ON NECESSARY OPTIMALITY CONDITIONS IN MULTIFUNCTION OPTIMIZATION WITH PARAMETERS

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1 ACTA MATHEMATICA VIETNAMICA Volume 25, Number 2, 2000, pp ON NECESSARY OPTIMALITY CONDITIONS IN MULTIFUNCTION OPTIMIZATION WITH PARAMETERS PHAN QUOC KHANH AND LE MINH LUU Abstract. We consider multifunction optimization problems with parameters. Necessary optimality conditions of the Fritz John and Kuhn- Tucker types are obtained with relaxed differentiability assumptions on the state variable and convexlikeness assumptions on the parameter. 1. Introduction The range of application of multifunctions, i.e., set-valued functions is very large. An optimization theory involving multifunctions was established by Corley [1, 2, 3]. In [3] the Fritz John necessary optimality condition is extended to multifunction optimization. In [11, 12, 14] there are improvements of the above result in various cases. This note is devoted to considering necessary optimality conditions for multifunction optimization problems with parameters of the following form. Let X, Y, Z and W be Banach spaces, Y and Z being ordered by convex cones K and M, containing the origins and with int K, int M, respectively. Let U be an arbitrary set. Let F, G be multifunctions of X U into Y and Z, respectively. Let p be a (single-valued) map of X U into W. The problem under consideration is min F (x, u), G(x, u) M, p(x, u) = 0; (P ) Received August 13, 1998; in revised form December 7, Key words and phrases. Multifunction optimization, Fritz John and Kuhn-Tucker necessary optimality conditions, Lusternik theorem, uniform K-differentiability in a direction, K-convexlikeness, K-strong lower semicontinuous with T. This work was supported in part by Fonds International de Coopération Universitaire FICU (AUPELF-UREF) and by the National Basic Research Program in Natural Sciences.

2 126 PHAN QUOC KHANH AND LE MINH LUU or min F (x, u), G(x, u) ( M), p(x, u) = 0. ( P ) Here min means that we are looking for a (Pareto) minimum or a weak minimum. A multifunction F : X Y is said to have a (global) minimum at ( ; f 0 ), where f 0 F( ), on a set A X if (1) F(A) f 0 Y \ (( K) \ K). If (1) is replaced by (2) F(A) f 0 Y \ ( int K). then ( ; f 0 ) is called a (global) weak minimum of F on A. If there is a neighborhood N of such that one has (1) or (2) with F(A) replaced by F(A N), then ( ; f 0 ) is called a local minimum or local weak minimum, respectively, of F. Since U is an arbitrary set, on considering local minima or local weak minima of (P ) and ( P ) we adopt for U the trivial topology consisting of only two sets and U, and for X U the product topology. Problems with parameters of the types (P ) and ( P ) are often met in practical situations. Such situations usually require that the assumptions imposed on x and on u should be different. For instance, in control problems differentiability assumptions imposed on the control u should be much weaker than that on the state variable x. (P ) coincides with ( P ) if G is single-valued. For both constraints G(x, u) M and G(x, u) ( M) become then the inequality constrainst G(x, u) 0. (For the sake of simplicity the notation is used commonly for various ordering in various spaces if no confusion may occur.) When F and G are single-valued, optimization problems with parameters were considered, e.g. in [5, 6, 7, 8, 9, 10, 13]. In the present note we restrict ourselves to the case where p is single-valued since in most applications equality constraints represent state equations (being often differential equations), initial and boundary conditions. Let Z stand for the topological dual to Z. The dual cone M of M is M := { µ Z : µ, z 0 z M }.

