JOURNAL OF INDUSTRIAL AND Website: http://AIMsciences.org MANAGEMENT OPTIMIZATION Volume 3, Number 2, May 2007 pp. 233–255
EXISTENCE OF CLOSED GRAPH, MAXIMAL, CYCLIC PSEUDO-MONOTONE RELATIONS AND REVEALED PREFERENCE THEORY
A. C. Eberhard
School of Mathematical and Geospatial Sciences Royal Melbourne Institute of Technology G.P.O. Box 2476V Melbourne, Australia 3001. J-P. Crouzeix
LIMOS, Universit´eBlaise Pascal Boite Postale 206 F-63174 AUBIERE Cedex, France (Communicated by Adil Bagirov)
Dedicated to Alex Rubinov, colleague and friend.
Abstract. We investigate a multifunction x 7→ N˜f (x) derived via normal cones to the level sets S˜(x) := {x′ | f(x′) < f(x)} for an important class of pseudo–convex functions. It is shown that x 7→ N˜f (x) is simultaneously both a maximally cyclically pseudo–monotone and a maximally pseudo-monotone relation within neighbourhoods on which f is nonconstant. The relevance of this work to the problem of construction of a utility function from observations of revealed preferences of a consumer is discussed.
1. Introduction. A function f : D → IR := IR ∪{−∞, +∞} (where D is a convex subset of IRn with interior) is called quasi–convex when the level sets S(x) := {x′ ∈ D | f(x′) ≤ f(x)} (or equivalently S˜(x) for S˜(x) := {x′ | f(x′)
2000 Mathematics Subject Classification. Primary: 47H05, 90A40, 26A51; Secondary: 90A10, 52A01. Key words and phrases. utility functions, normal cone mappings, cyclic quasi/pseudo- monotonicity, revealed preferences. The first author’s research was supported by the ARC Discovery grant no. DP0664423.
233 234 A. EBERHARD AND J-P. CROUZEIX
N˜f (x)), generated as normal cone relations to the level sets S(x) (resp. S˜(x)) or an associated subdifferential, see ([2], [3], [4], [5], and [16]). Many obstacles are present in the study of arbitrary quasi–convex functions, one of these being the failure of Nf or N˜f to have a closed graph even for the subclass of (general) pseudo–convex functions f. In this paper we will identify a suitable subclass which does yield a satisfactory theory in many respects includ- ing this closedness property for N˜f . This class was essentially introduced in [8] and implicitly used in [17]. The importance of identifying subclasses with agree- able properties is motivated by the desire to provide a suitably restrictive class with strong properties, but yet still broad enough to provide candidates for util- ity functions for consumer preference theory. Since utility functions are normally assumed to be locally ”nonsatiated” we may focus on the study of pseudo–convex functions that have no ”flats” (i.e. we have no open sets within which the function is constant). We are able to deduce the existence of locally uniform directions of strict descent from more elementary notions. This desirable behaviour of utility functions is often assumed a priori. Consequently, we concern ourselves with the most economic characterization of this class of function and its analysis. Classes of quasi–convex functions that possess ”flats” are of importance in optimization theory and variational inequalities, and their treatment have motivated the introduction of a modified level set in [6]. Such considerations are outside the stated scope of this paper. In [18] issues of maximality of pseudo–monotone operators are considered. In this paper, the maximality of a class of pseudo–monotone operators is established under the assumption of local Lipschitzness of the underlying functions. Maximal- ity is defined indirectly, avoiding the explicit introduction of conic valued operators. We show in this paper that not only is conic valuedness unavoidable for pseudo– monotone operators, when maximality is introduced in a natural way, but also that maximal pseudo–monotone extensions of pseudo–monotone operators always exists. Unfortunately, they may not have a closed graph, an observation prompting some authors to suggest an incompatibility between maximality and graph closedness. We introduce the subclass of lower semi–continuous, solid, pseudo–monotone func- tions to resolve this issue. This class of functions have normal cone relations to their level sets that are simultaneously maximally pseudo–monotone, maximally cyclically pseudo–monotone and possessing a closed graph. This situation is sim- ilar to the subdifferential of a convex function which is simultaneously maximally monotonic and maximally cyclical monotonic with closed graph. The problem of revealed preference can be viewed as being an integration problem where one has access only to the differential information provided by N˜f without a-priori knowing f. The desired goal is to obtain f from only the knowledge of the multi–functions N˜f . The need to restrict attention to subclasses that yield closed graph multi–functions become evident when one considers issues of approximation and numerical stability of constructions formed by sampling such multi–functions. These issues are not evident in the study of this problem for differentiable utilities as in [14], motivating the inclusion of nondifferentiabilty in this study. A full treatment of this problem is delayed to a subsequent paper but we provide a discussion of these issues in the last section.
2. A suitable sub-class of quasi–convex functions. Our notation is consistent with the classic textbook [24]. Denote by C the closure of the set C, by int C its interior, Cc its complement and the cone generated by the origin and the base C CLOSED GRAPH PSEUDO-MONOTONE RELATIONS 235 by cone C := ∪µ≥0µC. A δ > 0 neighbourhood of a point x is denoted by Bδ (x) := {x′ ∈ IRn |kx − x′k <δ}. The boundary of a set C is denoted by bd C := C ∩ (int C)c. The negative polar of a set C ⊆ IRn is given by Co := {z∗ ∈ IRm | hz∗, xi≤ 0, ∀x ∈ C}. Denote the convex hull of a set C by co C := { i µixi | i µi = 1, µi ≥ 0 and xi ∈ C}. We denote the extended reals as IR =P [−∞, +∞]P (that is f may assume both +∞ or −∞, as we do not consider addition nor multiplication of such functions). In this sense IR is really viewed as a totally ordered, continuous lattice with top and bottom. Let a function f be defined on a closed convex subset D of IRn where int D 6= ∅. At some times in the analysis we will allow the possibility that f(x) = −∞ but this will be eliminated for the main development. Place dom f = {x | f (x) < +∞}. For f : D → IR, let arg min f := {x ∈ D | f(x) ≤ inf f}, where inf f := inf{f(x) | x ∈ D} and define, for x∈ / arg min f, S(x) := {x′ ∈ D | f(x′) ≤ f(x)}, S˜(x) := {x′ ∈ D | f(x′) We note also that when x ∈ S˜(x) then as S˜(x) ⊆ S(x) we have K˜ (x) ⊆ K(x) and hence Nf (x) ⊆ N˜f (x). A few comments are needed here regarding the definition of N˜f (x) which differs o from that used in [15]–[17] where N˜f (x) is defined to be cone (S˜(x) − x) (which coincides with our definition when x ∈ S˜(x) and S˜(x) is convex). Our definition is consistent with that used in [2]–[5]. It should be noted that Nf (x) = N˜f (x) = o K˜ (x) when S(x) ∩ V = S˜(x) ∩ V for some convex neighbourhood V of x or when x ∈ int {y ∈ D | f(y)= f(x)} . n In the latter case Nf (x)= {0} since K(x) = IR . In particular it is quite common for N˜f (x)\{0} = ∅ for quasi–convex functions and certainly when using (1) we 236 A. EBERHARD AND J-P. CROUZEIX cannot claim that N˜f (x)\{0} 6= ∅ densely on its domain (see Proposition 11 of [17]) o while such an assertion may be made when using cone (S˜(x) − x) as the normal cone relation. Let K denote the set of all closed cones in IRn with vertex at the origin. We define the effective domain of a cone valued multifunction Γ : D → K by dom Γ := {x ∈ D | Γ(x)\{0} 6= ∅} . Definition 1. A function f : D → IR is said to be quasi–convex if and only if S(x) is convex for all x ∈ D. We could have equivalently defined quasiconvexity to be f(z) ≤ max {f(x),f(y)} for all z ∈ [x, y] := {x′ | x′ = λx + (1 − λ)y for all λ ∈ [0, 1]} and all x, y ∈ D. When f is lower semi–continuous we equivalently could define quasiconvexity as S˜(x) being convex for all x ∈ D (see [3]). In the following we assume D is a closed convex set with interior. Definition 2. We say a function f : D → IR is solid if it is not a constant function on D and for each λ> inf f the set {x′ ∈ D | f(x′) ≤ λ} has non–empty interior. To simplify notation we will from this point onwards drop reference to the con- dition x ∈ D when specifying level sets i.e. {x′ ∈ D | f(x′) ≤ λ}≡{x′ | f(x′) ≤ λ}. In reference [8] it is demonstrated that lower semi–continuous, solid quasi-convex functions possess normal cones N˜f (x) which have the following upper semi–continuity property. Recall that a set V ⊆ D is relatively open in D when there exists an open set O ⊆ IRn such that V = D ∩ O. Definition 3. We say a relation Γ : D → K is C -upper semi–continuous at x ∈ D if for each open, cone K, such that Γ(x) ⊆ K ∪{0} there is an open neighbourhood V (relative to D) of x such that Γ(x′) ⊆ K ∪{0} for all x′ ∈ V . It is also noted in [8] that this property is equivalent to the relation Γα,β(x) := {y ∈ Γ(x) | 0 < α ≤ kyk ≤ β} being upper semi–continuous in the usual sense on domΓ (i.e. for every open set W ⊃ Γα,β(x) there is a neighbourhood V of x ′ ′ such that W ⊃ Γα,β(x ) for all x ∈ V ). It is worth noting that we must always remove the origin and so in essence consider Γ(·)\{0} and for x∈ / dom Γ we have Γ(x)= {0}⊆ K ∪{0} for any open cone K. Definition 4. We say a quasiconvex function f : D → IR is locally in the class Π aroundx ¯ if there is a convex neighbourhood V ofx ¯ for which S(x) ∩ V = S˜(x) ∩ V (2) for all x ∈ V . If this holds for allx ¯ ∈ D with f(¯x) > inf f then we say f ∈ Π (or f is globally in the class Π). This means that locally S(x) and S˜(x) have the same closure and so Nf (x) and N˜f (x) coincide and int K(x) and int K˜ (x) coincide. If (2) holds, then V does not intersect arg min f since S˜ (x)= ∅ for x ∈ arg min f. It is also clear when f is solid, lower semi–continuous and also locally in the class Π around x¯ then int S˜(¯x) 6= ∅. Indeed from the definition of solidness we only need note that when f is locally in the class Π around x we have f (x) > inf f which follows from the observation that f (x) = inf f implies S˜ (x) = ∅ (while x ∈ S (x)). We now prove a series of small Lemmas outlining some immediate consequences of this definition. CLOSED GRAPH PSEUDO-MONOTONE RELATIONS 237 Lemma 1. Suppose f is solid, quasi-convex and globally in the class Π then S(x)= S˜(x). Proof. Suppose S(x) 6= S˜(x) then there exists y ∈ S (x) with y∈ / S˜(x). Take a sufficiently small neighbourhood V of y such that V ∩ S˜(x)= ∅ then V ∩ S˜(x)= ∅ and as y ∈ S (x) we have y ∈ V ∩ S(x) 6= ∅ giving the contradiction V ∩ S(x) 6= V ∩ S˜(x). Lemma 2. Suppose f is solid, quasi-convex and locally in the class Π then if (2) holds for a given convex neighbourhood V then we must also have S(x) ∩ V ′ = S˜(x) ∩ V ′ for any convex neighbourhood V ′ of x with V ′ ⊆ V . Proof. Because the different convex sets are solid and the intersections are not empty, one has S(x) ∩ V ′ = [S(x) ∩ V ] ∩ V ′ = [S(x) ∩ V ] ∩ V ′ = [S˜(x) ∩ V ] ∩ V ′ = S˜(x) ∩ V ′ as required. Proposition 1. Suppose f : D → IR is lower semi–continuous, solid quasiconvex but not a function which is locally in Π within a neighbourhood V of x¯ with V ∩ arg min f = ∅. Then f is constant on some open nonempty subset V ′ ⊆ V ∩ S(¯x). Proof. Because f is lower semi-continuous and in view of lemma 2, some open convex neighbourhood W ofx ¯ exists such that W ⊂ V and c Ω= W ∩ S˜(¯x) ∩ W ∩ S(¯x) 6= ∅. Because Ω is the intersection of an open set with a solid convex set, its interior V ′ is not empty. One has V ′ ⊂ [W c ∪ [S˜(¯x)]c] ∩ W ∩ S(¯x). Next, because W is open and S(¯x) is closed, V ′ ⊂ [W c ∪ [S˜(¯x)]c] ∩ W ∩ S(¯x). It follows that V ′ ⊂ [S˜(¯x)]c ∩ W ∩ S(¯x) and thereby f(x)= f(¯x) for all x ∈ V ′. Lemma 3. Suppose f : D 7→ IR is lower semi–continuous, solid quasiconvex but not locally in Π within a neighbourhood V of x¯ ∈ / arg min f. Then for some open ′ ′ ′ ′ ′ subset V ⊆ V ∩ S(x) we have Nf (x )\{0} = ∅ and N˜f (x )\{0} = ∅ for x ∈ V ′ (i.e. V ∩ dom Nf = ∅). Proof. Take V ′ as in Proposition 1. Then if x′ ∈ V we have f(x′) = f(¯x) and so ′ ′ ′ ′ n ′ V ⊆ S(x ) := {y | f(y) ≤ f(x )} implying K(x ) = IR and so Nf (x ) = {0}. For x′ ∈ V ′ we have f(x′) = f(¯x) and so S˜(x′) ∩ V ′ = ∅ and hence x′ ∈/ S˜(x′) for all ′ ′ ′ ′ ′ x ∈ V . Note that when x∈ / S˜(x ) we have N˜f (x ) ⊆ {0} and so N˜f (x )\{0} = ∅. This shows that the points x′ ∈/ arg min f around which f is locally in the class Π, are contained in the effective domain dom N˜f . 