Elements of Quasiconvex Subdifferential Calculus

Journal of Convex Analysis Volume 7 (2000), No. 2, 243–269

Elements of Quasiconvex Subdiﬀerential Calculus

Jean-Paul Penot Universitéde Pau, Facultédes Sciences, Département de Mathématiques, 64000 Pau, France. e-mail: [email protected]

Constantin Z˘alinescu∗ Université“Al. I. CuzaÔ Ia¸si,Facultéde Mathématiques, Bd. Copou Nr. 11, 6600 Ia¸si,Romania. e-mail: [email protected]

Received July 7, 1998 Revised manuscript received April 24, 1999

A number of rules for the calculus of subdifferentials of generalized convex functions are displayed. The subdifferentials we use are among the most significant for this class of functions, in particular for quasiconvex functions: we treat the Greenberg-Pierskalla’s subdifferential and its relatives and the Plastria’s lower subdifferential. We also deal with a recently introduced subdifferential constructed with the help of a generalized derivative. We emphasize the case of the sublevel-convolution, an operation analogous to the infimal convolution, which has proved to be of importance in the field of quasiconvex functions. We provide examples delineating the limits of the rules we provide.

Keywords: harmonic sum, incident derivative, lower subdiﬀerential, normal cone, quasiconvex function, subdiﬀerential set, sublevel-convolution

1991 Mathematics Subject Classiﬁcation: 26B25, 26A51, 26A27, 90C26

1. Introduction It is the purpose of the present paper to give some calculus rules for known subdifferentials which are used for generalized convex functions. We feel that without such rules, a number of problems involving generalized convex functions can hardly be dealt with. Some rules already exist (see [10], [5] for instance); but we endeavour to make a systematic study. In fact we concentrate on the most usual subdifferentials: the subdifferentials of the Greenberg-Pierskalla type and the lower subdifferential of Plastria. We also deal with a notion introduced in [5] which has not received much attention yet, although it seems to be well adapted to the case of quasiconvex functions. The operations we consider are the most useful ones: composition with a linear map into the source, composition with an increasing function at the range, suprema, performance functions. We also deal with an operation which has attracted some attention during the last few years, the sublevel-convolution h := f3g of the two functions f, g given by

h(w) := inf f(w x) g(x): x X ∈ − ∨ This operation, which is the analogue of the inﬁmal convolution of convex analysis (the

∗The contribution of this author was done during his stay at Universit´ede Pau, spring 1998.

ISSN 0944-6532 / $ 2.50 c Heldermann Verlag 244 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus

sum being replaced by the supremum operation ) is of fundamental importance for quasiconvex analysis inasmuch as the usual sum does∨ not preserve quasiconvexity whereas supremum does (see [1], [11], [12], [13], [14], [L-V]...). Moreover, one can check that the strict sublevel sets of h are given by

[h < r] = [f < r] + [g < r];

whereas the sublevel sets satisfy

[h r] = [f r] + [g r] ≤ ≤ ≤ whenever the infimum is attained in the formula defining h (then one says that the sublevel-convolution is exact). The interest of such an operation for regularization pur- poses is shown elsewhere (see [11], [9]). Our study relies on techniques from convex analysis which are described in our paper [8]. Section 2 is devoted to the case of subdifferentials of Greenberg-Pierskalla type. Such subdifferentials are easy to deal with, but they differ drastically from the usual Fenchel subdifferential as they are cones. The lower subdifferential of Plastria which is studied in section 3 also suffers from such a difference, but to a less extent, as it is a shady subset of the dual (i.e. is stable under homotheties of rate greater than 1), hence is unbounded or empty. The last section is devoted to a subdifferential which is a kind of compromise between the Plastria subdifferential and the incident subdifferential. For a Lipschitzian quasiconvex function, it has always nonempty values. A number of subdifferentials which have been proposed for the study of generalized quasiconvex functions have not been considered here for the sake of brevity (see [5] for a recent survey). However, we hope the present study will enable the reader to get a more precise idea of the possibilities and the limitations of what is called “quasiconvex analysisÔ in [6].

2. Subdiﬀerentials of normal type

Hereafter, f is a function defined on a Hausdorff locally convex space X with dual X∗, ; ; f x with values in R or R := R and we suppose that ( 0) is finite. As ∪ {∞} ∪{−∞ ∞} in [8] we denote by R+ (resp. P) the set of nonnegative (resp. positive) real numbers. We devote this section to some rules for the calculus of subdifferentials of Greenberg- Pierskalla type. These subdifferentials are among the simplest concepts. They are akin to the normal cones to the corresponding sublevel sets [f f(x )] : ≤ 0 @νf(x ) := N([f f(x )]; x ); 0 ≤ 0 0 where the normal cone to a subset C of X at x X is given by 0 ∈

N(C; x ) := x∗ X∗ x∗; x x 0 x C : 0 { ∈ | h − 0i ≤ ∀ ∈ } We also consider the case of the star subdifferential introduced in [5], or rather a slight modification of it (it differs from the definition in [5] by 0 when x0 is not a minimizer). ~ { } It is as follows: x∗ @ f(x ) iff x∗; x x 0 for each x [f < f(x )]. In other terms 0 ∈ 0 h 0 − 0i ≤ ∈ 0 ~ @ f(x0) := N([f < f(x0)]; x0): J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdifferential calculus 245

