On Optimality Conditions for Pseudoconvex Programming in Terms of Dini Subdifferentials

Transcription

1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 18, HIKARI Ltd, On Optimality Conditions for Pseudoconvex Programming in Terms of Dini Subdifferentials Jaddar Abdessamad Mohamed First University MATSI Laboratory, Oujda, Morocco Copyright c 2013 Jaddar Abdessamad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We characterize the minima of pseudoconvex functions on a convex set using Dini subdifferentials and normal cones. We consider both the cases of a general constraint set and that of a set defined by inequalities. Mathematics Subject Classifications: 46N10, 26A51, 26A27. Keywords: Optimization, pseudoconvexity, Dini subdifferential, Normal cone. 1 Introduction Nonsmoothness is one of the main concerns of modern Optimization research. When the objective function is not smooth enough, we have to use some kind of generalized derivatives or some subdifferential notion to overcome the difficulties arising from the lack of the gradient. The literature is very rich. Our aim in this paper is to characterize minima of pseudoconvex functions in terms of Dini upper-subdifferentials of Kuhn-Tucker type. The Dini upper-derivative gives good results with quasiconvex functions and as we will see with pseudoconvex functions [1, 3, 4, 6, 7]. Dini derivatives are associated to small subdifferentials while sharing good properties with bigger ones (Fenchel-Moreau,... ) [1,5]. The results of this paper were mainly inspired by the reading of an interesting paper ( [5]) introducing an appropriate subdifferential for quasiconvex functions by Daniilidis, Hadjisavvas and Martínez-Legaz.

2 892 A. Jaddar The paper is organized as follows. We begin by recalling the definitions of Dini-subdifferentials, descent directions and pseudoconvexity. We then characterize in section 2 minimizers of pseudoconvex functions on a closed convex set C using Dini-subdifferentials and normal cones. In section 3, we treat the particular case when the constraint set C is defined respectively by one and finitely many inequalities involving pseudoconvex functions. Let us recall some definitions and properties that will be used in the sequel. Dini derivatives and subdifferentials Let f : R n R {+ } be a function. The Dini upper-derivative of f at a in the direction d is defined by f +(a; d) = lim sup t 0 The Dini upper-subdifferential is f(a + td) f(a). t { {a R n ; a ; d f D+ + f(a) = (a; d), d Rn } if a dom (f) if a dom (f) Crouzeix [3, Proposition 16] has proved that d f + (a; d) is quasiconvex if f is quasiconvex around a. Descent directions The vector d R n is called a descent direction of the function f at a if and only if there exists T>0 such that f(a + td) <f(a), for all 0 <t<t. The descent directions of f at a is a cone denoted by Df(a). Dini descent directions Set D + f(a) ={d R n ; f + (a; d) < 0} The cone D + f(a) is called the cone of Dini descent directions of f at a. It is convex when f is quasiconvex.

3 Optimality conditions for pseudoconvex programming 893 Pseudoconvex functions A function f : R n R {+ } is pseudoconvex for the Dini upper-subdifferential if for all x, y R n, the following implication holds: ( ) x D+ f(x) : x,y x 0 = f(x) f(y). The pseudoconvexity has also been defined using generalized derivatives. Nevertheless, we will adopt in the sequel this definition. Using the definition of pseudoconvexity, we can see easily that for a pseudoconvex function f, we have 0 D+ f(x) x is a global minimum of f. Moreover, according to Aussel [1], for any continuous function f, f is pseudoconvex { f is quasiconvex, 0 D+ f(x) x is a global minimum 2 An optimization problem for pseudoconvex functions Consider the following problem: (P) { min f(x), where f is pseudoconvex, x C, C is closed convex. Let x 0 C, the upper Dini tangent cone at x 0 C is defined by T C (x 0 )={ d R n ; δ >0 such that t ]0,δ[; x 0 + tdu C }. Whereas N C (x 0 ), the polar of T C (x 0 ), N C (x 0 )={ u R n ; u, d 0, d T C (x 0 ) }, is called the normal cone to C at x 0 Notice that because C is closed convex, the indicatrice function δ C is convex lower semicontinuous (l.s.c.) and + δ C (x 0 ) coincide with the classical Fenchelmoreau subdifferential furthermore + δ C (x 0 )=N C (x 0 ). Indeed x D+ δ C (x 0 ) x R n, x,x x 0 δ C (x) δ C (x 0 ) x C, x,x x 0 O x N C (x 0 ). Before proceeding to our main Theorem, we need the following lemma

