Characterizations of Quasiconvex Functions and Pseudomonotonicity of Subdifferentials

Somayeh Yazdani

Department of Mathematics, Faculty of Basic Sciences, Nour Branch, Islamic Azad University, Nour, Iran. Received: January 6, 2015 Accepted: March 10, 2015 ABSTRACT

In this paper we introduce the concepts of quasimonotone maps and pseudoconvex functions. Moreover, a notion of pseudomonotonicity for multi mappings is introduced; it is shown that, if a function f is continuous, then its pseudoconvexity is equivalent to the pseudomonotonicity of its generalized subdifferential in the sense of Clarke and Rockafellar and prove that a lower semicontinuous function on an infinite dimensional space is quasiconvex if and only if its generalized subdifferential is quasimonotone. KEYWORDS: Quasiconvex functions, pseudoconvex functions, quasimonotone maps, pseudo monotone maps, generalized subdifferentials.

I. INTRODUCTION

As far as we know quasiconvex functions were first mentioned by Finetti in 1949,[ Finetti,1949]. Since then much effort has been focused on the study of this class of functions for they have much in common with convex functions. One of the most important properties of convex functions is that their level sets are convex . Mapping of the monograph theory in the beginning of methods were obtained in spaces with more banach efforts and evaluation background lightness for the relationship between monograhy on a map and the linkage of premise was created. This study was performed to uniform of non-linear mappings and differential relations and was paid uniformity. The purpose of this paper is to declare these powerful tools and new applications of it in a function in the analysis of the last few decades. At the beginning we offer uniform mapping types and generalized uniformity and we show the mapping of the gradient for generalized uniform properties that can be generalized convexity properties about the main functions performed (infrastructure) stated. Thus, the classification of different kinds of generalized convex functions will be achieved. We will communicate between the mapping concepts of generalized uniformity on some set of generalized convexity. In addition to this, the concept of pseudo-uniformity will be introduced to set the mapping We show if a function is continuous, then the convexity is equivalent to the following pseudo-pseudo differential on the uniformity of the generalized from the perspective of clark and Rockefeller. There is a characterization of the convexity of a function : f X → R via its Clarke generalized subdifferential ↑ ∗ ↑ ∂ f : namely, X is convex if and only if ∂ f is monotone. In order to establish characterizations of generalized convexities of a function via natural properties of its generalized subdifferential (Penot, Quang, 1990), some concepts of generalized monotonicity for multimappings have been introduced in (Penot, Quang,1990). One of the most important results in the papers referred above is the characterizations of quasiconvex functions by the quasimonotonicity of their generalized subdifferentials. The paper is structured as follows. In the next section, we stablish some notation and recall the definitions and some results presented. In section 3 we introduce the concept of quasimonotonicity for multimappings from a vector space to its dual. We show that in the case of linear operators there is no difference between this concept and the monotonicity known in convex analysis and prove that a function is quasiconvex if and only if its generalized subdifferential is quasimonotone. In section 4 we develop a definition of pseudoconvexicity for nondifferentiable functions. It is shown that, if f is continuous and pseudoconvex, then it is quasiconvex and invex. In section 5, we introduce a definition of pseudomonotonicity for multimappings. It is shown that, if a function is continuous, then its pseudoconvexicity is equivalent to the pseudomonotonicity of its generalized subdifferential.

*Corresponding Author: Somayeh Yazdani , Department of Mathematics, Faculty of Basic Sciences, Nour Branch, Islamic Azad University, Nour, Iran. E-mail: [email protected]

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2. Basic Definitions and Preliminary Resullts ∗ If X is a banach space with dual space, graph F( from X to X ) nominate monotone. (Hadjisavvas, Schaible, 2012) IF ∗ ∗ 〈xy , – x 〉+〈 yx , – y 〉≤ 0, ∀ xyX , ∈∀∈ , x∗ Fx , ∀∈ y ∗ Fy . 1 ( ) ( ) ( )

Monograph f is quasiconvex if the sublevel sets are convex. fxtyx + − ≤ max fxfy , , ∀xy , ∈ X , ∀∈ t 0 ,1. ( ( )) { ( ) ( )} [ ]

∗ Whenever X is a topological vector space with dual space and K ⊆ X is not null and convex, monographic T: KÃ X is pseudo monotone, if:

