ), instead of Problem (PS).
In each of Algorithms 2-4, the number of distinct Pareto points that can be obtained is at most (N + 1). This is in contrast with Algorithm 1, which may obtain more distinct Pareto points, as illustrated in the numerical experiments to follow.
5
Numerical Experiments
In this section, we demonstrate the working and main features/advantages of Algorithm 1 on test problems, including problems with a disconnected Pareto front and/or disconnected domain. We have chosen the objective and constraint functions of the problems we study to be differentiable, so that we can use powerful optimization methods and associated software to solve the single-objective (scalarized) problems in Steps 4(a)(i)-(ii) of Algorithm 1, and likewise Step 4 of Algorithms 2–4, efficiently. Otherwise, our theoretical results and techniques are also applicable to non-differentiable problems. We compare the performance of Algorithm 1 with the performances of Algorithms 2–4. We solve Problems (Pw1 ) and (Pw2 ), Problems (PS), (MPS), and (Pw ) by using Algencan, version 2.3.7, which is a popular software, based on augmented Lagrangian methods; see [19, 20]. We use AMPL [21] as an optimization modelling language which employs Algencan as a solver.
5.1
Test Problem 1
We modify a nonconvex bi-objective problem from [22] by introducing an elliptic constraint in the fourth line of the description of the problem below, so that not only the Pareto front, but also the feasible region (or domain) itself is disconnected. min
(x1 , x2 )
subject to −x21 − x22 + 1 + 0.1 cos(16 arctan(x1 /x2 )) ≤ 0 , (x1 − 0.5)2 + (x2 − 0.5)2 − 0.5 ≤ 0 , 1.69 x21 + 1.01 x22 − 2.6 x1 x2 − 0.02 ≥ 0 , 0 ≤ x1 , x2 ≤ π .
12
The original problem in [22] has earlier been studied by various other researchers in [4, 6, 23]. It was also studied in a modified nonsmooth form in [15]. In all three Algorithms, we take the utopia point to be u = (−1, −1). In two separate runs with N := 6 and N := 15, Algorithms 1-3 generate the points illustrated in Figures 2 and 3, respectively. Note that the exact Pareto front and feasible region (as a shaded area) are also exhibited in these figures. In the case when N = 6, Algorithm 1 generates seven Pareto points, which are spaced relatively evenly in the front, and among these generated points we have all the end points of the Pareto curve, inner and outer ones, as seen in Figure 2(a). Algorithm 2 generates six and Algorithm 3 only three Pareto points, as seen in Figures 2(b)-(c). Algorithms 2 and 3 cannot generate both inner end points in the front. Moreover, the two non-Pareto points generated by Algorithm 3, which are shown by two squares in Figure 2(c), cannot be “weeded-out” because they are not dominated by the other computed points. Therefore the approximation of the front generated by Algorithm 3 is not good. In the case when N = 15, Algorithm 1 is more successful: it generates 13 Pareto points, which are spread relatively evenly in the front, resulting in a reasonable approximation of the front, as can be seen in Figure 3(a). Algorithm 2 finds 11 Pareto points; however, these points are not only fewer than those found by Algorithm 1 but also are spread less evenly; see Figure 3(b). One of the inner end points of the front can still not be recovered. Algorithm 3 can find 12 points, but this is at the additional expense of producing two non-Pareto points (indicated by plusses on small circular dots), although these two points can now be weeded out; see Figure 3(c). Furthermore, when the ray corresponding to a weight does not intersect the feasible set Algorithm 3 fails to find a solution to Problem (MPS). This is a significant disadvantage, because the computational effort, before deciding whether or not a solution exists, is in general very large. When the computational resources are scarce, and so it is necessary to do minimal computation to generate an approximation of the Pareto front, Algorithm 1, implementing our proposed scalarization technique, seems to be doing a better job than existing scalarization techniques.