3 ON NECESSARY OPTIMALITY CONDITIONS 127 In later considered situations a feasible point (, u 0 ) and some g 0 G(, u 0 ) ( M) will be fixed. Then we set M 0 := { γ(z + g 0 ) : γ R +, z M }, M 0 := { µ M : µ, g 0 = 0 } = (M 0 ). For a multifunction F : X Y its graph is gr F := { (x, y) X Y : y F(x) } and its domain is dom F := {x X : F(x) }. We recall that the Clarke derivative of F at ( ; f 0 ) gr F, denoted by DF( ; f 0 ), is a multifunction of X into Y whose graph is grdf( ; f 0 ) = { (v, w) X Y : (x n, f n ) F (, f 0 ), t n 0 +, (v n, w n ) (v, w); n, f n + t n w n F(x n + t n v n ) }, where F means that (x n, f n ) gr F and (x n, f n ) (, f 0 ). Recall also that DF(, f 0 ) is always a closed convex process, i.e. a multifunction whose graph is a nonempty closed convex cone. For a single-valued map p : X U W, p x (, u 0 ) will stand for the Fréchet derivative of p(., u 0 ) at. In our consideration only the following directional differentiability with respect to x is imposed on F and G. Definition 1. Let X and Y be Banach spaces, Y being ordered by a convex cone K. A multifunction F : X Y is called uniformly K- differentiable in the direction x X at (, f 0 ) gr F if for each neighborhood V of zero in Y there is a neighborhood N of x and a real γ 0 > 0 such that γ (0, γ 0 ), x N, f F( + γx), f DF( ; f 0 )x 1 γ (f f 0) f V K. F is said to be uniformly K-differentiable at ( ; f 0 ) if this differentiability holds for all directions x in dom DF ( ; f 0 ). Note that the parameter set U in (P ) and ( P ) is equipped with no structure. However, the following extension to multifunctions of the convexlikeness introduced by Fan [4] is needed.

4 128 PHAN QUOC KHANH AND LE MINH LUU Definition 2. A multifunction F : U Y where U is a set and Y is a vector space ordered by a convex cone K, is said to be K-convexlike in (U 1, U 2 ) with U 1 U, U 2 U if (u 1, u 2 ) U 1 U 2, f 1 F(u 1 ), f 2 F(u 2 ), γ [0, 1], u U, f u F(u), γf 1 + (1 γ)f 2 f u K. Definition 3. Let X, Y and F be as in Definition 1. Let dom F and T F( ) be nonempty. Then, F is called K-strong lower semicontinuous (K-s.l.s.c.) with T at if for each neighborhood V of zero in Y, there is a neighborhood N of such that x N, f x F(x), f x T V K. Notice that, if T = F( ) and F is K-s.l.s.c. with T, then F(.) + K is l.s.c. in the usual sense (for multifunctions). 2. Main results First we present necessary optimality conditions for local weak minima of ( P ). Of course these necessary conditions hold also for local minima. Theorem 1 (Fritz John necessary condition). Assume that (i) p(., u 0 ) is continuously differentiable at and p x (, u 0 ) is onto, where (, u 0 ) is feasible for ( P ); (ii) F (., u 0 ) and G(., u 0 ) are uniformly K-differentiable at (, u 0 ; f 0 ) and uniformly M-differentiable at (, u 0 ; g 0 ), respectively, where f 0 F (, u 0 ), g 0 G(, u 0 ) ( M); (iii) for each u u 0, F (., u) (G(., u), respectively) is K-s.l.s.c. with F (, u) (M-s.l.s.c. with G(, u)) at. Moreover, F (., u 0 ) (G(., u 0 )) is K-s.l.s.c. with f 0 ( M-s.l.s.c. with g 0, respectively) at ; (iv) for each x in a neighhorhood V of, (F, G, p)(x,.) is K M {0}-convexlike in (U, {u 0 }). Moreover, (F, G, p)(,.) is K M {0}- convexlike in (U, U). If (, u 0 ; f 0 ) is a local weak minimum of ( P ), then there exists such that, for all (x, u) X U, (λ 0, µ 0, ν 0 ) K M 0 W \ {0}