238 A. EBERHARD AND J-P. CROUZEIX Lemma 4. Suppose f : D → IR is lower semi–continuous, solid quasiconvex and locally in Π around x ∈ D ∩ (argmin f)c. Then x ∈ S˜(x). Proof. Let V be a neighbourhood for which (2) holds. Suppose to the contrary that x∈ / S˜(x). Take a neighbourhood V ′ ⊆ V of x such that V ′ ∩ S˜(x) = ∅. Then S˜(x) ∩ V ′ ⊆ V ′ ∩ S˜(x)= ∅ and hence S˜(x) ∩ V ′ is properly contained in S(x) ∩ V ′ (as this latter set always contains x), a contradiction to Lemma 2. It follows from lemma 4 and definition (1) that when f : D → IR is lower semi– continuous, solid quasiconvex and locally in Π around x ∈ D we have o ∗ ′ ∗ ′ N˜f (x)= {x | hx − x, x i≤ 0 for all x ∈ S˜(x)} = cone (S˜(x) − x) , since N˜f (x) must correspond to the usual normal cone to a convex set S˜ (x) at a point x ∈ S˜ (x). Clearly Nf (x) has closed convex conical images. In [8] it is shown that N˜f (x) has a closed graph. For completeness we provide a proof since → our definition of N˜f differs from the classical one. A multifunction Γ : D → K is ′ said to be closed at x ∈ D when lim supx′(∈D)→x Γ (x ) ⊆ Γ (x). Lemma 5. Let f : D → IR be lower semicontinuous, solid quasi–convex and locally c in the class Π around x ∈ D ∩ (argmin f) . Then N˜f is closed at x. Thus when c f ∈ Π then N˜f has a closed graph relative to D ∩ (argmin f) . ∗ ∞ Proof. Note that as x∈ / argmin f we have f(x) > −∞. Let {(xk, xk)}k=0 ⊆ ∗ ∗ ∗ Graph N˜f converging to (x, x ). When x = 0 then immediately x = 0 ∈ N˜f (x). ∗ Now suppose x 6= 0. By definition N˜f (xk)= {0} when xk ∈/ S˜(xk) and hence the ∞ only non–trivial case to consider is when we have a subsequence of {xkm }m=0 where ˜ ∗ ∗ xkm ∈ S(xkm ) for which both xkm → x and xkm → x 6= 0. By Lemma 4 we have o ∗ ∗ x ∈ S˜(x). Thus to show x ∈ N˜f (x) we must show x ∈ cone (S˜(x) − x) or hy − x, x∗i≤ 0, when f(y) Let f(y) < f(x). Since f is lower semicontinuous at x and xkm → x we have ∗ ˜ lim infm f(xkm ) ≥ f(x) and hence f(xkm ) >f(y) for m large. Then xkm ∈ Nf (xkm ) ∗ ∗ implies hy − xkm , xkm i≤ 0. Taking limits as m → ∞, we obtain hy − x, x i≤ 0. o In [8] it is shown that Φ(x) := cone (S˜(x) − x) is actually C–upper semi– continuous when f is solid, lower semi–continuous and quasi–convex. This proof ∗ ∗ implies Φα,β (x) := {x ∈ Φ(x) | α ≤kx k≤ β} (for 0 < α ≤ β < +∞) is a locally uniformly bounded multifunction on D ∩(argmin f)c i.e. there exists M, δ > 0 such that Φα,β(Bδ(x)) := ∪{Φα,β(y) | y ∈ Bδ(x)}⊆ BM (0). Lemma 6. Suppose f : D → IR with f ∈ Π is solid, lower semi–continuous and quasi–convex function. Then x 7→ N˜f (x) is C–upper semi–continuous at each point in D ∩ (argmin f)c. Proof. As (N˜f )α,β (for 0 < α ≤ β < +∞) has a closed graph we only need show that it is also locally uniformly bounded to deduce upper semi–continuity (see [7]). o This follows immediately on noting that N˜f (x) ⊆ cone (S˜(x) − x) . α,β α,β CLOSED GRAPH PSEUDO-MONOTONE RELATIONS 239 o Remark 2. If x ∈ argmin f then S˜(x)= ∅ and so cone (S˜(x) − x) = ∅o = IRn. n This prompts one to define N˜f (x) = IR when x ∈ argmin f (a closed set when f is lower semi–continuous) and so extend the C–upper semi–continuity of N˜f to the whole of D. We will adopt this convention from now on and hence we have N˜f a C-upper semi–continuous relation on any set V where f is solid, quasi–convex, lower semi–continuous and in the class Π on V . It has been observed by [8] that when int {x | f(x) < λ} 6= ∅ andx ¯ ∈ / {x | f(x) < λ} for inf f<λ Proof. Because d0 belongs to the interior of K(¯x), there exist di ∈ int K(¯x), λi ∈ n n (0, +∞), i = 1, 2, · · · ,n such that d0 = i=1 λidi,1= i=1 λi and the n vectors di − d0 are linearly independent. For eachPi ≥ 1, becausePdi belongs to the interior of K(¯x), there exists ti > 0 and xi =x ¯ + tidi such that f(xi) / K+ˆx 1 xi=¯x+tidi W > ) x¯+d O 0 y x¯+t0d0 {x|hx∗,xi≥α} z - {x|f(x)≤m} C 6 x¯ + U xˆ=¯x−tdˆ 0 {x|f(x)≤f(¯x)} and so x0 − x¯ = t0d0 and, because f is quasiconvex, f(x0) ≤ m. Next, because f(¯x + t0d0) ≤ m ξi (xi − xˆ)= ξi x¯ + tidi − x¯ − tdˆ 0 Xi=1 Xi=1 n 1 1 = λidi + ntdˆ 0 = d0 + ntdˆ 0 = d0. 1+ ntˆ 1+ ntˆ Xi=1 Because f is quasiconvex and lower semi–continuous, m< f (¯x) there exist x∗ ∈ IRn and α ∈ IR such that sup[hx∗, xi : f(x) ≤ m ] <α< hx∗, x¯i. (5) Now (4) impliesx ¯ + t0d0 ∈ {f ≤ m } and xi − x¯ are linearly independent, one has ∗ ∗ ∗ α − hx,¯ x i hx¯ + t0d0, x i < α ⇒ hx , d0i≤ < 0. t0 Define C by C = {x =x ˆ + d : d ∈ K, hx∗, xi≥ α} = (K +ˆx) ∩{x | hx∗, xi≥ α} . Then, C is a convex neighbourhood ofx ¯ due to (5) and d0 ∈ int K implying x¯ =x ˆ + td0 ∈ int(ˆx + K). Let x ∈ C and d ∈ K such that x + d ∈ C. Then f(x) >m and since ∗ B := co {x1,...,xn}⊆{x | hx , xi < α} CLOSED GRAPH PSEUDO-MONOTONE RELATIONS 241 is base for the conex ˆ+K there exists t> 1 (because x+d ∈ C with C convex) such that x + td belongs to the convex hull of points xi. From f(x) > m = max f(xi) and quasi–convexity we can deduce that f (x + td) ≤ max f (xi)= m Corollary 1. Suppose f : D → IR¯ is lower semi–continuous, solid, quasi–convex and f(¯x) > inf f. Assume that int K(¯x) 6= ∅ and let some d0 ∈ int K(x). Then f is locally in Π around x¯ if and only if there is a neighbourhood V of x¯ such that x, x + td0 ∈ V with t> 0 imply f(x) >f(x + td0). Proof. Let V be a neighbourhood ofx ¯ and x, y ∈ V ⊆ C be such that y = x + td0 with t> 0 with C and K as in Proposition 2. Define Wx,y = (y − K) ∩ (x + K) ∩ C. Then Wx,y is convex, has non-empty interior and f(y) ≤ f(z) ≤ f(x) for any z ∈ Wx,y. If f is not locally in Π aroundx, ¯ by Proposition 1, any neighbourhood V ofx ¯ contains some neighbourhood V ′ and a x ∈ int V ′ ⊆ C from which we may obtain y and a Wx,y with f(x) = f(y), f is constant on Wx,y. Conversely, assume that for any neighbourhood V ofx ¯ there are x, y ∈ V and t > 0 such that f(x)= f(y) and y = x+td0. Define Wx,y as above, f is constant on Wx,y ∩V (since f(y) ≤ f(z) ≤ f(x) for any z ∈ Wx,y) and f thereby is not locally in Π aroundx ¯ ˜ since z∈ / S (x) ∩ V ⊂=6 S (x) ∩ V ∋ z for all z ∈ Wx,y ∩ V . In [2] and [15] the authors use the following concepts. Definition 5. We say a function f : D → IR is: 1. Radially continuous on a convex set D if it is continuous on each line segment [x, y] ⊆ D. 2. Radially nonconstant if for all x, y ∈ D, x 6= y, there exists z ∈ [x, y] with f(z) 6= f(x). Many lower semi–continuous solid quasiconvex functions can fail to be radially nonconstant and indeed this may also fail for functions that are also globally in Π. All that one needs to have is the subset {y | f(y)= f(x)} of the set S(x) to contain a nontrivial line segment. The concept of radial continuity implies the following assumption. Suppose inf f > −∞. Take y ∈ arg min f 6= ∅ and f(¯x) > inf f. Then when f is radially continuous we must have some z ∈ [¯x, y] with f(¯x) >f(z) > inf f (recall that [¯x, y] ⊆ S(¯x) and so f(¯x) ≥ f(z) for all z ∈ [¯x, y] but we cannot have f constant on [¯x,y] since this induces a discontinuity at y). This prompts us to define the following: Definition 6. We say a function f : D → IR is inwardly nonconstant iff for all f(¯x) > inf f there exists a z ∈ D with f(¯x) >f(z) > inf f. Clearly radially continuous quasiconvex functions that attain their infimum are