Journal of Applied Analysis Vol. 6, No. 1 (2000), pp. 139 148 A CHARACTERIZATION OF STRICT LOCAL MINIMIZERS OF ORDER ONE FOR STATIC MINMAX PROBLEMS IN THE PARAMETRIC CONSTRAINT CASE A. W. A. TAHA Received June 18, 1999 and, in revised form, December 27, 1999 Abstract. We present a new version of first order necessary optimality conditions for a static minmax problem with inequality constraints in the parametric constraint case. These conditions, after some modification, turn out to characterize strict local minimizers of order one for the given problem. 1. Introduction The concept of a strict local minimizer of order m was considered by Cromme, under the name strongly unique minimizer, in a study of iterative numerical methods (see [2]). Such minimizers play an important role in stability studies (see, e.g., [5], [8]). Some results concerning characterizations of such minimizers for standard nonlinear programming problems with both inequality and equality constraints have been obtained, for m = 1 or m = 2, in [7], [9], [12]. These results were derived under the presence 1991 Mathematics Subject Classification. 49K35, 49J52. Key words and phrases. Minmax problem, optimality conditions, Clarke s subdifferential. ISSN 1425-6908 c Heldermann Verlag.
140 A. W. A. Taha of constraint qualifications, leading to statements in which there is no gap between the necessary and sufficient conditions. Consider the following nonlinear programming problem: min {f(x) x S}, (1.1) where f :R n R and S is a nonempty subset of Rn defined by S := { x Rn gi(x) 0, i I }, (1.2) with I := {1,..., p} and g i :Rn R (i I). A special problem of the form (1.1) (1.2) is the static minmax problem in the parametric constraint case in which the objective function f is given by where φ :Rn Rm R, and f(x) := sup φ(x, y) y Z(x) Z(x) := { y Rm (x,y) 0, w(x,y) =0 }, for some d :Rn Rm Rk, w :Rn Rm Rl. In particular, if Z (x) and the sup above is attained at some point, we can write f as We can formulate problem (1.1) (1.3) as follows: f(x) := max φ(x, y). (1.3) y Z(x) min x S max y Z(x) φ(x, y), as a typical two-stage minmax problem, in which the maximizing playery Z(x) acts after the minimizing player-x S and with full knowledge of the choice of the minimizing player. Such problems arise in operations research (see [3], [4], [6]) In problem (1.1) (1.3), if the constraint set Z(x) does not depend on x (i.e., if for some set Y, Z(x) = Y, x S), then the problem becomes much simpler and is called the static minmax problem in the nonparametric constraint case. Recently, a characterization of strict local minimizers of order one for a nonsmooth static minmax problem with inequality constraints, in the nonparametric constraint case, has been investigated by Studniarski and the author in [10]. In this paper, we apply the ideas of [10] to derive a similar characterization for the more general problem (1.1) (1.3) (i.e., in the parametric constraint case). However, this will require imposing differentiability assumptions on the maximization problem which are stated below. max {φ (x, y) y Z (x)}, (1.4)
A characterization of strict local minimizers 141 For problem (1.4), we define the Lagrangian L(x, y, u, v) := φ(x, y) u T d(x, y) v T w(x, y), and we denote by P (x) and K(x, y ), respectively, the set of maximal solutions and the set of Kuhn-Tucker vectors associated with y P (x): P (x) := {y Z(x) φ(x, y ) = f(x)}, K(x, y ) := {(u, v ) Rk Rl yl(x,y,u,v ) =0, u T (x,y ) =0, d(x, y ) 0, w(x, y ) = 0, u 0}. We also define the index set of active inequality constraints I(x, y) := {i {1,..., k} d i (x, y) = 0}. For a given point x 0 Rn, let us assume that: (a 1 ) φ, d, w are continuously differentiable on Rn Rm ; (a 2 ) Z(x 0 ) is not empty, and Z is uniformly compact at x 0 (see [6, p. 17]); (a 3 ) for every y P (x 0 ), the gradients y d i (x 0, y ), i I(x 0, y ), y w i (x 0, y ), i = 1,..., l, are linearly independent. Let us note