5 ON NECESSARY OPTIMALITY CONDITIONS 129 (3) λ 0, D x F (, u 0 ; f 0 )x + F (, u) f 0 + µ 0, D x G(, u 0 ; g 0 )x + G(, u) g 0 + ν 0, p x (, u 0 )x + p(, u) R +. Proof. It is clear that (3) is true for x dom D x F (, u 0 ; f 0 ) dom D x G(, u 0 ; g 0 ) since the leff-hand side of (3) is empty. So we may assume that x belongs to this intersection. However, for the simplicity of notation, we still write x X. Let C be the set of all (y, z, w) Y Z W such that (x, u) X U, f x D x F (, u 0 ; f 0 )x, f u F (, u), g x D x G(, u 0 ; g 0 )x, g u G(, u), (4) (5) (6) f x + f u f 0 y int K, g x + g u g 0 z int M, p x (, u 0 )x + p(, u) = w. We claim that C is convex. Indeed, if (y i, z i, w i ) C, i = 1, 2, then, with the notations defined similarly as the terms in (4)-(6), for all γ [0, 1] one has (7) γf x 1 + (1 γ)f x 2 + γf u 1 + (1 γ)f u 2 f 0 γy 1 + (1 γ)y 2, (8) γg x 1 + (1 γ)g x 2 + γg u 1 + (1 γ)g u 2 g 0 γz 1 + (1 γ)z 2, (9) p x (, u 0 )(γx 1 + (1 γ)x 2 ) + γp(, u 1 ) + (1 γ)p(, u 2 ) = γw 1 + (1 γ)w 2. By the convexity of the Clarke derivative, setting x := γx 1 + (1 γ)x 2 one sees that f x := γf x 1 + (1 γ)f x 2 g x := γg x 1 + (1 γ)g x 2 D x F (, u 0 ; f 0 )x, D x G(, u 0 ; g 0 )x. The K M {0}-convexlikeness of (F, G, p)(,.) in (U, U) yields a u U, f u F (, u) and g u G(, u) such that

6 130 PHAN QUOC KHANH AND LE MINH LUU f u γf u 1 + (1 γ)f u 2, g u γg u 1 + (1 γ)g u 2, p(, u) = γp(, u 1 ) + (1 γ)p(, u 2 ). Consequently, (7)-(9) together show that (y, z, w) := γ(y 1, z 1, w 1 ) + (1 γ)(y 2, z 2, w 2 ) belongs to C, i.e. C is convex. Next we show that int C. Assume that (y, z, w) C. Then for a small and fixed ε > 0, (10) (11) (12) f x + f u f 0 y + εb Y int K, g x + g u g 0 z + εb Z int M, p x (, u 0 )x + p(, u) = w, where B X, B Y and B Z are open unit balls in X, Y and Z, respectively. By the uniform K-differentiability of F (., u 0 ) assumed in (ii), δ > 0, γ 0 > 0, x x + δb X, γ (0, γ 0 ), f F ( + γx, u 0 ), (13) 1 γ (f f 0) f x ε 4 B Y K. Now for each x x + δ 2 B X, from f x D x F (, u 0 ; f 0 )x it follows that, for the given ε, γ (0, γ 0 ), x x + δ 2 B X, f F ( + γx, u 0 ), (14) f x 1 γ (f f 0) ε 4 B Y. (13) and (14) together show that for all x ˆx + δ 2 B X, f x D x F (, u 0 ; f 0 )x, γ (0, γ 0 ), x x+ δ 2 B X, f F ( +γx, u 0 ), (15) f x f x = f x 1 γ (f f 0) + 1 γ (f f 0) f x ɛ 2 B Y K. Using (10), (15) one sees that, for all x x + δ 2 B X and f x D x F (, u 0 ; f 0 )x,