inwardly nonconstant. By the results of [10] we have any radially continuous quasi– convex function on a finite dimensional space (with a domain with non-empty in- terior) actually continuous. It is also clear from simple examples that inwardly 242 A. EBERHARD AND J-P. CROUZEIX nonconstant, solid, quasiconvex functions that are only lower semi–continuous may have discontinuities. Thus inward nonconstantness appears to be a much more natural concept than those in Definition 5. Proposition 3. Suppose f : D → IR is lower semi–continuous, solid quasi– convex and inwardly nonconstant. Let x¯ be such that f(¯x) > inf f. Then there exists a λ such that inf f<λ The class we will concern ourselves with from now on will be the class of lower semi–continuous and solid quasi–convex functions. Most of the analysis of these functions concerns their behavior around points which are locally in the class Π. When a lower semi–continuous and solid quasi–convex function is globally in Π we say it is lower semi–continuous, solid and pseudo–convex. This terminology will be justified in the next section. 3. Maximal cyclic pseudo-monotone relations. We now consider the problem of characterization of the monotonicity properties of the normal cone N˜f (x) to S˜(x) for lower semi–continuous, solid and pseudo–convex. Throughout this section we will assume f : D → IR¯ is lower semi–continuous and solid quasi-convex. Following the work of [8] when f is locally of class Π around x we may make the following change of origin and basis of the local coordinate system around x. Using Proposi- tion 2 we consider the direction d of strict monotonicity of f to be the nth vector in the canonical basis and x the origin. Now the neighbourhood of Corollary 1 may be taken to have the form V = Y × T where Y and T are open convex neighbour- hoods of the origin in IRn−1 and IR respectively and t 7→ f(y,t) is decreasing. Set λ¯ = f(x) ≡ f(0, 0). Let λ0 = inf{f(y,t) | (y,t) ∈ Y × T }. For λ > λ0 define g(y, λ) = inf{t | f(y,t) ≤ λ}. (6) We require such functions to be proper and so demand λ > inf f (where inf f = −∞ is possible). The function g possesses a number of nice properties. We will summarize some proved in [8] that are relevant in the subsequent development. Proposition 5. Suppose f ∈ Π is lower semi–continuous and solid. There exists an open convex neighbourhood Y of zero in IRn−1 such that g(y, λ¯) ∈ T for all y ∈ Y . Moreover, for all (y,t) ∈ Y × T = X ⊆ IRn, it holds, 1. g(y, λ1) ≥ g(y, λ2) whenever y ∈ Y , λ0 < λ1 < λ2; 2. for all λ > λ0, g(·, λ) is convex; 3. there exists a λˆ > λ0 such that g(·, λ) is finite on Y for all λ ∈ (λ0, λˆ]; 4. for all y ∈ Y we have g(y, λ) → g(y, λ¯) whenever λ → λ¯; 5. g(y,f(y,t)) = t; 6. S˜((y,t)) ∩ (Y × T ) = S((y,t)) ∩ (Y × T ) = epi g(·, λ) ∩ (Y × T ) with λ = f(y,t). Indeed, epi g(·, λ) depends on λ and ∗ ∗ 7. Nf ((y,t)) = {(ky , −k) | y ∈ ∂yg(y, λ), k ≥ 0} when λ = f(y,t) and ∂ de- notes the usual subdifferential of convex analysis. We see that we may define ∗ ∗ x ∗ Nˆf (y,t) := y := | (x , −α) ∈ Nf (y,t) α and obtain Nf ((y,t)) = ∂yg(y, λ) when λ = f(y,t) (i.e. g parameterizes the convex boundary of the level sets S(y,t)). We may recover f from g as the following proves. b The usual convention of sup ∅ = −∞ is used and so f(y,t)= −∞ is possible. Proposition 6. Suppose f is lower semi–continuous in t and g defined as in (6) then for (y,t) ∈ Y × T we have f(y,t) = sup {λ | g(y, λ) >t} . (7) 244 A. EBERHARD AND J-P. CROUZEIX Proof. By the definition of g (in (6)) it follows that f(y,t) < λ implies g(y, λ) ≤ t so the contrapositive gives sup {λ | g(y, λ) >t}≤ f(y,t). Now suppose sup {λ | g(y, λ) >t} The direction of strict decrease imparts some extra properties to the functions g(y, ·). Proposition 7. Suppose f ∈ Π is lower semi–continuous and solid. If t 7→ f(y,t) is strictly decreasing and finite then f(y,g(y, λ)) = λ. Proof. By Proposition 5 we have λ 7→ g(y, λ) is continuous. Consider t := g(y, λ), for some λ ∈ IR and by the continuity of g in λ we have f(y,t) = sup {λ′ | g(y, λ′) >t} = sup {λ′ | g(y, λ′) ≥ g(y, λ)}≥ λ> −∞. Suppose f(y,t) > λ then there would exist λ′ > λ such that g(y, λ′) >g(y, λ). But by the monotonic non-increase of g with λ we have g(y, λ′) ≤ g(y, λ) The concept of “locally in the class Π” is actually pseudo–convexity in a localized form and allows one to carry out analysis usually reserved for such functions locally. As we are assuming f is solid quasi–convex then int K(x) 6= ∅. Corollary 4. Suppose f : D → IR¯ is lower semi–continuous and solid, quasi– convex. Then if f ∈ Π we have both f(x′) S˜((0, 0)) ∩ (Y × T )= S((0, 0)) ∩ (Y × T ) = epi g(·, λ) ∩ (Y × T ) we have (y′,t′) ∈ bd S((0, 0)) ∩ (Y × T ) = Graph g(·, λ) ∩ (Y × T ) , implying t′ = g(y′, λ). But this implies f(y′,g(y′, λ)) = f(y′,t′)= λ = f(0, 0) a contradiction, establishing that f(x′) Indeed within each neighbourhood where f is locally in the class Π we may utilize the local structure encoded in the level curve function g(y, λ) at the level λ = f(y,t). Thus we may take some guidance from the properties of these functions. Under the assumption of Corollary 4 (which include lower semi-continuity and solidness) we have proved that if f ∈ Π then f(x′) Chaining the inequalities together we find f(x0) ≤ f(x1) ≤···≤ f(xp) (12) which implies x0 ∈ S(xp) and (via quasi–convexity and the definition of Nf (xp) ) that ∗ ∗ hzp, x0 − xpi≤ 0 for all zp ∈ Nf (xp). (13) ∗ When f is pseudoconvex and for some i we have hzi , xi+1 − xii > 0 then one of the inequalities in (12) is strict implying that the inequality in (13) is also strict. ∗ To see this use the contrapositive of (9) to deduce that hzi , xi+1 − xii > 0 implies xi+1 − xi ∈/ K(xi) and so f(xi) < f(xi+1). Thus on chaining the inequalities ∗ together we have f(x0) < f(xp). Now using (10) it follows that hzp , x0 − xpi < 0 ∗ for all zp ∈ Nf (xp)\{0}. This prompts us to define the following. n Definition 7. A relation Γ : D → IR is called cyclically pseudo–monotone of order ∗ ∗ p if for all i =0,...,p − 1 and (xi, xi ) ∈ Graph Γ, with xi 6= 0 , we have ∗ ∗ ∗ hxi , xi+1 − xii≥ 0 implies hxp, x0 − xpi≤ 0 for all xp ∈ Γ(xp). (14) A relation is maximal cyclically pseudo–monotone of order p if it is cyclically pseudo–monotone of order p and its graph is not properly contained in the graph of any other cyclically pseudo–monotone relation ∆ of order p with dom ∆ = dom Γ. A relation which is cyclically pseudo–monotone of all orders p ≥ 1 is called cyclically pseudo–monotone and maximally cyclically pseudo–monotone if its graph is not properly contained in the graph