5.2
Test Problem 2
We adapt the second test problem from [24, 25], by restricting (and disconnecting) the feasible set in the first constraint below: min
(f1 (x1 ), f2 (x1 , . . . , xn )) where f1 (x1 ) := x1 and f2 := g h , g(x2 , . . . , xn ) := 1 + 9/(n − 1) ni=2 xi , h := 1 − (f1 /g)2 ,
subject to the constraints −x21 − 1.09 (g h)2 + 2.6 x1 (g h) + 0.18 ≤ 0 , (x1 − 0.5)2 + (g h − 0.5)2 − 0.5 ≤ 0 , 0 ≤ xi ≤ 1, i = 1, . . . , n . In all of Algorithms 1-3, we take the utopia point to be u = (−1, −1). We take n = 100, giving us a 100-variable problem. We set N := 30. The points found by Algorithms 1-3 are shown in Figure 4. The gap in the feasible set and Pareto front is more prominent 13
1.2 1
f2 0.8 0.6 0.4 0.2 0
0
0.5
1
1.5
f1 (a) Points found by Algorithm 1
1.2
1.2
1
1
f2 0.8
f2 0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.5
f1
1
0
1.5
(b) Points found by Algorithm 2
0
0.5
f1
1
(c) Points found by Algorithm 3
Figure 2: Test Problem 1 – A comparison of Algorithms 1-3, with N = 6.
14
1.5
1.2 1
f2 0.8 0.6 0.4 0.2 0
0
0.5
f1
1
1.5
(a) Points found by Algorithm 1
1.2
1.2
1
1
f2 0.8
f2 0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.5
f1
1
0
1.5
(b) Points found by Algorithm 2
0
0.5
f1
1
(c) Points found by Algorithm 3
Figure 3: Test Problem 1 – A comparison of Algorithms 1-3, with N = 15.
15
1.5
1.2 1
f2 0.8 0.6 0.4 0.2 0
0
0.5
f1
1
(a) Points found by Algorithm 1
1.2
1.2
1
1
f2 0.8
f2 0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.5
f1
0
1
(b) Points found by Algorithm 2
0
0.5
f1
1
(c) Points found by Algorithm 3
Figure 4: Test Problem 2 – A comparison of Algorithms 1-3, with n = 100 and N = 30. for this problem. Algorithm 1 yields 21 Pareto points, while Algorithms 2 and 3 yield 16 and 19 Pareto points, respectively. Algorithm 1 generates evenly distributed points and thus resulting in a good approximation of the Pareto front; see Figure 4(a). With Algorithm 2, the upper portion of the Pareto front is approximated rather poorly; see Figure 4(b). Algorithm 3 produces four non-Pareto points, three of which (shown by plusses on circular points) are weeded out, still leaving one non-Pareto point in the approximation of the front; see the point shown by a square in Figure 4(c). Again, as in Test Problem 1, the computational time spent for finding the non-Pareto points is a liability. Moreover, as in Test Problem 1, when a ray corresponding to a weight does not intersect the feasible set Algorithm 3 fails to produce a solution to Problem (MPS), and this constitutes an even worse waste of computational resources. Furthermore, the number of points computed in the front is still less than that computed with Algorithm 1.
16
4
4
3.5
3.5
3
3
f2 2.5
f2 2.5
2
2
1.5
1.5
1 0
1
2
f1
3
1 0
4
1
2
f1
3
4
(b) Points found by Algorithm 4
(a) Points found by Algorithm 1
Figure 5: Test Problem 3 – A comparison of Algorithms 1 and 4, with N := 15.
5.3
Test Problem 3
The third test problem is from [6], where the well-known weighted-sum method is illustrated. We use this problem to compare our proposed scalarization implemented in Algorithm 1 and the weighted-sum scalarization implemented in Algorithm 4. The test problem is given by
2 min 1 + x21 , x1 − 4x1 + x2 + 5 subject to the constraints
x21 − 4x1 + x2 + 5 ≥ 3.5 , x1 ≥ 0, x2 ≥ 0 .