〈∗ , – 〉≥⇒〈 0 ∗ , – 〉≥ 0 xyx yyx ∗ ∗ ∗ The spaces X and X are paired in duality by the continuous bilinear from 〈xx, 〉 : = xx () Given ε >0, x ∈ X we denote by B x ε the closed ball centered at x with radius ε . Let → = −∞ +∞ be a given function. ( , ) f : X R :{ , } Assume that the value of the function is finite at a point x∈ X . The Clarke-Rockafellar generalized derivative of at x f in the direction v is defined by: ↑ = [(f y+ tu ) − α ] f( x , v ) supε >0 lim(,)yα ↓ sup t ↓ 0 infu ∈ B (,) v ε , f x t α ↓ →α → α ≥ Where (y , ) f x means that y x , fx( ) , fy( ) .When f is locally lipschitzian [Rockafellar,1980], this derivative coincides with the Clarke directional derivative, which is defined by 0 [fz (+ tu ) − fz ( )] f(,)lim x v = sup . z→ xu, → v t↓0 t The circa-subdifferential (or Clarke-Rockafellar subdifferential ) of f at x is ↑ ∗∗∗ ↑ ∂fx( ) { =∈ x X : 〈 xv , 〉≤ fxv (, ), ∀∈ vX } , ↑ with the convention that ∂ f( x ) is empty if f is not finite at x . We need the following lemma, which was established in [Luc,1993].

3. Quasimonoton maps ∗ ∗ , ∈ x∈ Fx , y ∈ Fy F is quasi monotone if x y X , ( ) ( ) ∗ ∗ min{〈xyx , −〉〈 , yxy , −〉≤ } 0 . ∗ ∗ F is monotonous if y , x , yx ,∈ X

∗, ∗ , 0 〈xyx −〉+〈 yxy −〉≤ Obviously each monograph is semi monographic but It is not established viceversa.

Proposition 3.1. Let X be a real Hilbert space and A a linear operator on X Then the following conditions are equivalent: (1) A is monotone; (2) A is quasimonotone; (3) the symmetric part of A is positive semidefinite.

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Lemma 3.1. Assume that f is lower semicontinuos and that fb()> fa () . Then, there exists a sequence {xk } in X ∈∗ ∈∂ ↑ converging to some x0 [ a,b) , xk fx( k ) such that, for any c= a + tb − a t ≥1 k x∗ c x ( ) with and for every ,one has 〈k , – k 〉 > 0 . Theorem 3.1. is quasiconvex if and only if ∂ is quasimonotone. f f

4. Pseudoconvex Functions The following definition extends a classical notion to nondifferentiable functions; see [Luc,1993 Definition 2.6] for instance. Definition 4.1 . A function f: X→ R is said to be pseudoconuex if ∗ ↑ ∗ ∀∈ , xyXfy :( ) < fx( ) ⇒∀∈∂ x fxxy (): 〈 , – x 〉< 0. (2) ↑ Definition 4.2. The function is said to be invex if the condition ∈ ∂ ensures that x is a global minimize of . f 0f ( x ) f Invexity is a kind of generalized convexity investigated in many papers in recent years; see [Luc,1993] and their references. It is clear that, if is Pseudoconvex, then is invex. In the sequel, we say that that is radially continuous if its f f f restriction to line segments is continuous; in fact, we use a weaker property in the proof below. ↑ Proposition 4.1. Let be lower semicontinuous (l.s.c). If is quasiconvex, then it is ∂ − in the sense f f quasiconvex that the following condition holds: ∗ ↑ ∗ ∀∈ , :xyXfy( ) < fx( ) ⇒∀∈∂ x fxxy (): 〈 , – x 〉≤ 0. (3) Conversely, if is l.s.c and radially continuous, and if condition (3) holds, then is quasiconvex. f f ∗ ↑ Proof. Assume that is quasiconvex l.s.c and that, contrary of (3), there exits ∈ ∈ ∂ such that f , xyXx , fx () fy( ) > fx( ), (4) ∗ 〈x , y – x 〉 > 0. (5) z X From Lemma 3.1 and (10), it followes that there exists a sequence {i } in converging to some ∗ ↑ z∈ ( xyz , ] ,i ∈ ∂ fz ( i ) such that ∗ 〈zi , – x z i 〉 > 0. (6) Frome (11), it follows that there exists ε > 0, ∗ 〈xzx ,–′ 〉> 0. ∀∈ zBz ′ (,).ε (7) ↑ Taking i so large that zi ∈ B( z ,ε ) , we see that (6) and (7) show that ∂ f is not quasimonotone. So from [Hassouni, Theorem 3.2], we get that is not quasiconvex, which is a contradiction. f Conversely, suppose that f is l.s.c., radially continuous, satisfies (3), and f is not quasiconvex; i.e., there exists z∈( x , y ) such that fz( ) > max{ fxfy( ) ,( )} . Let s> max{ fxfy( ) ,( )} , sfz < ( ), and let rˆ := sup{ r ∈ [0,1]: fxrzx ( +−≤ ( )) s }, xˆ:= x + rzx ˆ ( − ). As is l.s.c., we have ˆ ≤ and xˆ ≠ z . Moreover, as is radially continuous and ≥ for w∈ xˆ z we f f( x ) s f f( w ) s ( , ], ∗ ↑ have f( xˆ )≥ s . From Lemma 3.1, it follows that there exists a sequence {a i } converging to a∈ [ xzaˆ , ) ,i ∈ ∂ fa ( i ) , such that