of any other cyclically pseudo–monotone relation ∆ with dom ∆ = dom Γ. Clearly a pseudo–convex function possesses a normal cone multi-function Nf which is cyclically pseudo–monotone. A radially continuous quasi–convex function has Nf \{0} cyclically quasi–monotone on dom Nf . The following observation was communicated by N. Hadjisavvas. n Lemma 8. If a relation Γ : D → IR is cyclically pseudo–monotone of order p then we also have: ∗ ∗ ∗ ∃i such that hxi , xi+1 −xii > 0 implies hxp, x0 −xpi < 0 for all xp ∈ Γ(xp)\{0} . ∗ Proof. Suppose that for some j ∈ {0,...,p − 1} we have hxj , xj+1 − xj i > 0. We ∗ ∗ wish to show that hxp, x0 − xpi < 0 for all xp ∈ Γ(xp)\{0}. If not, then there exists ∗ ∗ xp ∈ Γ(xp) such that hxp, x0 − xpi ≥ 0. Define ki, i = 0,...,p by ki = i + j +1 for i =0,...,p − j − 1 and ki = i − p + j for i = p − j,...,p. Finally set yi = xki , ∗ ∗ and yi = xki for i = 0,...,p. This forms a cycle beginning at y0 = xj+1 and ∗ ending at yp = xj . Then we have hyi ,yi+1 − yii ≥ 0 for all i = 1,...,p − 1 but ∗ hyp,y0 − ypi > 0, a contradiction to cyclic pseudo–monotonicity of Γ. Thus we have p = 1 cyclically pseudo–monotonicity corresponding to x∗ ∈ Γ(x)\{0} , such that hx∗,y − xi > 0=⇒ hy∗,y − xi > 0, ∀y∗ ∈ Γ(y)\{0} . If we take the contrapositive we arrive at the equivalent statement y∗ ∈ Γ(y)\{0} , s.t. hy∗, x − yi≥ 0=⇒ hx∗, x − yi≥ 0, ∀x∗ ∈ Γ(x)\{0} . (15) A p = 1 cyclically pseudo–monotonicity operator will be called pseudo–monotone. CLOSED GRAPH PSEUDO-MONOTONE RELATIONS 247 Remark 3. Note that we only consider non-zero elements from the images of Γ. This is critical. It then follows from definitions that when Γ is pseudo–monotone then so is F (x) := Γ(x) ∪{0}. The following example due to N. Hadjisavvas gives 2 2 a concrete example if this phenomena. Let Γ : IR → IR be given by {(0, α) | α> 0} if y ≥ 0 Γ(x, y) := {(β,β) | β > 0} if y < 0 This is pseudo–monotone and the implication in (15) may be used to show that it is not possible to add another point ((x0,y0), (κ, λ)) with (κ, λ) 6= (0, 0) which is distinct from the points in the images of Γ to obtain a larger pseudo–monotone relation. Thus F (x) := Γ(x)∪{(0, 0)} is maximal pseudo–monotone. Note that this results in the images of Γ becoming closed convex cones but still Γ is not C-upper semi–continuous. Thus maximal pseudo–monotone relations are not automatically C–upper semi–continuous. n Remark 4. Given a pseudo–monotone multifunction Γ : E → IR we may restrict it to a subset D ⊆ E to obtain another pseudo–monotone multifunction ΓD : n n D → IR . Given Γ : D → IR pseudo–monotone we may extend Γ as a pseudo- monotone multifunction to a set E=6⊃ D by defining Γ(x) for x ∈ D ΓE(x)= . (16) {0} for x ∈ E ∩ Dc Of course maximality may not be preserved in such restrictions nor extensions and the effective domain is unaltered by the extension (16). Another concept with the name cyclically quasi (resp. pseudo) monotone is introduced in [16]. We state these now in order to make a comparison. n Definition 8. We call Γ : D → IR cyclically quasi-monotone if for every positive integer p and x0, x1,...,xp (and xp+1 = x0) there exists an i ∈{0, 1, 2,...,p} such that ∗ ∗ hxi , xi+1 − xii≤ 0, ∀xi ∈ Γ(xi). (17) We now observe that (17) is implied by p cyclical pseudo-monotonicity as given above in (14). Indeed suppose (14) holds. Take x0, x1,...,xp (and xp+1 = x0). ∗ Now if we suppose that for all i ∈ {0, 1, 2,...,p − 1} we have hxi , xi+1 − xii > 0 ∗ ∗ ∗ for some xi ∈ Γ(xi) then (14) implies hxp, x0 − xpi ≤ 0 for all zp ∈ Γ(xp) and we have (17) holding for i = p. Otherwise for some i ∈{0, 1, 2,...,p − 1} we have ∗ ∗ hxi , xi+1 − xii≤ 0 for all xi ∈ Γ(xi) and (17) holding. Thus we have: n Proposition 8. A multifunction Γ : D → IR is cyclically quasi-monotone when- ever Γ is cyclically pseudo–monotone. The authors of [16] also define a cyclic pseudo–monotonicity. n Definition 9. A multifunction Γ : D → IR is D-H cyclically pseudo–monotone if for all x0, x1,...,xp (and xp+1 = x0) the following implication holds: ∗ ∗ ∃i ∈{0, 1, 2,...,p} , ∃xi ∈ Γ(xi): hxi , xi+1 − xii > 0 ∗ ∗ =⇒ ∃j ∈{0, 1, 2,...,p} , ∀xj ∈ Γ(xj )\{0} : hxj , xj+1 − xj i < 0. (18) n Proposition 9. If a multifunction Γ : D → IR is D-H cyclically pseudo-monotone if and only if it is cyclically pseudo–monotone (in the sense of this paper). 248 A. EBERHARD AND J-P. CROUZEIX n Proof. Suppose Γ : D → IR is D-H cyclically pseudo-monotone. We first show that ∗ Γ is cyclically quasi–monotone. Suppose for all i =0,...,p−1 (any p) and (xi,zi ) ∈ ∗ ∗ Graph Γ, with zi 6= 0, we have hzi , xi+1 − xii≥ 0. Now if for some i =0,...,p − 1 ∗ ∗ ∗ we have hzi , xi+1 − xii > 0 then (18) implies hzp, x0 − xpi < 0 for all zp ∈ Γ(xp) as ∗ desired. Otherwise we must have hzi , xi+1 −xii = 0 for all i =0,...,p−1 and some ∗ ∗ ∗ zi ∈ Γ(xi). Suppose contrary to thesis that hzp , x0 − xpi > 0 for some zp ∈ Γ(xp). ∗ Then invoking (18) again we have hzi , xi+1 − xii < 0 for some i = 0,...,p − 1 a clear contradiction. Thus we have established Γ is cyclically pseudo-monotone of all orders. n Next suppose Γ : D → IR is cyclically pseudo monotone. Now suppose there ∗ ∗ exists an i ∈{0, 1, 2,...,p} and a xi ∈ Γ(xi) with hxi , xi+1 −xii > 0. If we suppose ∗ ∗ that there does not exist j 6= i for which hxj , xj+1 − xj i < 0 for all xj ∈ Γ(xj )\{0} ∗ ∗ then for all j 6= i we have some xj ∈ Γ(xj )\{0} for which hxj , xj+1 − xj i ≥ 0 for ∗ all j ∈ {1,...,p}. But by Lemma 8 we must then have hxp, x0 − xpi < 0 for all ∗ xp ∈ Γ(xp)\{0} and since x0 = xp+1, this is a clear contradiction. Thus (18) must hold. A number of papers have been written to extend such connections between quasi (resp. pseudo) monotonicity and D-H cyclical quasi–(resp. pseudo–) monotonicity of some subdifferential and the quasi–(resp. pseudo–) convexity of the associated function (see, for instance, [13], [2], [15] and [16]). All such studies do not character- ize the properties required for such multifunctions to be some kind of subderivative of a quasi (resp. pseudo) convex function but rather assume the multifunction is from the outset a subdifferential of some kind. Part of the reason for this is that there exist counter examples to the maximality of such subdifferentials for simple quasiconvex functions. From [16] we take an example first supplied by D.