As the utopia point we choose u = (−1, −1), and set N := 15. Algorithm 1 yields the Pareto points shown in Figure 5(a), while Algorithm 4 yields the Pareto points shown in Figure 5(b). Obviously, the approximation obtained by Algorithm 1 is much better than the one obtained by Algorithm 4. The points found by Algorithm 4 are not uniformly distributed. Moreover, they also fail to approximate the upper two-thirds of the front. The performances of Algorithms 2 and 3 for this test problem are the same as that of Algorithm 1.
6
Conclusions
We have proposed a new scalarization technique and an algorithm for generating an approximation of the Pareto front (or the efficient set) of nonconvex multiobjective optimization problems, in particular, problems with a disconnected Pareto front and/or a disconnected feasible set. The numerical experiments we did with such problems suggest that our proposed scalarization method is more successful compared to the existing Pascoletti-Serafini and modified Pascoletti-Serafini scalarization techniques. Our new technique seems to be particularly useful when it is mandatory to approximate the Pareto front via a small number of points (conceivably because of the computational burden). This was demonstrated in Test Problems 2 and 3. 17
Although the proposed algorithm and test problems involved only two objective functions, we have in fact presented the theoretical results for the new scalarization for any number of objective functions. The next step as future work would be to implement the new scalarization technique in an algorithm involving three or more objective functions. It is worth considering the special case when Problem (P) is convex, and, in particular, the case when Problem (P) is linear, i.e., the objective functions are linear and the inequality constraint functions are affine. For linear (P), our proposed scalarization (Pkw ) is also a linear problem, for which powerful solution methods and software are available in the literature. For the simple linear problem, min (x1 , x2 ), s.t. 1 ≤ x1 + x2 − 1 ≤ 2 and x1 , x2 ≥ 1/4, the Pareto front contains two weak Pareto segments, namely, one with x∗1 = 1/4 and 3/4 < x∗2 ≤ 7/4, and the other with 3/4 < x∗1 ≤ 7/4 and x∗2 = 1/4. Even for this simple case, the weighted sum or the Pascoletti-Serafini scalarization might fail in finding a point in the weak segments (see [15, pp. 1093–1094] for a general discussion). Our method, on the other hand, is useful in finding weak efficient solutions (as well as efficient solutions) in the front. In the case when Problem (P) is linear, the constraint set is a convex polytope, and so it is conceivable to think that more specialized results can be obtained for our new scalarization method. However, one has to keep in mind that our scalarization approach has been illustrated to be advantageous in particular for problems with a disconnected Pareto front and/or a disconnected feasible set. Minimization over the Pareto front has been an active research area, see, e.g., [28, 3032] and the references therein. In this case, one has to minimize an additional (single) objective function in the space of Pareto minima. Because the Pareto front is a very large (in fact, infinite) set, it is desirable not to perform the difficult task of constructing (even an approximation of) the Pareto front for the minimization of the additional objective function. By Theorem 3.1 and Proposition 3.2, our proposed scalarization is a bijection from the space of weights to the set of weak efficient points. So, instead of carrying out minimization over the Pareto front, one may perform the minimization over the space of weights, which is much more desirable than carrying out minimization over the Pareto front directly. An estimate on how accurate an approximate Pareto front is is provided by Yang and Goh [33] for the case when the Pareto front is described by a convex curve (in R2 ). In the case when the Pareto front is given by a nonconvex curve, an estimate of the above mentioned accuracy is given by Liu et al [34]. More precisely, these papers give an estimate on the accuracy of the approximate Pareto front in terms of Hausdorff distance: for the given Hausdorff distance error between the exact Pareto front and the approximate Pareto front, they can provide an upper bound on the number of scalarized problems that are needed to be solved to reach the given accuracy. These estimates might be used to limit the number of Pareto points (N + 1) that one needs to find in order to approximate the front up to a required accuracy.
Acknowledgments M. M. Rizvi acknowledges support by a UniSA President’s Scholarship and the School of Mathematics and Statistics at the University of South Australia. The authors would like to thank the Editors and the two reviewers for their constructive comments, which improved the paper. 18
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