optimality conditions when compared to convex problems (cf. 6]). Problem (2.1) is said to satisfy the Cottle constraint quali cation at ^x if either gj(^x) < 0 for all j ...
Tangent and Normal Cones in Nonconvex Multiobjective Optimization Kaisa Miettinen and Marko M. Makela Department of Mathematics, University of Jyvaskyla, P.O. Box 35, FIN-40351 Jyvaskyla, Finland
Abstract. Trade-o information is important in multiobjective optimization.
It describes the relationships of changes in objective function values. For example, in interactive methods we need information about the local behavior of solutions when looking for improved search directions. Henig and Buchanan have generalized in Mathematical Programming 78(3), 1997 the concept of trade-o s in convex multiobjective optimization problems. With the help of tangent cones they de ne a cone of trade-o directions. In this paper, we examine the possibility of extending the results of Henig and Buchanan for nonconvex multiobjective optimization problems. We carry out the generalization in the sense of Clarke's nonconvex analysis.
1 Introduction The need of trading o plays an elementary role when searching for desirable solutions for multiobjective optimization problems, where it is impossible to meet all the objectives simultaneously. Trading o is of particular value in interactive methods (see, e.g. [1], [10]) since the knowledge of how changes in some objective function values a ect the others is useful. So far, ways of generating trade-o information have been suggested based on either using certain scalarizing functions and under rather restricting assumptions (see, e.g. [1], [3], [12], [14]) or relying on special characteristics of the problems solved (like linearity, see, e.g. [4]). Another approach has been to create scalarizing functions that generate solutions satisfying pre-speci ed bounds on trade-o information (see, e.g. [7], [8]). Recently, the trade-o ideas have been generalized into a cone of tradeo directions by Henig and Buchanan [6] for convex problems. This is a promising approach because of its independency of the scalarizing function used and its minor presumptions set to the problem treated. In [6], the cone of trade-o directions is de ned as a Pareto optimal surface of a tangent cone located at the point considered. The calculation of trade-o directions is based on the characterizations of tangent cones and their polar cones, normal cones. Special attention is paid to proper Pareto optimality which is equivalent to the nonemptiness of the cone of trade-o directions.
The treatment in [6] has its basis on classical convex analysis in the sense of Rockafellar [11]. Clarke [2] has generalized this theory for a nonconvex case. Our intention here is to examine how the results in [6] can be generalized in the spirit of Clarke. Giving up convexity brings along the need of dealing with local instead of global analysis. It is well-known that suciency relations usually necessitate convexity. However, there is no guarantee that even convex necessary results would be valid for nonconvex problems. In what follows, we begin by de ning optimality concepts. For clarity of notations, we concentrate only on global analysis. However, the results are valid also for local optima. According to Clarke we de ne tangent and normal cones for nonconvex sets. The results given in [6] are then studied under the nonconvexity assumption.
2 Foundations We consider a multiobjective optimization problem of the form (2.1)
minimize ff1(x); f2 (x); : : : ; fk (x)g subject to x 2 S = fx 2 Rn j (g1 (x); g2 (x); : : : ; gm(x))T 0g:
where we have k objective functions fi : Rn ! R and m constraint functions gi : Rn ! R. The decision vector x belongs to the closed (nonempty) feasible set S Rn . In the following, we denote the image of the feasible set by Z Rk . The set Z is called a feasible criterion set and its elements are termed criterion vectors , denoted by z = f (x) = (f1 (x); f2 (x); : : : ; fk (x))T . Thus, we have Z = f (S ). All the functions are assumed to be locally Lipschitz continuous. A function h : Rn ! R is locally Lipschitz continuous at a point x 2 Rn if there exist scalars K > 0 and > 0 such that jh(x1 ) h(x2 )j K kx1 x2 k for all x1 ; x2 2 B (x; ); where B (x; ) Rn is an open ball with a centre x and a radius . Problem (2.1) is convex if all the objective functions and the feasible set are convex. The sum of two sets A and E is de ned by A + E = fa + e j a 2 A; e 2 E g. The interior, closure, boundary and convex hull of a set A are denoted by int A, cl A, bo A and conv A, respectively. We denote the negative orthant of Rk by Rk = fd 2 Rk j di 0 for i = 1; : : : ; kg. As the concepts of optimality we employ Pareto optimality and proper Pareto optimality.