4 894 A. Jaddar Lemma 2.1. let f : R n R {+ }, x 0 C a solution of (P) and Then (0, 1) M M =(T C (x 0 ) R) T epi (f) (x 0,f(x 0 )). Proof. If x 0 C is a solution of (P), then we have Furthermore we can easily see that f + (x 0; u) 0, u T C (x 0 ) (2.2.1) M = N C (x 0 ) {0} + ( T epi (f) (x 0,f(x 0 )) ). By using (2.2.1), we can easily that (0, 1) M. Indeed, Let s show that ( ) ( ) f + (x 0; d) 0, d T C (x 0 ) (0, 1); (d, μ) 0, (d, μ) M because (d, μ) M implies (d, μ) T epi (f) (x 0,f(x 0 )), i.e. (x 0 +td, f(x 0 )+tμ) epi (f) for all t ]0,δ[, so f(x 0 +td) f(x 0 )+tμ. Hence, (f(x 0 + td) f(x 0 ))/t μ. Therefore f + (x 0; d) μ, so 0 f D + (x 0 ; d) μ which implies that (0, 1) M. Theorem 2.2. A necessary and sufficient condition for a point x 0 C to be a solution of (P) is that 0 + f(x 0 )+N C (x 0 ). Proof. ) Ifx 0 C is a solution of (P), then by lemma2.1, (2.2.1) holds and (0, 1) M we have that x N + f(x 0 ). Hence Since by [5, Proof of Proposition 21] (0, 1) = (x N, 0) }{{} + ( x N, 1) }{{} N C (x 0 ) {0} (T epi (f) (x 0,f(x 0 ))) + f(x 0 )={x X ;(x, 1) ( T epi (f) (x 0,f(x 0 )) ) }, we have that x N + f(x 0 ). So, 0 + f(x 0 )+N C (x 0 ). ) Suppose that 0 + f(x 0 )+N C (x 0 ), then there would exist x 1 + f(x 0 ) and x 2 N C(x 0 ), such that x 1 + x 2 =0. (2.2.2) Take x C, then x 2,x x 0 0. By (2.2.2), x 1,x x 0 0. By the pseudoconvexity of f, f(x 0 ) f(x). Remark 2.3. Notice that the necessary optimality condition does not require any kind of generalized convexity. The pseudoconvexity was used only to show that the condition is also sufficient.

5 Optimality conditions for pseudoconvex programming The Case of a Constraint Set Given by Inequalities Consider first the case where C = {x; h(x) 0}, h is pseudoconvex and continuous such that: h +(x 0 ; d) = h(x 0 ) = 0 and satisfies the Slater condition sup x,d, (3.3.1) x + h(x 0 ) z C, h(z) < 0 is satisfied. Remark 3.1. The condition (3.3.1) was supposed in [5, Proposition 18]. It is in particular satisfied whenever h is regular or (Pshenichnyi) quasisubdifferentiable at x 0 with nonempty subdifferential. Theorem 3.2. A necessary and sufficient condition for a point x 0 C to be a solution of (P) is that 0 + f(x 0 )+Cl ( R + + h(x 0 ) ). Remark 3.3. When + h(x 0 ) is compact, R + + h(x 0 ) is closed and a Lagrange multiplier appears. Lemma 3.4. The following equality holds true Cl(D + h(x 0 )) = E + h(x 0 ), where E + h(x 0 )={d R n ; h + (x 0; d) 0}. Proof of the lemma. : Consider a sequence (d n ) n D + h(x 0 ), such that d n d. So, D + h(x 0 ; d n ) < 0. Suppose by contradiction that h +(x 0 ; d) > 0. According to (3.3.1), there is x + h(x 0 ) such that x, d > 0. Since d n d, for n sufficiently large, x,d n > 0. By (3.3.1), we get the contradiction h + (x 0; d n ) > 0. : By contradiction, suppose that there is p E + h(x 0 ) but p Cl(D + h(x 0 )). Then, h +(x 0 ; p) = 0. Moreover, there is δ > 0 such that for all s R n, s <δ, we have h + (x 0; p + s) 0. Otherwise, p would be in Cl(D + h(x 0 )). By (3.3.1), for some (ε n ) n such that ε n 0, we have the existence of a sequence (x n ) n + h(x 0 ) with x n; p + s = x n; p + x n; s h + (x 0; p + s) ε n }{{} 0