that conditions (a 1 ) (a 2 ) ensure that, for each x in some neighborhood of x 0, the maximum in (1.4) is attained at some point y Z(x) and condition (a 3 ) implies that the Kuhn-Tucker vector set K(x 0, y ) is a singleton set for each y P (x 0 ) (see [6, Proposition 6.5.1]). Before stating the following theorem, we will need some notations and definitions which can be found in [1, Chapter 2]. For a locally Lipschitzian function f :Rn R, we denote by f(x) the generalized gradient of f at x. We say that f is regular (or subdifferentially regular) at x if the usual one-sided directional derivative f (x; d) exists for all d and is equal to the generalized directional derivative f (x; d). Below, the symbol co A denotes the convex hull of the set A and the dot between two vectors denotes the usual inner product in Rn. Theorem 1 ([6, Proposition 9.2.1]). Under assumptions (a 1 ) (a 3 ), f is locally Lipschitzian at x 0 and directionally differentiable at x 0, so we have f (x 0 ; d) = f (x 0 ; d) = f(x 0 ) = co max y P (x 0 ) xl(x 0, y, u, v ) d, d Rn, (1.5) y P (x 0 ) where (u, v ) is the unique element of K(x 0, y ). x L(x 0, y, u, v ), (1.6)
142 A. W. A. Taha We consider problem (1.1) (1.3) in which the objective function (1.3) satisfies assumptions (a 1 ) (a 3 ), while the functions defining inequality constraints (g i, i I) are locally Lipschitzian and regular in Clarke s sense. Section 2 is devoted to necessary optimality conditions which are satisfied by all local minimizers (not necessarily strict) for problem (1.1) (1.3). These conditions include a restriction on the number of nonzero multipliers. In Section 3, we show that the previous necessary conditions can be modified so as to obtain a characterization of strict local minimizers of order one. We conclude this section with a compilation of some notations and definitions that will be useful in the sequel. For x 0 Rn and δ > 0, we denote B(x0, δ) := {x Rn x x0 δ}. We say that x 0 S is a local minimizer for problem (1.1) if there exists ε > 0 such that f(x) f(x 0 ) for all x S B(x 0, ε). Let m 1 be an integer. We say that x 0 S is a strict local minimizer of order m for problem (1.1) if there exist ε > 0, β > 0 such that f(x) f(x 0 ) + β x x 0 m for all x S B(x 0, ε). Throughout the paper, we will use the following notations for a given x Rn : I(x) := {i I g i (x) = 0}, I 0 (x) := {0} I (x). 2. First order necessary optimality conditions It was shown in [6, Theorem 9.2.2] that Theorem 1 can be used to derive first order necessary optimality conditions for the static minmax problem (1.1) (1.3) in which the functions φ, d, w and g i are continuously differentiable. These conditions do not include a restriction on the number of nonzero multipliers. Therefore, in this section, we present a detailed proof of first order necessary optimality conditions for problem (1.1) (1.3), in which the functions φ, d, w are continuously differentiable and g i, i I, are locally Lipschitzian, with appropriate modifications which allow us to obtain the mentioned restriction. Let x 0 S. Consider the following unconstrained optimization problem: where h :Rn R is defined by min { h(x) x Rn }, (2.1) h(x) := max {f(x) f(x 0 ), g i (x) i I}. (2.2)
A characterization of strict local minimizers 143 Observe that h(x 0 ) = 0. The relationship between problems (1.1) (1.2) and (2.1) (2.2) is given in the following lemma. Lemma 2. If x 0 is a local minimizer for (1.1) (1.2), then x 0 is also a local minimizer for (2.1) (2.2). Proof. The proof is elementary (see, e.g., [11, Theorem 2.1]). The following lemma will be used in the proof of Theorem 4 below. Lemma 3. For any subsets A, B of Rn, we have Proof. See [10, Lemma 2]. co((co A) B) = co(a B). The following result is a generalization of [6, Theorem 9.2.2]. Theorem 4. Let x 0 be a local minimizer for (1.1) (1.3). Suppose that assumptions (a 1 ) (a 3 ) are satisfied and the functions g j, j I, are locally