Second-Order Optimality Conditions in Multiobjective Optimization Problems

Second-Order Optimality Conditions in Multiobjective Optimization Problems 1

B. AGHEZZAF

AND M. HACHIMI

Communicated by P. L. Yu

Abstract. In this paper, we develop second-order necessary and sufficient optimality conditions for multiobjective optimization problems with both equality and inequality constraints. First, we generalize the Lin fundamental theorem (Ref. 1) to second-order tangent sets; then, based on the above generalized theorem, we derive second-order neces- sary and sufficient conditions for efficiency.

Key Words. Multiobjective optimization, efficient solutions, constraint qualifications, second-order tangent sets, second-order necessary and sufficient conditions.

1. Introduction

Optimality conditions for multiobjective optimization problems have been studied extensively in the literature. Many efforts have been made to derive first-order necessary and/or sufficient conditions for a feasible solu- tion to be an efficient solution (Refs. 1-4). However, little work has been concerned with second-order optimality conditions for multiobjective opti- mization problems. Following Aghezzaf (Ref. 5), we investigate second- order optimality conditions for a multiobjective optimization problem with both equality and inequality constraints. First, we generalize the Lin funda- mental theorem (Ref. 1, Theorem 5.1) to second-order tangent sets; then, based on this generalized theorem, we derive second-order necessary condi- tions for efficiency.

1

The authors would like to thank Professor P. L. Yu and the referees for many valuable comments and helpful suggestions.

2

Professor, Departement de Mathematiques et d'Informatique, Faculte des Sciences Am Chock, Universite Hassan II, Casablanca, Morocco.

3

Graduate Student, Departement de Mathematiques et d'Informatique. Faculte des Sciences Ain Chock, Universite Hassan II, Casabalanca, Morocco.

37

0022-3239/99/0700-0037$16.00/0 C 1999 Plenum Publishing Corporation

To establish second-order necessary conditions, Aghezzaf 's Theorems 3.1 and 3.2 (Ref. 5) utilize the generalized Guignard second-order constraint qualification for a set described by the feasible region constraints of the original problem plus an additional group of constraints, derived from the problem objective functions. In this paper, we relax the above additional constraints. Furthermore, in the proof of our second-order necessary condi- tions, we do not require the assumption of weak convex inclusion as is done in Wang (Ref. 6, Theorem 3.4). Finally, for a certain class of multiobjective optimization problems, second-order sufficient conditions are also developed and proved.

The paper is organized as follows. In Section 2, we formulate the multi- objective optimization problem with both equality and inequality constraints and provide some definitions and basic results, which are to be used throughout the paper. In Section 3, we extend and demonstate the Lin fundamental theorem for second-order tangent sets. Using the latter gen- eralized theorem, we derive second-order necessary conditions for a feasible solution to be an efficient solution to a multiobjective optimization problem.

In Section 4, for a certain class of multiobjective optimization problems, we develop and prove second-order sufficient conditions for efficiency.

2. Preliminaries

In this section, we introduce some definitions and basic results, which are to be used throughout the paper. Let R n be the n-dimensional Euclidean space, and let x = ( x

, . . ., x

)

and y = (y

• • •, y

)

be any two vectors in R n ; here, the superscript T denotes the transpose of x. We denote the inner product of x and y by

and use the following conventions:

Also, for any two vectors x = ( x

, x

)

and y = (y

, y

)

in R ² , we use the following conventions:

here, the subscript lex means lexicographic order.

Now, let f: R n ->R l be any twice continuously differentiable l-vector function, and let y be any vector in R n ; we denote by Vf(x) [resp.

V 2 f(x)(y,y)] the lXn Jacobian [resp. l X H X n 3D-Hessian] of f at x e R n

whose i th row is Vf i (x) [resp. .y T V 2 f i (x)y], the gradient vector of f i at xeR n

[resp. V 2 f i (x), Hessian matrix of f i at xeR n ].

