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Existence of Closed Graph, Maximal, Cyclic Pseudo-Monotone Relations and Revealed Preference Theory

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Existence of Closed Graph, Maximal, Cyclic Pseudo-Monotone Relations and Revealed Preference Theory 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′) <f(x)}) are convex for all x ∈ D. Obviously convex functions are quasi–convex but quasi– convexity has a character quite distinct from convexity. This is evident in the fact that quasi–convex functions are not necessarily continuous on the interior of their ′ 1 domain and the lower Dini-derivative f (x, h) := lim inft↓0 t (f(x + th) − f(x)) may take the value −∞ [11]. Another significant difference is the lack of closure of the class of quasi–convex functions under addition (a closure property very important for convex functions). Thus, epigraphical analysis has been replaced by level set analysis and this has been developed by a number of researchers (for example [26], [21] and [8]). A number of authors have recently discussed the equivalence of various cyclical quasi/pseudo–monotonicity properties of the multifunctions Nf (x) (resp. 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′) <f(x)}, {x∗ | hd, x∗i≤ 0 for all d ∈ K(x)} when x ∈ dom f Nf (x) := , ∅ otherwise {x∗ | hd, x∗i≤ 0 for all d ∈ K˜ (x)} if x ∈ S˜(x) N˜f (x) := ∅ if x∈ / dom f . (1) {0} otherwise where the contingent cone to S(x) and S˜(x) at x are given by 1 K(x) = lim sup (S(x) − x) and t↓0 t 1 K˜ (x) = lim sup S˜(x) − x , t↓0 t where the limit supremum are in the sense of Kuratowski–Painlev´e. In finite di- mensions we can write ∗ n ∗ ∗ ∗ lim sup Ct = {x ∈ IR | ∃tn ↓ 0, ∃xn ∈ Ctn , with xn → x } . t↓0 Remark 1. Suppose x ∈ dom f ∩ (arg min f)c and S(x) is convex then ∗ ′ ∗ ′ o Nf (x)= {x | hx − x, x i≤ 0 for all x ∈ S(x)} = (cone (S(x) − 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= ∅} .
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