6 896 A. Jaddar Then, So, So, and x n; p + x n; s ε n. 0=h +(x 0 ; p) x n; p x n; s ε n h + (x 0; s) ε n. 0 h +(x 0 ; s) ε n, h + (x 0; s) 0, s B δ (0). By (positive) homogeneity, this holds for all s R n. So, 0 D + h(x 0 ), and by the pseudoconvexity of h, x 0 is a global minimum. A contradiction with the Slater condition. Hence, Cl(D + h(x 0 )) E + h(x 0 ), and finally, we have equality. Proof of Theorem 3.2. Since h(x 0 ) = 0 and by the pseudoconvexity of h, we can easily see that D + h(x 0 ) T C (x 0 ) E + h(x 0 ). So, by Lemma 3.4, Using (3.3.1), we can see that Cl(C(x 0 )) = T C (x 0 )=E + h(x 0 ). E + h(x 0 ) = { d; h + (x 0; d) 0 } = { d; x,d 0, x + h(x 0 ) } = [ + h(x 0 ) ] So, N C (x 0 )= [ + h(x 0 ) ] Suppose now that C = { x; h i (x) 0 }, I = {1, 2,...,k} is finite, (h i ) i I is a family of functions that are locally Lipschitz, regular i.e., + h i (x 0 )= h i (x 0 ) the Clarke subdifferential and the h i s satisfy (3.3.1).

7 Optimality conditions for pseudoconvex programming 897 By [5, Proposition 18], we have Cl + h i (x 0 ) = + h(x 0 ) (3.3.2) i I(x 0 ) where I(x) ={i; h(x) =h i (x)} and h(x) = max h i(x). According to [6] 1 i k h +(x; d) = max (h i) +(x; d). i I(x) Claim. The function h satisfies (3.3.1), i.e., h + (x 0; d) = sup x + h(x 0 ) x ; d. Indeed, for any x + h(x 0 ), x,d h +(x 0 ; d). And conversely, h + (x 0; d) = (h i ) + (x 0; d), for some i I(x 0 ) = sup x,d, for some i I(x 0 ) x + h i (x 0 ) sup x,d by (3.3.2), x + h(x 0 ) which proves the claim. Moreover, h is pseudoconvex. It suffices to show [1] that h is quasiconvex and satisfies the optimality condition ( 0 D + h(x) x is a global minimum of h ) We know that for all β R, S h (β) = i I(x) S hi (β). Since the h i are quasiconvex, the S hi (β) are convex. So S h (β) is convex, for all β R. Hence, h is quasiconvex. Suppose that 0 + h(x) =Cl + h i (x 0 ), then there are β i 0, i I(x 0 ) that are not all null such that for all i I(x), 0 = β i + h i (x). i I(x) So, for any y R, there is i I(x) and x i + h i (x), x i,y x 0. Since the h i s are pseudoconvex and h i (x) h i (y), h(x) h(y). Since y R was taken arbitrary, x is a global minimum of h, and hence h is pseudoconvex. So, by Theorem 3.2, we get the following result. Proposition 3.5. x 0 is a solution of (P) if and only if 0 D+ f(x 0 )+ D+ h i (x 0 ). i I(x 0 )

8 898 A. Jaddar References [1] Aussel, D. Subdifferential properties of quasiconvex and pseudoconvex functions: Unified approach. J. Optimization Theory Appl. 97, No.1, 29 45, (1998). [2] Clarke, F.H. Optimization and nonsmooth analysis. Wiley-Interscience, New York, (1983) [3] Crouzeix, Jean-Pierre Continuity and differentiability properties of quasiconvex functions on R n. in Generalized concavity in optimization and economics, Proc. NATO Adv. Study Inst., Vancouver/Can. 1980, , (1981). [4] Crouzeix, Jean-Pierre Some properties of Dini-derivatives of quasiconvex functions. in Giannessi, Franco (ed.) et al., New trends in mathematical programming. Homage to Steven Vajda. Dordrecht: Kluwer Academic Publishers. Appl. Optim. 13, 41 57, (1998). [5] Daniilidis, Aris; Hadjisavvas, Nicolas; Martínez-Legaz, Juan-Enrique An appropriate subdifferential for quasiconvex functions. SIAM J. Optim. 12, No.2, , (2002). [6] Giorgi, G.; Komlósi, S. Dini derivatives in optimization. II. Riv. Mat. Sci. Econ. Soc. 15, No.2, 3 24, (1992). [7] Komlósi, S. On generalized upper quasi-differentiability. in Nonsmooth Optimization: Methods and Applications, F. Gianessi ed., Gordon and Breach, London, , (1992). [8] Komlósi, S. Quasiconvex first-order approximations and Kuhn-Tucker type optimality conditions. Eur. J. Oper. Res. 65, No.3, , (1993). Received: November, 2012