Lipschitzian and regular. Then there exists a positive integer q, vectors y i P (x 0 ) together with scalars λ i 0, i = 1,..., q, µ j 0, j I, such that q 0 λ i x L(x 0, y i, u i, v i ) + µ j g j (x 0 ) (2.3) where (u i, v i ) is the unique element of K(x 0, y i ), i = 1,..., q, and µ j g j (x 0 ) = 0, j I. (2.4) Furthermore, if α is the number of nonzero λ i, and β is the number of nonzero µ j, then 1 α + β n + 1. (2.5) Proof. Let x 0 be a local minimizer for (1.1) (1.3); then x 0 is a local minimizer for (2.1) (2.2) by Lemma 2. Define g 0 (x) := f(x) f(x 0 ). We have h(x) = max{g i (x) i I 0 (x 0 )} and h(x 0 ) = 0 = g 0 (x 0 ). By assumptions (a 1 ) (a 3 ) and Theorem 1, f is locally Lipschitzian and regular at x 0, so that g 0 has the same properties. Since g j, j I, are locally Lipschitzian and regular at x 0 by hypotheses, then using [1, Propositions 2.3.2 and 2.3.12], we obtain 0 h(x 0 ) = co g j (x 0 ) = co f(x 0) g j (x 0 ). j I 0 (x 0 )
144 A. W. A. Taha Now, applying formula (1.6) to f(x 0 ), we deduce 0 co co x L(x 0, y, u, v ) y P (x 0 ) By Lemma 3, this is equivalent to 0 co x L(x 0, y, u, v ) y P (x 0 ) g j (x 0 ). g j (x 0 ). Hence, by Caratheodory s theorem, there exist scalars λ i > 0, i = 1,..., α, η s > 0, s = 1,..., r, such that and 0 = α λ i x L(x 0, y i, u i, v i ) + 1 α + r n + 1 (2.6) r η s w s for some y i P (x 0 ), s=1 w s g j (x 0 ). (2.7) (Here, we allow the case where α = 0, then the set of multipliers λ i is empty; similarly, if r = 0, then there are no multipliers η s ). For each j I(x 0 ), we define J(j) := {s {1,..., r} w s g j (x 0 )\ g t (x 0 )} and ζ j := t I(x 0 ) t<j { ( s J(j) η sw s ) /( s J(j) η s), if J(j), an arbitrary element of g j (x 0 ), if J(j) =, µ j := { s J(j) η s, if J(j), 0, if J(j) =. Then µ j 0 and ζ j g j (x 0 ) (by the convexity of g j (x 0 )) for all j I(x 0 ). Moreover, condition (2.7) implies that α 0 = λ i x L(x 0, y i, u i, v i ) + µ j ζ j. To obtain (2.3) and (2.4), let µ j = 0 for j I\I(x 0 ). Also, if α > 0, then we can take q = α, and all λ i are nonzero. However, if α = 0, then let q = 1, λ 1 = 0, and y 1 an arbitrary element of P (x 0 ).
A characterization of strict local minimizers 145 Let β be the number of nonzero µ j. Then it follows from our construction that β is not greater than the number r of all η i. Moreover, if r > 0, then β > 0. Hence, condition (2.6) implies (2.5). 3. Characterizations of strict local minimizers of order one We begin by reviewing some results for the standard nonlinear programming problem (1.1) (1.2). Throughout this section, we assume that the following constraint qualification is satisfied at x 0 : (a 4 ) For each η Rp satisfying the conditions the following implication holds: η i = 0, i I\I(x 0 ); η i 0, i I(x 0 ), z i g i (x 0 ) ( i I), η i zi = 0 η = 0. Theorem 5 ([10, Theorem 7]). Consider problem (1.1) (1.2) where the functions f and g i, i I, are locally Lipschitzian and possess one-sided directional derivatives at x 0 S. Suppose that assumption (a 4 ) holds. Then x 0 is a strict local minimizer of order one for (1.1) (1.2) if and only if where f (x 0 ; d) > 0, d C(x 0 )\{0}, C(x 0 ) := {d Rn g i (x 0; ) 0, i I(x0)}. (3.1) We now formulate the main result of this section. Theorem 6. Consider problem (1.1) (1.3). Suppose that assumptions (a 1 ) (a 4 ) are satisfied and the functions g j, j I, are locally Lipschitzian and regular. Then x 0 is a strict local minimizer of order one for (1.1) (1.3) if and only if the following conditions hold: (a) C(x 0 ) {d Rn \{0} max y P(x0) xl(x0,y,u,v ) = 0} =, where (u, v ) is the unique element of K(x 0, y ); (b) there exists a positive integer α, vectors y i P (x 0 ) together with scalars λ i > 0, i = 1,..., α, scalars µ j 0, j I, such that α 0 λ i x L(x 0, y i, u i, v i ) + µ j g j (x 0 ), (3.2) µ j g j (x 0 ) = 0, j I, (3.3)