7 ON NECESSARY OPTIMALITY CONDITIONS 131 f x + f u f 0 y + ε 2 B Y K int K int K. Similarly, for all x x + δ 2 B X and g x D x G(, u 0 ; g 0 )x, g x + g u g 0 z + ε 2 B Y int M. Considering (12), one sees from assumption (i) that p x (, u 0 )( δ 2 B X p(, u) contains an open neighborhood w + ε 1 B W of w. Now it is easy to check that (y + ε ( 2 B Y ) z + ε ) 2 B Z (w + ε 1 B W ) C, i.e. int C. If (16) C {( int K) ( int M 0 ) {0})} =, then by a standard separation theorem we obtain (3). So, it remains to prove (16). Suppose to the contrary that there are (ˆx, û) X U, f ˆx D xf (, u 0 ; f 0 )ˆx, g ˆx D xg(, u 0 ; g 0 )ˆx, f û F (, û) and gû G(, û) such that ) + (17) (18) (19) f ˆx + f û f 0 int K, g ˆx + gû g 0 int M 0, p x (, u 0 )ˆx + p(, û) = 0. Defining a new map P : X R W by (20) P(x, α) := αp( + x, û) + (1 α)p( + x, u 0 ), we see that P(0, 0) = 0, P (0, 0)(x, α) = p x (, u 0 )x+αp(, û) and P (0, 0)(ˆx, 1) = 0. Hence, by the Lusternik theorem, there exist t 0 > 0 and maps t ˆx(t), t ˆα(t) of [0, t 0 ] into X and R, respectively, such that ˆx(t) 0 and ˆα(t) 0 as t does and, for all t [0, t 0 ],

8 132 PHAN QUOC KHANH AND LE MINH LUU (21) P(t(ˆx + ˆx(t), t(1 + ˆα(t))) = 0. Setting x(t) := + t(ˆx + ˆx(t)), from (20) and (21) we obtain t(1 + ˆα(t))p(x(t), û) + (1 t(1 + ˆα(t)))p(x(t), u 0 ) = 0. Hence, by (iv), for all t small enough, there exists u(t) U such that p(x(t), u(t)) = 0. A contradiction to the minimality of (, u 0 ; f 0 ) will be achieved if we can show that for all sufficiently small t, there is f t F (x(t), u(t)) such that G(x(t), u(t)) ( M), f t f 0 int K. These two facts should be shown by the following common argument, which we write down explicitly only for G. By virtue of the assumption (iii), for all sufficiently small t and ε, there are gûx(t) G(x(t), û) and g u 0 x(t) G(x(t), u 0) such that (22) (23) gûx(t) gû εb Z M, g u 0 x(t) + g 0 εb Z M. In turn, the uniform M-differentiability of G(., u 0 ) assumed in (ii) ensures that, for all δ and t small enough, (24) g u 0 x(t) g 0 + tg ˆx + tδ 2 B Z M. According to (18) we choose δ so small that (25) g ˆx + gˆx g 0 + δb Z M int M 0. Estimating g u 0 x(t) + t(1 + ˆα(t))(gûx(t) gu 0 x(t) ) we get, by (22), (23) and the fact ˆα(t) 0 that t(1 + ˆα(t))(gûx(t) gu 0 x(t) ) = t(1 + ˆα(t))(gûx(t) gû g u 0 x(t) + g 0) + t(gû g 0 ) + tˆα(t)(gû g 0 ) t(1 + ˆα(t))(2ɛB Z M) + t(gû g 0 ) + tɛb Z t(gû g 0 + δ 2 B Z M)