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∗ , –〈ayai i 〉> 0, ∀ i .

Combined with (9), this inequality implies that fa()i ≤ fy () ; hence, by the lower semicontinuity of f , fa()≤ fy () we have a contradiction with fw()≥ s > fy () for each w∈ [, xˆ z ]. The following implications are straightforward consequences of the last proposition. Corollary 4.1 . If f is peudoconvex, l.s.c., and radially continuous, then f is quasiconvex. Conversrly, if f is quasicnvex, lower semicontinuous, and if ∗ ↑ ∗ fx( ) > fy( ) ⇒∀∈∂ x fxxyx(): 〈 , – 〉≠ 0, Then f is pseudoconvex. Remark 4.1 . In [Luc, 1993], pseudoconvexity is defined in the following way: f: X→ R is said to be pseudoconvex whenever x, y∈ domf with fx( ) > fy( ) , there exist β (x , y )> 0 and σ (x , y )∈ (0,1] such that fxfxyx()>+−+ (λ ( )) λβ (,), xy ∀∈ λσ (0,(,)]. xy It is easy to show that, if f is liocally Lipschitz, then the above statement coincides with (2). Let us also note the following consequence of pseudoconvexity. Proposition 4.2. If f is locally Lipschitzain and pseudoconvex, then f is semistricty quasiconvex, i.e., is such that fxfy( ) >⇒<( ) fzfx( ) ( ) , ∀∈ zxy ( , ). Proof. Suppose the contrary that, for some xyz, ,∈ X with z∈( x , y ) , one has fz( ) ≥ fx( ) > fy( ) . Then, as f is quasiconvex, for each w∈[ x , z ] we have fw( ) = fz( ) . ∗ ↑ ∗ ∗ Thus, for each such w and each w∈ ∂ f( w ) , one has 〈w, y – w 〉 < 0 , henc 〈w, z – x 〉 > 0 . Now the Lebourg ∗ ↑ mean-value theorem (Schaible, 1992) ensures the existence of some w∈(,), xz w ∈ ∂ fw () , with ∗ 〈wxz,– 〉= () fx − fz () = 0, a contradiction.

5. Pseudomonotone Multimappings Here, we propose a generalization for set-valued maps of the pseudomonotonicity property introduced in [Poliquin,1990 , Definition 2.5]. ∗ Definition 5.1. A mulimapping F: XÃ X is pseudomonotone if, for every pair of distinct points x, y∈ X , one has ∗∗ ∗ ∗ ∃∈xFx( ) : 〈 xyx , – 〉>⇒∀∈ 0 yFy( ) : 〈 yyx , – 〉> 0 . (8) Obviously, a pseudomonotone mapping is quasimonotone, but the converse is not true. Remark 5.1. See [Poliquin,1990, Proposition 2.5]. It is easy to see that (8) is equivalent to the following: ∗∗ ∗ ∗ ∃∈xFx( ) : 〈 xyx , – 〉≥⇒∀∈ 0 yFy( ) : 〈 yyx , – 〉≥ 0 . ∗ Proposition 5.1. The multimapping F: XÃ X is pseudomonotone if and only if, for every x∈ X for every direction d∈ X , there exists λλ12,∈R , λλ 21 ≥ such that, for each λ∈ [ λ1 , λ 2 ] the following relations hold: ∗ ∗ () ,−〈λvd 〉> 0, ∀∉ λλλ [,],1 2 ∀∈+ vFxd ( λ ), (9) a ∗ ∗ 〈〉=vd , 0, ∀∈λλλ (,),1 2 ∀∈+ vFxd ( λ ). (9) b ∗ Proof . Assume that F: XÃ X is pseudomonotone. Introducing Ht():=〈 Fx ( + tdd ), 〉 and T= { t ∈ RHt : () ∩ (0, +∞≠∅ ) }, S= { s ∈ RHs : () ∩−∞ ( ,0) ≠∅ }.