De nition 2.2. A decision vector x 2 S and the corresponding criterion vector z 2 Z are Pareto optimal if there does not exist another decision vector x 2 S such that fi (x) fi (x ) for all i = 1; : : : ; k and fj (x) < fj (x )
for at least one index j . The set of Pareto optimal criterion vectors is called a Pareto optimal set and denoted by P (Z ) = fz 2 Z j (z + Rk n f0g) \ Z = ;g:
De nition 2.3. Vectors x 2 S and z 2 Z are properly Pareto optimal if there does not exist another decision vector x and corresponding criterion vector z = f (x) 2 Z such that z 2 z C n f0g, in other words, (z + C n f0g) \ Z = ; for some convex cone C such that Rk n f0g int C . The properly Pareto optimal set is denoted by PP (Z ) = fz 2 Z j (z + C n f0g) \ Z = ;g:
De nition 2.3 of proper Pareto optimality was originally given by Henig [5]. An example of a properly Pareto optimal criterion vector is given in Figure 1. 111111 000000 1 0 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111
Figure 1. Proper Pareto optimality.
3 Tangent and Normal Cones Next we de ne two geometrical basic tools: tangent and normal cones for nonconvex problems. The cones are de ned at a vector in the feasible criterion set. Then we can establish some connections to proper Pareto optimality. De nition 3.1 (Clarke). The tangent cone of a set Z Rk at z 2 Z is given by the formula Tz (Z ) = fd 2 Rk j for all tj & 0 and zj ! z with zj 2 Z , there exists dj ! d with zj + tj dj 2 Z g: The normal cone of Z at z 2 Z is the polar cone of the tangent cone, that is, Nz (Z ) = Tz (Z ) = fy 2 Rk j yT d 0 for all d 2 Tz (Z )g: It is important to note that tangent and normal cones are convex even in the case Z is nonconvex.
Lemma 3.2. Tz (Z ) and Nz (Z ) are closed and convex cones such that 0 2 Tz (Z ) \ Nz (Z ). Proof. See Theorems 4.1.2 and 4.1.4 of [9]. De nition 3.1 is a generalization of the corresponding convex notions.
Lemma 3.3. If Z is convex, then Tz (Z ) = cl fd 2 Rk j there exists t > 0 such that z + td 2 Z g and
Nz (Z ) = fy 2 Rk j (z 0 z )T y 0 for all z 0 2 Z g:
Proof. See Theorem 4.1.5 of [9]. The next two theorems characterize the relations between properly Pareto optimal solutions and the corresponding tangent and normal cones.
Theorem 3.4. If z 2 PP (Z ), then Tz (Z ) \ Rk n f0g = ;: The necessary condition above is also sucient if Z is convex.
Proof. If z = f (x) 2 PP (Z ), then (3.5) (z + C n f0g) \ Z = ;: Let us suppose that there exists d^ 2 Tz (Z ) \ Rk n f0g. If tj & 0, then it follows from the de nition of the tangent cone that there exists dj ! d^ with z + tj dj 2 Z . Let C be a convex cone such that Rk n f0g int C . Since d^ 2 Rk n f0g int C and dj ! d^, there exist j0 such that 0 6= dj 2 int C C n f0g for all j j0 . Because C is a cone and tj > 0, we have tj dj 2 C n f0g for all j j0 . Then we have z + tj0 dj0 2 (z + C n f0g) \ Z; which is a contradiction with (3.5). Suciency follows from Theorem 1 of [6]. Figure 2 depicts the empty intersection in Theorem 3.4.