9 ON NECESSARY OPTIMALITY CONDITIONS 133 for all ε > 0 small enough. This together with (24) give b t δ δb Z and m t M such that (26) g u 0 x(t) + t(1 + ˆα(t))(gûx(t) gu 0 x(t) ) = g 0 + t(g ˆx + gû g 0 + b t δ m t ). Hence, in view of assumption (iv), there is g t G(x(t), u(t)) such that (27) g 0 + t(g ˆx + gû g 0 + b t δ m t ) g t + M. To verify that G(x(t), u(t)) ( M) for all t > 0 small enough, we assume to the contrary that t n 0 +, µ n M, µ n = 1 (then assume that µ n-weakly tends to µ M ), µ n, g tn 0. µ must belong to M 0. Indeed, if µ M 0, there would be β > 0 such that µ, g 0 < β. On the other hand, it follows from (27) that (28) µ n, g tn µ n, g 0 + t n µ n, g ˆx + gû g 0 + b t n δ m t n. Therefore, for sufficiently small t n, µ n, g tn < 0, which is a contradiction. Now since µ M 0, a glance at (25) yields, for large n, t n µ n, g ˆx + gû g 0 + b t n δ m t n < 0. Therefore (28) contradicts the fact that µ n, g tn 0. The above argument applied to F instead of G will give f t F (x(t), u(t)) such that (similarly as (27), (25)) f t f 0 K + t(f ˆx + f û f 0 + b t δ k t ) K int K = int K. Thus, (16) must hold and the proof is complete. Remark. The necessary condition (3) has a form of the classical multiplier rule. We can write this condition in the following equivalent form (a) (b) λ 0, D x F (, u 0 ; f 0 )x + µ 0, D x G(, u 0 ; g 0 )x + ν 0, p x (, u 0 )x R + x X; λ 0, f 0 + µ 0, g 0 + ν 0, p 0 { = min f(λ0,, u) + g(µ 0,, u) + p(ν 0,, u) }, u U where p 0 = p(, u 0 ) = 0,

10 134 PHAN QUOC KHANH AND LE MINH LUU f(λ 0,, u) = min { λ 0, y : y F (, u) }, g(µ 0,, u) = min { µ 0, z : z G(, u) }, and p(ν 0,, u) = ν 0, p(, u). This equivalent form was used in [5, 7, 8, 9, 10] instead of (3). Adding a constraint qualification of the Slater type we get from Theorem 1 an extension of the Kuhn-Tucker necessary condition as follows. Theorem 2. In additon to the assumptions of Theorem 1, assume that p x (, u 0 )X +p(, U) contains a neighborhood of zero in W and that there are ( x, ũ) X U, g x D xg(, u 0 ; g 0 ) x and gũ G(, ũ) such that Then, λ 0 0. g x + gũ g 0 int M 0, p x (, u 0 ) x + p(, ũ) = 0. Proof. Suppose λ 0 = 0. If µ 0 = 0, then ν 0 0. Since p x (, u 0 )X + p(, U) includes a neighborhood of zero, one can choose (x 1, u 1 ) X U such that ν 0, p x (, u 0 )x 1 + p(, u 1 ) < 0, contradicting (3). So µ 0 must be non zero. But, then ( x, ũ) does not satisfy (3). Thus, λ 0 0. Now we consider the problem (P). We shall see that with a small modification in the assumption (iv) the two theorems are still valid for (P ). We need the following definition Definition 4. Let F, G and p be defined as in the problem (P). Then (F, G, p)(x,.) is K M {0}-convexlike in (U, {u 0 }) strongly with respect to G if u 1 U, (f 1, g 1 ) F (x, u 1 ) G(x, u 1 ), (f 0, g 0 ) F (x, u 0 ) G(x, u 0 ), γ [0, 1], u U, g u G(x, u), f u F (x, u), γf 1 + (1 γ)f 0 f u K, γg 1 + (1 γ)g 0 g u M, γp(x, u 1 ) + (1 γ)p(x, u 0 ) = p(x, u). It is easy to check that Theorems 1 and 2 still hold for (P) if the convexlikeness in (U, {u 0 }) assumed in (iv) is replaced by the convexlikeness in (U, {u 0 }) strong with respect to G.