-T. Luc. Let f(x) = sign(x) |x| then ∇f is not maximally pseudo–monotone and also not maximally cyclicallyp pseudo–monotone. It must be stressed from the outset that such counter–examples do not apply to N˜f (·). As we will see a multifunction is only a candidate for a maximal cyclically quasi–monotone operator if it is closed–convex with conical images. Even if we form Γ(x) = cone ∇f(x) we find that Γ(0) = ∅ and for x 6= 0 we have Γ(x) = IR+ := {γ ∈ IR | γ ≥ 0}. Thus we may extend Γ to a larger cyclically quasi–monotone operator by defining Γ(0) = IR+. On the other hand the normal cone to the level–sets of f(x) = sign(x) |x| are defined every- where and the graph of N˜f is the closed half–space {γ ∈ IRp| γ ≥ 0} and hence N˜f is maximal. Thus maximality of the multifunction is in conflict with the use of the derivative (and indeed generalized subdifferentials). For this reason we avoid the use of subdifferentials in this paper. For a differentiable function pseudo–monotonicity of ∇f characterizes pseudo– convexity of f. We say (x, x∗) is pseudo–monotonically related to the graph of Γ if hx∗,y − xi > 0=⇒ hy∗,y − xi > 0 ∀y∗ ∈ Γ(y)\{0} and y ∈ D. (19) Note that for all x we always have the point (x, x∗) = (x, 0) pseudo–monotonically related to the graph of Γ since (19) holds vacuously. This justifies the use of the effective domain dom Γ = {x ∈ D | Γ (x) \{0} 6= ∅}. n Lemma 9. Let Γ : D → IR be a pseudo–monotone relation that contain all its pseudo–monotonically related points in the sense that if (x, x∗) is pseudo–monotonically related to the graph of Γ then x∗ ∈ Γ(x). Then Γ is maximally pseudo–monotone. CLOSED GRAPH PSEUDO-MONOTONE RELATIONS 249 Proof. Define Γ′(x)= {x∗ | (x, x∗) is pseudo–monotonically related to Graph Γ}. (20) Then by assumption Graph Γ′ ⊆ Graph Γ. By the pseudo–monotonicity of Γ and definitions Graph Γ ⊆ Graph Γ′ giving equality. Now suppose there exists a pseudo–monotone relation F with Graph F strictly containing Graph Γ. Then by the pseudo–monotonicity of F there exists a non–zero x∗ ∈ F (x) (and x∗ ∈/ Γ(x)) such that when hx∗,y − xi > 0 we have hy∗,y − xi > 0 for all y∗ ∈ F (x) ⊇ Γ(x). But this implies (x, x∗) is pseudo–monotonically related to the graph of Γ and so x∗ ∈ Γ(x), a contradiction. Thus Γ is maximal pseudo–monotone. At this point it is not clear whether one can contain a given pseudo–monotone re- lation in a maximal pseudo–monotone relation or if such maximal pseudo–monotone relations are unique. They do exist. Consider the following relation {0}× IR+ if y > 0 Γ(x, y)= {0}× IR− if y < 0 {0}× IR if y =0 One may show Γ is maximal pseudo–monotone by using the previous lemma. Indeed (1, 0) ∈/ Γ(0, 0) and ((0, 0), (1, 0)) is not pseudo–monotonically related to the graph ∗ of Γ since h(1, 0), (y1, 0) − (0, 0)i = y1 > 0 implies h(0,y2 ), (y1, 0) − (0, 0)i = 0 for ∗ ∗ (0,y2) ∈ Γ(y1, 0) and (0,y2) 6= (0, 0). One can similarly argue that enlarging Γ(x, y) at any point will introduce points that are not pseudo–monotonically related to the graph of Γ. n Lemma 10. Let Γ : D → IR be a pseudo–monotone relation with 0 ∈ Γ (x) for all x ∈ D. Then there exists a maximal pseudo–monotone relation that contains the graph of Γ. Proof. Given any pseudo–monotone relation Γ that does not contain all its pseudo– monotonically related points we may define a new relation F by adding one such point (y, yˆ∗) with y∗ 6= 0 i.e. F (y) := Γ(y) ∪{yˆ∗}. We claim that F is also pseudo– monotone. Using (15) we see that if this was not the case then we would have for some x∗ ∈ Γ(x)\{0} such that hx∗,y − xi ≥ 0 implying hyˆ∗,y − xi < 0. But as (y, yˆ∗) is pseudo–monotonically related to the graph of Γ we have hyˆ∗, x − yi > 0 implies hx∗, x − yi > 0 for all x∗ ∈ Γ(x)\{0}, a contradiction. Thus adding a pseudo–monotonically related point to the graph of any pseudo–monotonically relation preserves pseudo–monotonicity. We may increase the size of the graph of any relation by adding to the graph one pseudo–monotonically related point. Define the relation via the graph Graph Γˆ := {(x, x∗) | (x, x∗) is pseudo–monotonically related to Graph Γ}. When Γ is pseudo–monotone we have Graph Γ ⊆ Graph Γˆ along with Γˆ also pseudo– monotone. Now if Γˆ does not contain all points that are pseudo–monotonically to the graph of Γˆ then we create a relation with a larger graph by including another pseudo–monotonically related point to the new graph (essentially a re-iteration of the argument made in the first part of this proof). In this way we may create an increasing (by inclusion) chain of subsets of IR2n corresponding to graphs of pseudo-monotone multi–functions. By Zorn’s lemma there exists a maximal relation F (in the sense of inclusion of graphs). By definition Graph F contains all its pseudo–monotonically related points. As F must be pseudo–monotone we may apply Lemma 9 to deduce that F is maximal pseudo–monotone. 250 A. EBERHARD AND J-P. CROUZEIX n ∗ We say a graph Graph Γ is relatively closed in D × IR if whenever (xn,yn) ∈ Graph Γ converges to (x, y∗) with x ∈ D we have y∗ ∈ Γ(x). n Lemma 11. Suppose Γ : D → IR . 1. When Γ is maximally cyclically pseudo–monotone or a maximal pseudo–monotone relation then Γ has conic, convex images Γ(x). Thus when Γ has closed graph then Γ(x) must be closed, convex and conic for every x ∈ D. 2. If Γ is cyclically pseudo–monotone of order p then so must be coconeΓ. Proof. We first show that if Γ is cyclically pseudo–monotone of order p then so is the relation cone Γ(x)= ∪µ>0µΓ(x). ∗ ∗ For all i = 0,...,p − 1 take (xi,zi ) ∈ Graph cone Γ, with zi 6= 0, and note that ∗ ∗ ∗ there exits µi > 0 such that zi = µixi where xi ∈ Γ(xi). Then ∗ ∗ hzi , xi+1 − xii≥ 0 implies hxi , xi+1 − xii≥ 0 ∗ ∗ and the cyclic pseudo–monotonicity of Γ implies hxp, x0−xpi≤ 0 for any xp ∈ Γ(xp). So ∗ ∗ hzp , x0 − xpi = hµpxp, x0 − xpi≤ 0 ∗ for any zp ∈ cone Γ(xp). Hence if Γ does not have conical images we can properly contain its graph in the cyclic pseudo–monotone relation cone Γ. Next we show that if Γ is cyclically pseudo–monotone of order p then so is co Γ(x) ∗ ∗ where co denotes the convex hull operation. Take (xi,zi ) ∈ Graph co Γ, with zi 6=0 such that ∗ hzi , xi+1 − xii≥ 0 (21) for i = 0,....p − 1. By the Caratheodory theorem, for each i there exists ki ≥ 1, i i ∗ ∗ i ∗ µk > 0 with k µk = 1, xi,k ∈ Γ(xi) for k = 0,...,n such that zi = k µkxi,k. Thus by (21)P we have P n i ∗ µkhxi,k, xi+1 − xii≥ 0 kX=0 ∗ implying hxi,k, xi+1 − xii ≥ 0 for some k. Thus by cyclic pseudo–monotonicity ∗ ∗ of Γ we have hzp , x0 − xpi ≤ 0 for all zp ∈ Γ(xp). In particular this implies ∗ ∗ hzp, x0 − xpi≤ 0 for all zp ∈ coΓ(xp). Thus a maximal cyclically pseudo–monotone relation must have convex, conic images Γ(x). In the event that Γ also has a closed graph then Γ(x) must also be closed. The case of pseudo–monotonicity corresponds to p = 1. Remark 5. Suppose f quasi–convex, lower semi–continuous, solid