146 A. W. A. Taha where (u i, v i ) is the unique element of K(x 0, y i ), i = 1,..., α, and 1 α + β n + 1, where β is the number of nonzero µ j. (3.4) Proof. (i) Necessity. Suppose that x 0 is a strict local minimizer of order one for problem (1.1) (1.3). Then it is also a local minimizer for (1.1) (1.3); therefore, Theorem 4 implies that there exists a positive integer q, vectors y i P (x 0 ) together with scalars λ i 0, i = 1,..., q, scalars µ j 0, j I, such that conditions (2.3) (2.5) hold. Suppose that all λ i are zero; then condition (2.3) takes on the form 0 µ j g j (x 0 ). Then it follows from assumption (a 4 ) and equalities (2.4) that all µ j are zero, a contradiction with the left-hand inequality in (2.5). Therefore, at least one λ i must be nonzero, which means that condition (b) holds. Condition (a) follows from Theorem 5 and formula (1.5). (ii) Sufficiency. By Theorem 5 and formula (1.5), it suffices to show that f (x 0 ; d) = max xl(x 0, y, u, v ) d > 0 (3.5) y P (x 0 ) for all d C(x 0 )\{0}. Define ν i := λ i /λ, i = 1,..., α, where λ := α λ i > 0. Then, using condition (3.2) and the equality α ν i = 1, we deduce α 0 λ ν i x L(x 0, y i, u i, v i ) + µ j g j (x 0 ) λco y P (x 0 ) = λ f(x 0 ) + x L(x 0, y, u, v ) + µ j g j (x 0 ), µ j g j (x 0 ) where the last equality follows from (1.6). By (3.3), we have µ j = 0 for all j I\I(x 0 ). Therefore, 0 λ f(x 0 ) + µ j g j (x 0 ). Since f and g j, j I, are regular at x 0 by formula (1.5) and hypotheses, we may apply [1, Corollary 3, p. 40] to obtain 0 (λf + µ j g j )(x 0 ). (3.6)
A characterization of strict local minimizers 147 Now, take any d C(x 0 )\{0}. Our assumptions on f and g j imply that the function ψ := λf + µ jg j is regular at x 0 (see [1, Proposition 2.3.6 (c)]) Hence, from (3.6) and the definition of generalized gradient, it follows that 0 ψ (x 0 ; d) = ψ (x 0 ; d) = λf (x 0 ; d) + µ j g j(x 0 ; d) = λ max y P (x 0 ) xl(x 0, y, u, v ) d + µ j g j(x 0 ; d) λ max y P (x 0 ) xl(x 0, y, u, v ) d (3.7) where the last inequality is a consequence of (3.1). Now, the desired inequality (3.5) follows from (3.7) and (a). Acknowledgment. The author is extremely grateful to his supervisor Professor Marcin Studniarski for his valuable comments and continuing encouragement. References [1] Clarke, F.H., Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983. [2] Cromme, L., Strong uniqueness: a far-reaching criterion for the convergence of iterative procedures, Numer. Math. 29 (1978), 179 193. [3] Danskin, J.M., The theory of maxmin with applictions, SIAM J. Appl. Math., 14(4) (1966), 641 664. [4] Danskin, J.M., The Theory of Maxmin and Its Appliction to Weapons Allocation Problems, Springer-Verlag, Berlin, 1967. [5] Klatte, D., Stable local minimizers in semi-infinite optimization: regularity and second-order conditions, J. Comput. Appl. Math. 56 (1994), 137 157. [6] Shimizu, K., Ishizuka, Y. and Bard, J.F., Nondifferentiable and Two-Level Mathematical Programming, Kluwer Academic Publishers, Boston, 1997. [7] Still, G. and Streng, M., Optimality conditions in smooth nonlinear programming, J. Optim. Theory Appl. 90 (1996), 483 515. [8] Studniarski, M., Sufficient conditions for the stability of local minimum points in nonsmooth optimization, Optimization 20 (1989), 27 35. [9] Studniarski, M., Characterizations of strict local minima for some nonlinear programming problems, Nonlinear Anal. 30 (1997), 5363 5367 (Proc. 2nd World Congress of Nonlinear Analysts). [10] Studniarski, M. and Taha, A., A characterization of strict local minimizers of order one for nonsmooth static minmax problems, (submitted for publication). [11] Sutti, C., Monotone generalized differentiability in nonsmooth optimization, Riv. Mat. Sci. Econom. Social. 18 (1995), 83 89. [12] Ward, D.E., Characterizations of strict local minima and necessary conditions for weak sharp minima, J. Optim. Theory Appl. 80 (1994), 551 571.
148 A. W. A. Taha Abdul Whab A. Taha Faculty of Mathematics University of Lódź S. Banacha 22 90-238 Lódź, Poland e-mail: wahab@math.uni.lodz.pl