11 ON NECESSARY OPTIMALITY CONDITIONS Remarks We discuss the assumptions of Theorems 1 and 2. Observe first that (i) is rather strict but it is probably inevitable (in a certain sense) because it is usually imposed on the equality constraint in order to apply the Lusternik theorem and it is satisfied in various practial situations. The uniform differentiability on (ii) is an extension to multifunctions of the corresponding notion of Ioffe-Tihomirov [5]. The example below gives a multifunction that satisfies both (ii) and (iii). The K M {0}-convexlikeness assumed in (iv) is of course much weaker than the K M {0}-convexity often imposed in convex cases. Example. Let X = R, U = R and X. Let G : X U R defined by G(x, u) := (x ) 2 ( u [0, 1] 1). We verify (ii) and (iii) for (, u 0 ) := (, 1). (ii) For given ε > 0 and x X we will find δ > 0 such that, x ( δ, + δ), g D x G(, u 0 ; g 0 )x, γ > 0, (29) G( + γx, u 0 ) g 0 γg γ( ɛ, ɛ) R +, where g 0 G(, u 0 ) = {0}. First we show that D x G(, u 0 ; 0)x = {0} x X. By definition, g D x G(, u 0 ; 0)x means that g n 0, x n, t n 0 +, x n x, g n g, n, g n + t n g n G(x n + t n x n, u 0 ) [ (x n + t n x n ) 2, 0]. Choosing g n 0, x n yields g n [ t n x 2 n, 0] n. Therefore, since g n g, t n 0 + and x n x, g = 0. Then (29) becomes [ γ 2 x 2, 0] γ( ɛ, ɛ) R +, which always holds (for any δ > 0 and γ > 0). (iii) For any u u 0 and ε > 0 we can find δ > 0 such that x ( δ, + δ), G(x, u) G(, u) [0, (x ) 2 u ] (x ) 2 ( ε, ε) R +. (This property is even stronger than the R + -s.l.s.c. with G(, u).) Note ε that δ may be taken as u.

12 136 PHAN QUOC KHANH AND LE MINH LUU Now we check a property stronger than the R-s.l.s.c. with g 0 = 0 of G(., u 0 ). For any ε > 0, one can find δ > 0 such that, x ( δ, +δ), G(x, u 0 ) g 0 [ (x ) 2, 0] ( ɛ, ɛ) + R +. So one may choose δ = ε. Let us observe that Theorems 1 and 2 contain as special cases the corresponding results in [5, 9] for the problem with p x (, u 0 )X = W. Necessary optimality conditions for (P ) and ( P ) when p x (, u 0 )X has a finite codimention need further considerations and are the aim of our forthcoming research. References 1. H. W. Corley, Duality theory for maximizations with respect to cones, J. Mathematical Analysis and Applications 84 (1981), H. W. Corley, Existence and Lagrange duality for maximizations of set-valued functions, J. Optimization Theory and Applications 54 (1987), H. W. Corley, Optimality conditions for maximizations of set-valued functions, J. Optimization Theory and Applications, 58 (1988), Ky Fan, Minimax theorems, Proceedings Nat. Acad. of Sciences 39 (1953), A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, North-Holland Publishing Company, Amsterdam, Holland, Phan Quoc Khanh, Proper solutions of vector optimization problems, J. Optimization Theory and Applications 74 (1992), Phan Quoc Khanh, Invex-convexlike functions and duality, J. Optimization Theory and Applications 87 (1995), Phan Quoc Khanh, Sufficient optimality conditions and duality in vector optimization with invex-convexlike functions, J. Optimization Theory and Applications 87 (1995), Phan Quoc Khanh and Tran Hue Nuong, On necessary optimality conditions in vector optimization problems, J. Optimization Theory and Applications 58 (1988), Phan Quoc Khanh and Tran Hue Nuong, On necessary and sufficient conditions in vector optimization, J. Optimization Theory and Applications 63 (1989) Dinh The Luc, Theory of Vector Optimization, Springer Verlag, Berlin, Germany, Dinh The Luc and C. Malivert, Invex optimization problems, Bulletin Australian Mathematical Society 46 (1992), Tran Hue Nuong, Optimality conditions in cooperative differential games, Control and Cybernetics 18 (1989), Pham Huu Sach and B. D. Craven, Invexity in multifunction optimization, Numerical Functional Analysis and Optimization 12 (1991), Department of Mathematics-Computer Science, Vietnam National University at Hochiminh City Department of Mathematics University of Dalat