111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111
Figure 2. Empty intersection. Theorem 3.6. If z 2 PP (Z ), then Nz (Z ) \ int Rk 6= ;: The necessary condition above is also sucient if Z is convex. Proof. Let z 2 PP (Z ), then by Theorem 3.4 we have Tz (Z ) \ Rk nf0g = ;, which is equivalent to Tz (Z ) \ Rk = f0g: Then (Tz (Z ) \ Rk ) = f0g = Rk : On the other hand, by Corollary 16.4.2 of [11] and by the fact that the sum of closed convex cones containing zero is closed, we get (Tz (Z ) \ Rk ) = cl (Tz (Z ) + (Rk ) ) = cl (Nz (Z ) + Rk+ ) = Nz (Z ) + Rk+: Then for any p 2 Rk , there exist y 2 Nz (Z ) and d 2 Rk+ such that p = y + d. If in particular p 2 int Rk , then pi < 0 for every i = 1; : : : ; k. Because di 0 for all i = 1; : : : ; k, then yi < 0 for all i = 1; : : : ; k and thus y 2 Nz (Z ) \ int Rk : Suciency follows from results in [6] and references therein.
4 Generalizing the Cone of Trade-o Directions Henig and Buchanan [6] have generalized the notion of traditional trade-o s into a cone of trade-o directions for convex problems. This cone of trade-o directions is the Pareto optimal subset of a tangent cone. We generalize this cone for nonconvex problems. Let us denote the Pareto optimal surface of a tangent cone by PTz (Z ) = P (Tz (Z )): Note that in a convex case the cone of trade-o directions of [6] coincides with PTz (Z ). We can show that PTz (Z ) is nonempty for any properly Pareto optimal point.
Theorem 4.1. If z 2 PP (Z ), then PTz (Z ) 6= ;: The necessary condition above is also sucient if Z is convex.
Proof. Let z 2 PP (Z ) and let us suppose that PTz (Z ) = ;. Then by the de nition of Pareto optimality
(d + Rk n f0g) \ Tz (Z ) 6= ; for all d 2 Tz (Z ). Especially by choosing d = 0, we get (Rk n f0g) \ Tz (Z ) 6= ;; which by Theorem 3.4 is a contradiction to the assumption that z 2 PP (Z ). Suciency follows from Theorem 2 of [6]. We close this section by formulating a result enabling the generation of vectors in PTz (Z ). The proof of the corresponding Theorem 5 in [6] is here valid as such.
Theorem 4.2. If d 2 PTz (Z ), then dT y 0 for all y 2 Nz (Z ) and dT y = 0 for some y 2 Nz (Z ) \ Rk n f0g. The relations of the cones in Theorem 4.2 are illustrated in Figure 3.
Figure 3. Cones and perpendicular vectors of Theorem 4.2.
5 Necessity Conditions for Normal Vectors Let us continue with a collection of de nitions and results from nonsmooth analysis.
De nition 5.1 (Clarke). The subdi erential of a locally Lipschitz continuous function h : Rn ! R is a set @h(x) = f 2 Rn j h (x; v) T v for all v 2 Rn g; where
h(u + tv) h(u) h (x; v) = limu!sup x t t&0
is the generalized directional derivative of h at x in the direction v. For subdi erentials we have
@
(5.2)
k X i=1
wi fi (^x)
k X i=1
wi @fi (^x);
where wi 2 R for i = 1; : : : ; k. In addition, (5.3) n o @ i=1 max f (^ x ) conv @f (^ x ) j i such that max f (^ x ) = f (^ x ) : i i i i ;:::;k i=1;:::;k For details, see [2]. We bring this paper to an end with some optimality results concerning normal cones. In our nonconvex case we can establish somewhat weaker optimality conditions when compared to convex problems (cf. [6]). Problem (2.1) is said to satisfy the Cottle constraint quali cation at x^ if either gj (^x) < 0 for all j = 1; : : : ; m, or 0 2= conv f@gj (^x) j gj (^x) = 0g:
Theorem 5.4. Let z^ = f (^x) 2 PP (Z ) and let x^ 2 S satisfy the Cottle constraint quali cation. If y 2 Nz^(Z ) \ int Rk , then there exist 0 2 Rk , = 6 0, and 0 2 Rm such that j gj (^x) = 0 for every j = 1; : : : ; m, and m i @f (^x) + X j @gj (^x): i i=1 yi j =1 Proof. According to Theorem 3.6 there exists y 2 Nz^(Z ) \ int Rk whenever z^ 2 PP (Z ). This means that yi < 0 for every i = 1; : : : ; k. Let us de ne for every x 2 S a function