and is locally pseudo–convex. Then locally N˜f (x) = Nf (x) and hence by Lemma 5 Nf (x) has a closed graph locally. It has been shown in [9] that under these assumptions then ∗ Γ= N˜f is locally a singleton–direction (i.e. N˜f (x) = cone {x (x)}) on a dense set D(N˜f ) ∩ D which may be chosen as the dense set of points within D where f is differentiable. It is well known that the subgradient of a convex function is not only maximally cyclically monotone but also maximal monotone. A similar result holds for pseudo convex–functions on a closed convex domain with interior (i.e. f ∈ Π). CLOSED GRAPH PSEUDO-MONOTONE RELATIONS 251 Lemma 12. Suppose f : D → IR is lower semi–continuous, solid, quasi–convex on D and locally in the class Π within a neighbourhood V ⊆ D. Suppose in addition that → n for N˜f : D → IR and for all x ∈ V ; d ∈ int K˜ (x) and δ > 0 there exists 0 Proof. If we can show Graph N˜f contain all its pseudo monotonically related points we may deduce that N˜f (x) is maximal. We need to show that for x ∈ V and ∗ ∗ ∗ x ∈/ N˜f (x) (and hence x 6= 0) implies (x, x ) is not pseudo–monotonically related n ∗ to Graph N˜f ∩ (V × IR ). That is we need to find y ∈ V and y ∈ N˜f (y) such that hx∗,y − xi > 0 and hy∗,y − xi≤ 0. ∗ ∗ As x ∈/ N˜f (x) there exists a d ∈ int K˜ (x) such that hd, x i > 0. Define C = ∗ ∗ ∗ {z | hz , di < 0}. Then x ∈/ C and because d ∈ int K˜ (x) we have C ∪{0}⊇ N˜f (x). Note that N˜f must be C–upper semi–continuous on V (see Remark 2). By C-upper semi–continuity there exists a neighbourhood V of x such that for all z ∈ V we have N˜f (z) ⊆ C ∪{0}. Now take t> 0 such that y = x+td ∈ V and N˜f (x+td)\{0} 6= ∅. ∗ ∗ ∗ It follows that for any nonzero y ∈ N˜f (y) we have hy ,y − xi = thy , di < 0 while hx∗,y − xi = thx∗, di > 0. When f is locally in the class Π around x (which we call locally pseudo–convexity) we restrict the domain of f to a closed, convex neighbourhood V := D of x. It n is clear that when a relation Γ : D → IR is simultaneously maximally pseudo– monotone and cyclically pseudo–monotone then its must also be maximally cycli- cally pseudo–monotone. Theorem 1. Suppose f : D → IR is lower semi–continuous, solid, locally pseudo convex around x and V ⊆ D is a closed convex neighbourhood of x within which f ′ ′ is pseudo convex. Then x 7→ Nf (x ) is both maximal pseudo–monotone on int V and also (maximally) cyclically pseudo–monotone. ′ ′ ′ Proof. As f is in the class Π on V we have K(x ) = K˜ (x ) and hence Nf (x ) = ′ ′ N˜f (x ) for x ∈ V . Also note that as f is pseudo–convex on V it is globally in the class Π on V so a local minimum is a global minimum on V (by Corollary 3)). c Hence by Corollary 2 we have N˜f (v)\{0} 6= ∅ for all v ∈ V ∩ (argmin f) . For x ∈ n argmin f we again adopt the convention that N˜f (x) = IR and so N˜f (x)\{0} 6= ∅ ′ ′ ′ again. Thus we apply Lemma 12 to deduce that x 7→ N˜f (x )= Nf (x ) is maximal pseudo–monotone on int V . Next we note that as f is pseudo–convex on D and the discussion directly after Corollary 4 establishes Nf is also cyclically pseudo– monotone. As shown earlier in Lemma 11 a maximal cyclically pseudo–monotone relation or a maximal pseudo–monotone relation has convex, conic images. If these relations also have a closed graph then these images must also be closed. ′ ′ 4. The relevance to integration and utility theory. As x 7→ N˜f (x ) defines a relation that is simultaneously both maximal pseudo–monotone and a cyclically pseudo–monotone relation, it begs the question as to whether this property char- acterizes the normal cones to level sets of pseudo–convex functions much like the cyclical monotonicity characterizes the subgradient of a lower semi-continuous, con- vex function. Equality with N˜f is sought since this has a closed graph. We therefore 252 A. EBERHARD AND J-P. CROUZEIX restrict attention to relations with this closed graph property. Maximality of cyclic monotone operators and the closed graph property are closely related in the case of convex functions and the role of these properties is well illustrated in the integration theory for convex functions [23]. The maximality is essential for similar results to hold for lower semi–continuous solid pseudo–convex functions as can be seen by the following example from [5]. Define a lower semi–continuous function f : IR2 → IR by 1 if x< 0 and y > 0 0 if xy ≥ 0 f(x, y)= −x if x> 0, y< 0 and − y ≥ x y if x> 0, y< 0 and − y ≤ x which is not quasi–convex but has Nf \{0} pseudo–monotone. First note that N˜f is not a maximal pseudo–monotone relation. As N˜f (x)= {0} for all points in {(x, y) | x< 0 and y < 0}∪{(x, y) | x> 0 and y > 0} we can properly contain Nf in a maximal pseudo–monotone relation which corre- sponds to the N˜g where g is a pseudo–convex function −x when − y ≥ x g(x, y)= . y when − y ≤ x The role of maximality is less evident in the case of differentiable pseudo–convex functions and indeed this case has already been studied in [14]. In mathematical economics many models presume the existence of a utility func- tion u : K → R (the extended real line) which reflects the preference structure with respect to possible consumption. Here K is a cone which may be interpreted as a set of possible consumption vectors of an economy. It is natural to assume that u is non-decreasing and so u(x1) ≤ u(x2) when x1 ≤K x2 in the order de- fined by the cone K. In standard applications K is the non negative orthant of the underlying space X = IRn and so we will exclusively deal with the finite di- mensional case. The dual cone K∗ := {x∗ ∈ X∗ | hx∗, xi≥ 0 for all x ∈ K}, where hx∗, xi denoted the inner product in IRn, denotes a set of price vectors for com- modities and so hx∗, xi indicates the value of the consumption represented by x under the price vector x∗. When X = IRn we may associate h·, ·i with the usual ∗ n Euclidean inner product and K = K = IR+. Now v represents the preference structure in that y is preferred to x if u(y) ≥ u(x) with respect to the upper level sets y ∈ S−u(x) := {z ∈ K | u(z) ≥ u(y)}. Usually such sets are assumed convex and thus u is a quasi–concave function (i.e. has convex upper level sets S−u(x)). Such assumptions on first appearance may appear convenient but there is a body of theory to justify these assumptions. From [22] we quote some observations. In [22] a number of properties are assumed to define a relation R with which is associated a weak preference structure. That is, y is at least as good as x if yRx. Among these assumptions are those of local nonsatiation which claim that in every neighbourhood of U of x there