1 ABOUT STATIONARITY AND REGULARITY IN VARIATIONAL ANALYSIS 2 ALEXANDER Y. KRUGER To Boris Mordukhovich on his 60th birthday Abstract. Stationarity and regularity concepts for the three typical for variational analysis classes of objects { real-valued functions, collections of sets, and multifunctions { are investi- gated. An attempt is maid to present a classification scheme for such concepts and to show that properties introduced for objects from different classes can be treated in a similar way. Furthermore, in many cases the corresponding properties appear to be in a sense equivalent. The properties are defined in terms of certain constants which in the case of regularity proper- ties provide also some quantitative characterizations of these properties. The relations between different constants and properties are discussed. An important feature of the new variational techniques is that they can handle nonsmooth 3 functions, sets and multifunctions equally well Borwein and Zhu [8] 4 1. Introduction 5 The paper investigates extremality, stationarity and regularity properties of real-valued func- 6 tions, collections of sets, and multifunctions attempting at developing a unifying scheme for defining 7 and using such properties. 8 Under different names this type of properties have been explored for centuries. A classical 9 example of a stationarity condition is given by the Fermat theorem on local minima and max- 10 ima of differentiable functions. In a sense, any necessary optimality (extremality) conditions de- 11 fine/characterize certain stationarity (singularity/irregularity) properties. The separation theorem 12 also characterizes a kind of extremal (stationary) behavior of convex sets. 13 Surjectivity of a linear continuous mapping in the Banach open mapping theorem (and its 14 extension to nonlinear mappings known as Lyusternik-Graves theorem) is an example of a regularity 15 condition. Other examples are provided by numerous constraint qualifications and error bound 16 conditions in optimization problems, qualifying conditions in subdifferential calculus, etc. 17 Many more properties which can be interpreted as either stationarity or regularity have been 18 introduced (explicitly and in many cases implicitly) and investigated with the development of op- 19 timization theory and variational analysis. They are important for optimality conditions, stability 20 of solutions, and numerical methods. 21 There exist different settings of optimization and variational problems: in terms of single-valued 22 and multivalued mappings and in terms of collections of sets. It is not surprising that investigating 23 stationarity and especially regularity properties of these objects has attracted significant attention. 24 Real-valued functions and collections of sets were examined respectively in [18, 21, 27{30, 33] and 25 [3,5{7,14,21,26{32,34,39,41{43,48]. Multifunctions represent the most developed class of objects. 26 A number of useful regularity properties have been introduced and investigated - see [1, 2, 9, 12, 27 13,20{22,24,28{30,36{40,44{47] and the references therein - the most well recognized and widely 28 used being that of metric regularity. 29 In this paper, which continues [30{32], an attempt is maid to present a classification scheme for 30 such concepts and to show that, in accordance with the cited above words by Borwein and Zhu, 31 properties introduced for objects from different classes can be treated in a similar way. Further- 32 more, in many cases the corresponding properties appear to be equivalent. 2000 Mathematics Subject Classification. 90C46, 90C48, 49K27; Secondary: 58C, 58E30. Key words and phrases. subdifferential, normal cone, optimality, extremality, stationarity, regularity, multifunc- tion, slope, Asplund space. 1 2 ALEXANDER Y. KRUGER 33 First of all, stationarity and regularity properties are mutually inverse. For example, the equality 0 34 f (¯x) = 0 for a real-valued differentiable atx ¯ function f is a stationarity condition, while the 0 35 inequality f (¯x) 6= 0 can be considered as a regularity criterion. Thus, such properties always go 36 in pairs. Given one condition (stationarity or regularity), its negation automatically describes its 37 opposite counterpart. 38 It seems natural to distinguish between primal space properties and those defined in terms of dual 39 space elements. Metric conditions are primal space properties while their characterizations in terms 40 of normal cones or coderivatives are dual conditions. In some cases primal and dual conditions 41 are equivalent, and dual conditions provide complete characterizations of the corresponding primal 42 space properties. However, there are cases when equivalences do not hold, and one has necessary 43 or sufficient conditions. 44 Another natural way of classifying stationarity and regularity properties is to distinguish between 45 basic (\at a point") and more robust strict (\near a point") conditions. In the latter case one 46 can speak about approximate stationarity and uniform regularity. For instance, dual conditions 47 formulated in terms of usual Fr´echet derivatives or Fr´echet subdifferentials/normals belong to 48 the first group, while conditions in terms of strict derivatives or limiting subdifferentials/normals 49 belong to the second one. Metric regularity of multifunctions is a typical example of a primal space 50 uniform regularity property. 