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Then T′ =∈ t RHt ⊂ +∞ { : () (0, )}, S′ = { s ∈ RHt : () ⊂−∞ ( ,0)}, we observe that, for any tTt∈′, ′ > t , one has t′∈ T ′ ; and for s∈ Ss, ′ < s , one has s′∈ S ′ Moreover, s< t for each (st , ) ∈ S′ × T ′ . The result follows by setting ′ ′ λ1:= supS , λ 2 : = inf T . ∗ Conversely, suppose that conditions (9) hold, and let a pair of distinct points (x , y ) ∈ X and x∈ F( x ) be such that ∗ 〈x , y – x 〉 > 0 . By taking d = y – x in (9), we get 0 ≥ λ2 ; hence, for any λ∈ [ λ1 , λ 2 ] , we have (1−〈λ )F ( y ) , d 〉> 0 , and (8) follows. ↑ Theorem 5.1. Assume that is l.s.c. and radially continuous. If ∂ is pseudomonotone, then is pseudoconvex and f f f conversely. Proof. Suppose that f is radially continuous and pseudoconvex. Consider x, y∈ X such that ∗ ↑ ∗ ∃∈∂x fx( ) : 〈 xyx , – 〉> 0. (10) 0 ↑ ↑ − > ∃>ε ∃x → x ∃ t ↓ First, we show that ∉ ∂ f( y ) Indeed, (10) imolies f(, xy x ) 0 ; hence, 0,{}i ,{}0 i such that + − > inf [(fxii tu ) fx ()]/ ii t 0. u∈ By( − x ,ε )

Taking i so large that xi ∈ B( x ,ε ) , one has y− xi ∈ By( − x ,ε ) and we can deduce that

fx(ii+ tyx ( − i )) > fx (). i As f is radially continuous and pseudoconvex, it follows from Corollary 4.1 that f is quasiconvex; hence , the last inequality implies

fx()i< fx ( ii + tyx ( − i )) ≤ fy (). ↑ So frome (2), we deduce that 0 ∉ ∂ f( y ) . Now , from (10) it follows that there exists ε > 0 such that ∗ 〈xyx ,–′ 〉> 0, ∀∈ yBy ′ (,),ε which together with (2) shows that ∀y′ ∈ By(,):()ε fy ′ ≥ fx (). (11) ∗ ↑ ∗ Now, if (8) does not hold [ i.e., if there exsits y∈∂ f y such that 〈y , x – y 〉 ≥ 0 ], then from (2) one has ( ) ≥ . Combining the last inequality with (11), it follows that is a local minimizer of ; hence, fx() fy () y f ↑ 0 ∈ ∂ f( y ) , and we get a contradiction. ↑ Suppose now that ∂ f is pseudomonotone. Consider x, y∈ X such that fx()().> fy (12) z ∈∗ ∈∂ ↑ from Lemma 3.1, it follows that there exists a sequence {i } converging to some zyxz[ , ), i fx( i ) , such that ∗ 〈zxzi , – i 〉> 0, ∀ i . (13) Hence from (8), it follows that ∗ ↑ ∗ ∀∈∂x fx( ) , 〈 xxz ,–i 〉> 0, which implies that ↑ 0∉ ∂ f( x ) , (14)

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∗ ↑ ∗ ∀∈∂x fx( ) , 〈 xxz , – 〉≥ 0. (15) ∗ ↑ ∗ Now, suppose the contrary of (2), that there exists x∈∂ f( x ) with 〈x , y – x 〉 ≥ 0 . Combined with (15), the last ∗ ∗ inequality shows that 〈x , y – x 〉 = 0 . So, using (14) to pick u∈ X with 〈x , u 〉 > 0 , setting y:= y + tu with t > 0 small enough, (12) and the radial continuity of f ensure that fx()> fy () and ∗ 〈x , x – y 〉 < 0 . (16)

Once more, applying Lemma 3.1 to the points x, y it follows that there exists a sequence {zi } converging ∈∗ ∈∂ ↑ z∗ x z z z to zyxz[ , ), i fz( i ) , such that 〈i ,– i 〉 > 0 . But (16) implies that, for i close enough to , one has ∗ 〈zi ,– x z i 〉 < 0 , and the last inequalities contradict (8).

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