exists a y such that we do not have xRy. In terms of the possible utility associated with such a relation we are assuming that there exists a y such that u(y) > u(x). This is essentially an assumption that u is locally non-constant and from an economic view–point the consumer has choices available locally. The CLOSED GRAPH PSEUDO-MONOTONE RELATIONS 253 remaining assumptions placed on the relation R include the standard ones of re- flexivity, transitivity, continuity and convexity of the set R(x) := {y ∈ K | yRx} (which is related to the quasi-concavity assumption on u). The consumer is assumed to choose a consumption bundle in his/her budget BG(x∗, w) := {x ∈ K | hx∗, xi≤ w} that is at least weakly preferred to all elements in BG(x∗, w). As BG(x∗, w) = BG(λx∗, λw) for all λ > 0 we may as well assume that w = 1 (unit wealth) and ∗ ∗ denote BG(x , 1) = BG(x ). A preference R induces a demand relation DR in ∗ ∗ ∗ that (x, x ) ∈ DR if and only if x ∈ BG(x ) and xRy for all y ∈ BG(x ). For ∗ ∗ a nonsatiated relation we have (x, x ) ∈ DR implies hx , xi = 1. Since usually only a subset Dv ⊆ DR is observable, revealed preference theory tries to relate assumptions on R with conditions on ∗ ∗ ∗ Du(x ) := {x ∈ BG(x ) | u(x) ≥ u(y) for all y s.t. hx ,yi≤ 1} = {x ∈ BG(x∗) | u(x)= v(x∗)} where v(x∗) := sup {u(x) | hx, x∗i≤ 1 with x ≥ 0} (22) ∗ ∗ and the assumption of convexity of S−u(x) ensures that x ∈ Du(x ) have hx , xi =1 ∗ ∗ and Du(x ) is a proper subset of DR. The indirect utility function v(x ) assigns to any price vector x∗ the greatest utility that the consumer may achieve given a ∗ constraint of a unit of wealth. The customers choices x ∈ Du(x ) are indeed the observables of the theory, not u itself. Thus the issue of how to reconstruct u from such observations arises. Related issues include the study of the properties v(x∗) ∗ and Du(x ) must possess in order for it to be associated with a valid utility u. It may be shown that v must be a non–increasing, quasi-convex function [20] (i.e. ∗ ∗ ∗ ∗ ∗ the lower level sets Sv(x ) := {z ∈ K | v(z ) ≤ v(x )} are convex) and (possibly) extended–real–valued function. Indeed in [20] a minimal set of assumptions on v are derived which ensures that it is associated with a utility u via (22). The generality used in [20] exceeds that which is required here and so we will concern ourselves with v extended–real–valued, quasi–convex, non–increasing and lower semi–continuous. When u is associated with v via (22) then under mild assumptions one may recover u via the duality formula u(x) = inf {v(x∗) | hx, x∗i≤ 1 with x∗ ≥ 0} . (23) ∗ A yet unanswered problem is that of what properties Du(x ) requires in order ∗ that there be a valid utility u producing this set Du(x ) of observed preferences for this consumer? This problem is referred to in the literature as the problem of revealed preferences. A more difficult problem is that of providing a constructive ∗ way of finding u from Du(x ). Even more arduously we can ask whether it is possible ∗ to construct an algorithm to approximate u using only some selections x ∈ Du(x )? We would now like to argue that the problem of revealed preferences reduces to the problem of constructing a pseudo–convex −u function from its normal cone ∗ information Γ = N−u (or at least some subset Γ ⊆ N−u). If x ∈ Du(x ) attains the supremum in (22) then u(x)= v(x∗) and as x∗ satisfies hx∗, xi = 1 we have x∗ achieving the infimum in (23). In this way we see that when the duality between u and v holds then the x∗ achieving the infimum in (23) for a given x are precisely −1 ∗ −1 ∗ ∗ ∗ Du (x). Thus when x ∈ Du (x) we have hy , xi ≤ 1 = hx , xi implies v(y ) ≥ ∗ ∗ ∗ ∗ ∗ ∗ v(x ). That is hy − x , −xi≥ 0 implies v(y ) ≥ v(x ) or −x ∈ Nv(x ). Similarly ∗ ∗ ∗ ∗ when x ∈ Du(x ) we have hy, x i≤ 1= hx, x i implies u(y) ≤ u(x) or hy−x, −x i≥ 254 A. EBERHARD AND J-P. CROUZEIX ∗ ∗ 0 implies −u(y) ≥−u(x). Thus x ∈ Du(x ) may be seen to imply −x ∈ N−u(x). One could imagine an experiment in which the prices for a commodity bundle are varied in order to observe when a consumers choice (or revealed preference) x changed. In this way an inner approximation Γ of N−u(x) could be made. We may say that the preference structure is non-satiated when there exists a direction d ∈ K such that hd, x∗i < 0 for all x∗ ∈ Γ(x) and all x ∈ K. This is certainly implied by u being lower semi–continuous, solid and pseudo–convex as was shown in Corollary 2. The pseudo–convexity assumption is crucial in obtaining this symmetric duality and so deserves some justification. We say that D is weakly rationalized by a preference R if D ⊆ DR. Usually there is postulated an axiom of “weak revealed preference” which in its modified form in [22] corresponds to an assumption of pseudo-monotonicity of the relation defined by −Du. This axiom may be arrived ∗ ∗ at as follows. The observation of (x, x ) ∈ Du with hx ,y − xi ≤ 0 means that x has been chosen at price x∗ when y could have been chosen (i.e. is in budget). When y 6= x this means x is a revealed preference to y. Consistent behavior should demand that y must not be a revealed preference of x or hy∗, x − yi > 0 for all ∗ ∗ −1 (y,y ) ∈ Du. That is for any x ∈ Du (x)\{0} we have ∗ ∗ ∗ −1 hx ,y − xi≤ 0 implying hy ,y − xi < 0 for all y ∈ Du (y)\{0} . (24) ∗ −1 ∗ −1 ∗ ∗ ∗ As x ∈ Du (x) and y ∈ Du (y) imply hx, x i =1= hy,y i we have hx ,y − xi = ∗ ∗ ∗ ∗ ∗ ∗ hy, x − y i and hy ,y − xi = hx, x − y i. Hence (24) implies for any x ∈ Du(x ) we have ∗ ∗ ∗ ∗ ∗ hy, x − y i≤ 0 implying hx, x − y i < 0 for all y ∈ Du(y ) which is just pseudo–monotonicity of Γ = −Du. Such axioms (and other related ones) go back to [25], see also [1]. The strong axiom of revealed preferences [19] ∗ ∗ corresponds to an iteration of the above argument and holds if for all x0,...,xp ∈ ∗ ∗ ∗ K such that there exist xi ∈ Du(xi ) with hxi , xi+1 −xii≤ 0 for i =0,...,p−1 and ∗ ∗ xj 6= xi for i 6= j we have hxp, xp − x0i < 0 for all xp ∈ Du(xp). Once again we may ∗ ∗ ∗ ∗ argue that hxi, xi i = 1 gives hxi , xi+1 − xii = hxi+1, xi − xi+1i for all i and hence ∗ restate the strong axiom of revealed preferences as the existence of xi ∈ Du(xi ) for i =0,...,p − 1 such that ∗ ∗ ∗ ∗ ∗ hxi+1, xi − xi+1i≤ 0 implies hx0, x0 − xpi < 0, for all x0 ∈ Du(x0)\{0} . Reversing the numbering we see that this implies cyclic pseudo–monotonicity of −Du ⊆ N−u. When −u is lower semi–continuous, solid and pseudo–convex then N−u = N˜−u is maximally pseudo–monotone and hence maximality of −Du would ensure −Du = N−u = N˜−u. Acknowledgements. 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