51 The properties can be defined in terms of certain constants which in the case of regularity prop- 52 erties provide also some quantitative characterizations of these properties. It will be demonstrated 53 in the subsequent sections that such constants are convenient when establishing interrelations 54 between the properties. 55 Obviously not all existing stationarity and regularity properties are discussed in the paper. 56 Only those typical properties have been chosen which better illustrate the classification scheme 57 described above. The content of this paper is not expected to surprise those working in the area of 58 variational analysis. However, the author believes that some relations presented in it can be useful 59 when dealing with specific problems. 60 The remaining three sections are devoted to our three main objects of interest: real-valued 61 functions, collections of sets, and multifunctions respectively. 62 In Section 2, we consider stationarity and regularity properties of real-valued functions. The 63 main feature of this class of objects compared to the two others is that, in the nondifferentiable 64 case, one can (and should) distinguish between properties of functions \from below" (from the point 65 of view of minimization) and \from above" (from the point of view of maximization). The terms 66 inf-stationarity and inf-regularity are used in the paper in the first instance, and sup-stationarity 67 and sup-regularity in the second one. The \combined" properties are considered as well. A number 68 of stationarity and regularity properties as well as constants characterizing them are introduced. 69 The relations between these constants are summarized in Theorem 2. It can be interesting to 70 note that while two different basic primal space constants are in use, the corresponding strict 71 constants coincide for lower semicontinuous functions on a complete metric space. If, additionally, 72 the space is Asplund, they coincide with the appropriate dual space strict constant. This result 73 (Theorem 2(ix)) improves [33, Theorem 4]. Special attention is given to the differential and convex 74 cases when most of the constants and properties coincide. 75 In Section 3, collections of sets are considered. The stationarity properties discussed here ex- 76 tend the concept of locally extremal collection introduced in [34] while the relation between the 77 corresponding primal and dual constants formulated in Theorem 4(vi) extends the extremal prin- 78 ciple [34, 41]. This result improves [30, Theorem 1]. The corresponding regularity properties are 79 discussed as well as their relations with other properties of this kind: metric inequality (local linear 80 regularity) [4, 19, 20, 43, 48] and Jameson's property (G) [5, 42]. 81 The last Section 4 is devoted to multifunctions with the main emphasis on their regularity 82 properties. The constants characterizing these properties are defined along the same lines. Metric 83 regularity is treated as an example of a uniform primal space regularity property corresponding to 84 similar properties of real-valued functions and collections of sets. The relations between different 85 constants, including the equality of primal and dual strict constants, are summarized in Theorem 6. 86 Finally, relations are established between the multifunctional regularity/stationarity constants and 87 the corresponding constants defined in the preceding sections for the other two main classes of 88 objects of the current research { real-valued functions and collections of sets. ABOUT STATIONARITY AND REGULARITY IN VARIATIONAL ANALYSIS 3 89 Mainly standard notations are used throughout the paper. Br(x) denotes a closed ball in a 90 metric space with centre at x and radius r. A closed unit ball in a normed space is denoted by 91 B. If Ω is a set then int Ω and bd Ω are respectively its interior and boundary. If not explicitly 92 specified otherwise, when considering product spaces we assume that they are equipped with the 93 maximum-type distances or norms: d (x1; y1); (x2; y2) = max d(x1; x2); d(y1; y2) , k(x; y)k = 94 max(kxk ; kyk). Sometimes, in products of normed spaces, the following norm depending on a 95 parameter γ > 0 will be used: k(x; y)kγ = max(kxk ; γ kyk). 96 2. Real-Valued Functions 97 2.1. Extremality, stationarity, and regularity. The classical criterion characterizing extremum 98 points of real-valued functions is given by the famous Fermat theorem. 99 Theorem 1 (Fermat). If a differentiable function f has a local minimum or maximum at x¯ then 0 100 f (¯x) = 0. 0 101 This assertion provides a dual (f (¯x) is an element of the dual space!) necessary condition for 102 a local minimum or maximum. It is well known that it actually characterizes a weaker property 103 called stationarity.