In this paper, we consider the following multiobjective optimization problem:

where f: R n -»R l g: R n ->R m , h: R n -»R q are twice differentiable functions.

Due to the conflicting nature of the objectives, an optimal solution that simultaneously minimizes all the objectives is usually not obtainable. Thus, for Problem (P), the solution is defined in terms of an efficient solution (see Ref. 7).

The following Definition 2.1 states the concept of efficiency for a feasible solution of Problem (P); Definition 2.2 refers to the widely used concept of tangent cone; Definition 2.3 introduces an approximating cone to the feas- ible region of Problem (P) at any of its points.

Let x e A be any feasible solution to Problem (P), and let E be the subset of indexes defined by

Definition 2.1. A point xeA is called an efficient solution to Problem (P) if there is no xeA such that f(x)<f(x).

Definition 2.2. The tangent cone to A at xeA is the set defined by

where o(t n ) is a vector satisfying ||o(t n )||t n -»0.

Definition 2.3. The linearizing cone to A at xeA is the set defined by

3. Second-Order Necessary Conditions

In order to derive the second-order necessary conditions for efficiency, we define two kinds of second-order approximation sets to the feasible region

(P) min f(x),

(Ref. 8). These are more precise approximations and they can be considered as extensions of T 1 (A, x) and L 1 .

Definition 3.1. The second-order tangent set to A at xeA is the set defined by

where o(t 2 ) is a vector satisfying \\o(t 2 )\\/t 2 ->0.

Definition 3.2. The second-order linearizing set to A at x is the set defined by

The y-sections of L 2 and T 2 ( A , x ) will be denoted by L 2 ( y ) and :T 2 (A,x)(y);that is,

and the following lemma holds.

Lemma 3.1. See Ref. 8. Let x be any feasible solution to Problem (P).

Then, we have

Second-Order Constraint Qualification. We say that A satisfies a second-order Abadie constraint qualification (ACQ) at xeA if

A first-order sufficient conditions for efficiency is that the following

system has no nonzero solution y:

The Kuhn-Tucker type condition for efficiency is equivalent (Refs. 2-4) to the inconsistency of the following system:

The gap between (5)-(7) and (8)-(10) is caused by the following directions:

A direction y which satisfies (11)-(14) is called a critical direction.

For the sake of simplicity, we use the following notation:

In the following, we present the Lin fundamental theorem (Ref. 1). Then, using a small example, we show that in some situations this theorem fails to determine whether a given solution is efficient or not.

Theorem 3.1. Fundamental Theorem. If xeA is (weak) efficient solu- tion to Problem (P), then

where

Now, we consider the following example.

Fig. 1. Point x = (0,0)

is not efficient for Problem (PI).

Example 3.1.

(P1) min f(x) = ( x ¹ , x ² ) ,

The situation is shown schematically in Fig. 1.

Let us consider the feasible solution x = (0, 0) ^T . Then, we have

At this stage, we cannot say whether x is an efficient solution for Problem (P). Example 3.1 shows that Theorem 3.1 is not applicable to such a simple case. Now, we generalize the above theorem to remedy the situation.

Theorem 3.2. If xeA is an efficient solution to Problem (P), then the following result holds:

where

Proof. Clearly, the theorem holds trivially if z=f(x) is an isolated point off(A). Now, assume that z=f(x) is an accumulation point off (A), and let (y, z) be any element of T 2 ( f ( A ) , z). Therefore, there exists z n e f ( A ) and t n -»+0 such that

Now, suppose that (y, z)eQ; since the sequence z n can be chosen so that z n /z for large n (z is an accumulation point), for each i we have

(a) If y i < 0, from (16) we get

Hence, there exists N i such that

Hence, there exists M, such that

To sum up:

Now, if we set

then we get

Recall that z n e f ( A ) ; thus, there exists x n eA such that z n = f ( x n ) and

But this contradicts the fact that z is an efficient solution, and this completes the proof. D

(b) If y i = 0, hence z i <0, from (9) we get

Returning to Example 3.1, and now using our result, we have

since

Therefore, by Theorem 3.2, we conclude that x is not efficient solution.