02
k X
F (x) = i=1 max ;:::;k
1 yi (fi (x) z^i yi ) :
This function attains its minimum at x^ 2 S . If this was not the case, there would exist x 2 S such that 1 (f (x ) z^ y ) < max 1 (f (^x) z^ y ) = 1: max i=1;:::;k
yi i
i
i
i=1;:::;k
yi i
i
i
This means that y1 (fi (x ) z^i yi ) < 1 for every i = 1; : : : ; k. In other words, fi (x ) < z^i for every i = 1; : : : ; k, which contradicts the proper Pareto optimality of z^. Now we know that F (^x) F (x) for every x 2 S . Applying necessary Karush-Kuhn-Tucker optimality conditions (see, e.g. [9]) we know that there exists 0 P2 Rm such that j gj (^x) = 0 for every j = 1; : : : ; m and 0 2 @F (^x) + m j =1 j @gj (^x): According to (5.2) and (5.3) we have 1 @F (^x) conv @ (f (^x) z^ y ) j i = 1; : : : ; k i
conv conv
i i yi i 1 @ (f (^x) z^ y ) j i = 1; : : : ; k i i y i
i
1 @f (^x) j i = 1; : : : ; k : y i i
>From the we know that there exist 0 2 Rk Pkde nition of convex hulls P such that i=1 i = 1 and @F (^x) ki=1 y @fi (^x). This completes the proof. Under the convexity assumption the Cottle constraint quali cation is equivalent to the so-called Slater constraint quali cation, that is, there exists at least one x 2 int S . We state the next theorem from [6] in order to facilitate the comparison between convex and nonconvex cases. Theorem 5.5. Let problem (2.1) be convex and satisfy the Slater constraint quali cation. If z^ = f (^x) 2 Z , then y 2 Nz^(Z ) if and only if there exist 0 2 Rm such that j gj (^x) = 0 for every j = 1; : : : ; m and i i
02
k X i=1
yi @fi (^x) +
m X j =1
j @gj (^x):
We write down the following general optimality results which can directly be derived from the theorems above. Corresponding results for Pareo optimality are given, for example, in [13]. The convexity part is derived in [6].
Corollary 5.6. Let x^ 2 S satisfy the Cottle constraint quali cation. A necessary condition for z^ = f (^x) 2 PP (Z ) being valid is that there exist 0 2 Rk , 6= 0, and 0 2 Rm such that j gj (^x) = 0 for every j = 1; : : : ; m and
02
k X i=1
i @fi (^x) +
m X j =1
j @gj (^x):
If problem (2.1) is convex, then > 0. Further, the condition is also sucient.
6 Discussion In this paper, we have studied the possibility of generalizing the convex results of [6] into nonconvex cases. The starting point of our treatment was to utilize the convexity property of Clarke's tangent cones in nonconvex cases. This enables the exible generalization of the convex results. However, using tangent cones in de ning trade-o directions in nonconvex cases may blur the original idea (see Figure 2). In this sense, starting from feasible directions would re ect the trade-o idea better. Unfortunately, the cone of feasible directions is nonconvex whenever the feasible criterion set is not locally convex. This means that handling trade-o directions on the basis of feasible directions requires other tools and further research.
7 Conclusions We have studied how results originally given in [6] concerning the cone of trade-o directions, tangent and normal cones, and proper Pareto optimality can be generalized from convex to nonconvex problems. Some of the results can be extended analogously. Nevertheless, the necessary optimality result concerning normal cones and properly Pareto optimal solutions di ers from its convex counterpart in both assumptions and formulation.
Acknowledgements This research was supported by the grants 22346 and 8583 of the Academy of Finland.
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