Now, we are in a position to state the primal form of our second-order necessary conditions.

Theorem 3.3. Let x be an efficient solution to Problem (P), and assume that the second-order (ACQ) holds at xeA. Then, the following system has no solution (y, z):

Proof. Let (y, z) be any element of T 2 (A, x). Then, there exist x n eA and t n ->+0 such that

By the Taylor expansion,

which implies that

Since x is an efficient solution to Problem (P), using Theorem 3.1, we have

where

Also, using the assumption, we obtain

Hence, the following system has no solution (y, z):

The proof is now complete.

In the following, for simplicity, we rewrite (17)-(19) as

It may be noticed that Theorem 3.3 contains the first-order optimality condi- tions for efficiency (Refs. 3 and 4). In fact, by setting y = 0, the first-order optimality conditions are embedded in (17)-(19).

Now, we state the dual form of Theorem 3.3.

Theorem 3.4. Let x satisfy the assumptions made in Theorem 3.2.

Then, for each critical direction y, there exist multipliers Ae R l , u e R m , veR q

such that

Proof. Let y be a critical direction. Then, the system

has no solution z. This is equivalent to

which has no solution zeR n , t e R. By the Motzkin theorem for the alterna- tive (Ref. 9), there exist multipliers £eR, AeR l , neR m , v e K q such that

Since (A, £) > 0 implies

there exist multipliers AeR l , ueR m , veR q such that either (28)-(30) or (31)- (33) below hold:

Let us assume that (31)-(33) does not hold. It is equivalent to the inconsist- ency of the system

or

By the Motzkin theorem of the alternative (Ref. 9), there exist z and t>0 satisfying

Since (26) and (27) have no solution, we have t = 0; hence,

On the other hand,

because y is critical. Thus, it holds that

for any sufficiently small e>0, which contradicts the first-order necessary conditions for efficiency. This completes the proof. D

Now, we turn to discussing second-order sufficient conditions.

4. Second-Order Sufficient Conditions for Efficiency

In this section, we will give the second-order sufficient conditions for a feasible solution xeA to be an efficient solution to Problem (P).

Theorem 4.1. Suppose that f,g are quasiconvex, h is quasilinear, and all are twice continuously differentiable at xeA. If for each critical direction y=0, there exist AeR l , ueR m , veR q such that

then x is an efficient solution to Problem (P).

Proof. Assume that, for each critical direction y=0, there exist AeR l , ueR m , veR q such that (34)-(36) hold and that x is not efficient solution to Problem (P). Then, there is xeA such that

Using the quasiconvexity of f and g, the quasilinearity of h, and (37), we obtain

Two cases are to be considered.

Case 1. If Vf(x)(x-x)<0, for d=x-x, the following system is consistent:

By the Motzkin theorem of the alternative, the following system is inconsistent:

thereby contradicting (34) and (36).

Case 2. If V f r ( x ) ( x - x ) = 0, for at least one r e { 1 , . . . , l}, then d=

x — x is a nonzero critical direction. Then, taking

and from the quasiconvexity of f, we have

hence,

Similarly,

Using the assumption, there exist AeR l , ueR m , v e R q such that (34)-(36) hold. Multiplying (38), (39), (40) by A, u, v and summing, we get

From the expressions (34), (36), and t>0, we obtain

Therefore, we have

5. Conclusions

In this paper, we have developed two second-order necessary conditions and a second-order sufficient condition for multiobjective optimization prob- lems with both equality and inequality constraints. We have generalized the Lin fundamental theorem to second-order tangent sets, which is required in the necessary conditions. Another result, less important but worth mention- ing, is the relaxation of the assumption of weak convex inclusion made in Wang (Ref. 6, Theorem 3.4), which was not required for the proof of our second-order necessary conditions for efficiency. Finally, the results derived here might be utilized to design new algorithms for solving multiobjective optimization problems and might also be useful for sensitivity analyses.

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