Both definitions are variants of the classical Greenberg-Pierskalla’s subdifferential, given x @ f x x ; x x < x f < f x by 0∗ ∗ ( 0) iff 0∗ 0 0 for each [ ( 0)]. These subdifferentials are cones and∈ thus theyh are large− i and quite different∈ from the usual Fenchel subdifferential. Obviously ~ ν ~ @∗f(x ) @ f(x );@ f(x ) @ f(x ): 0 ⊂ 0 0 ⊂ 0 ~ Observe that x0 is a minimizer of f iff 0 @∗f(x0) iff @∗f(x0) = X∗, iff @ f(x0) = X∗. In ν ∈ ~ contrast, one always has 0 @ f(x0), 0 @ f(x0). Thus, although these subdifferentials are similar, they have distinct∈ features. Let∈ us give a criterion for their coincidence up to zero. Here f is said to be radially u.s.c. (upper semicontinuous) at some point x if for any y X the function t f(x + ty) is u.s.c. at 0. ∈ 7→ ~ Lemma 2.1. If @∗f(x0) is nonempty, then @ f(x0) = cl(@∗f(x0)): If f is radially u.s.c. ~ at each point of [f < f(x0)] then @ f(x0) = @∗f(x0) 0 . If there is no local minimizer 1 ~ ν ∪ { } of f in f − (f(x0)) then @ f(x0) = @ f(x0):

Proof. The first assertion is easy. For the second assertion, it suffices to observe that a linear form which is nonnegative on an absorbant subset is 0: 1 If there is no local minimizer of f in f − (f(x0)) then the sublevel set [f f(x0)] is ~≤ contained in the closure of the strict level set [f < f(x )]. Thus, any x∗ @ f(x ) being 0 0 ∈ 0 bounded above by x∗; x on [f < f(x )] is also bounded above by x∗; x on [f f(x )]: h 0 0i 0 h 0 0i ≤ 0 x∗ N([f f(x )]; x ): 0 ∈ ≤ 0 0 We observe that each subdifferential @? among the three we consider is homotone, i.e. ? ? given f, g such that f g and f(x0) = g(x0) one has @ f(x0) @ g(x0). The first assertion of the next lemma≤ follows from this observation. ⊂

Lemma 2.2. Let (fi)i I be an arbitrary family of functions ﬁnite at x0 and let f := ∈ ? ~ ν infi I fi. Suppose I(x0) := i I fi(x0) = f(x0) is nonempty. Then, for @ = @∗;@ ;@ ∈ one has { ∈ | } @?f(x ) @?f (x ): 0 ⊂ i 0 i I(x ) ∈\0 ~ If I(x0) = I, then equality holds for @∗;@ ; if moreover I(x) is nonempty for each x X (in particular if I is ﬁnite) then equality holds for @ν. ∈

Proof. For the second assertion, one observes that if I(x ) = I and if x [f < f(x )] 0 ∈ 0 (resp. x [f f(x0)] and if I(x) is nonempty), then, for some i I one has x [fi < f (x )] (resp.∈ x≤ [f f (x )]). ∈ ∈ i 0 ∈ i ≤ i 0 A similar result holds for suprema. Here we denote by co A (resp. coA) the convex hull (resp. the closed convex hull) of A.

Proposition 2.3. Let (fi)i I be an arbitrary family of functions ﬁnite at x0 and let f f I x∈ i I f x f x @? := supi I i. Suppose ( 0) := i( 0) = ( 0) is nonempty. Then, for = ~ ν ∈ { ∈ | } @∗;@ ;@ one has

@?f(x ) co @?f (x ) : (2.1) 0 ⊃  i 0  i I(x ) ∈[0   246 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus

ν Equality holds for @ if I is ﬁnite and equal to I(x0), if Ci := [fi fi(x0)] is convex for i I x k I x C ≤ C X each ( 0) and if either for some ( 0) one has k ( i=k int i) = , or is a ∈ ∈ ∩ 6 6 ∅ C XI C Banach space, each i is closed and = R+ ∆ i I i ,T where ∆ is the diagonal of XI : − ∈ Q ¡

When I(x0) has two elements j; k only, the qualification condition of the preceding statement can be rewritten in the simpler (and more familiar) form X C C : = R+( j k) − ? Proof. Again the first assertion is a consequence of homotonicity. When @ f(x0) is closed, one can replace co by co in (2.1). The second assertion is a consequence of Lemma 2.1 and f f x f f x of the calculus of normal cones ([15] for instance) since [ ( 0)] = i I [ i i( 0)] ≤ ∈ ≤ when I = I(x0) is finite. T Proposition 2.4. Let f : X R , g : Y R and M : X Y R , M x; y f x g y → z ∪{∞}x ; y X→ Y∪{∞} r f ×x →g y∪{∞} ( ) = ( ) ( ). Consider 0 := ( 0 0) such that 0 := ( 0) = ( 0) R. ? ∨ ~ ν ∈ × ∈ Then for @ = @∗;@ ;@ one has

@?M(x ; y ) @?f(x ) @?g(y ); 0 0 ⊃ 0 × 0 ? ν with equality for @ = @ : When x0 and y0 are not local minimizers of f and g, respectively, equality holds for @? = @~; and

~ ~ @∗M(x ; y ) = @∗f(x ) @ g(y ) @ f(x ) @∗g(y ) : 0 0 0 × 0 ∪ 0 × 0 ¡ ¡ Proof. The ﬁrst assertions are easy consequences of the preceding result. Let z∗ := ~ (x∗; y∗) @ M(z ). Thus ∈ 0

x x ; x∗ + y y ; y∗ 0 x [f < f(x )]; y [g < g(y )]: (2.2) h − 0 i h − 0 i ≤ ∀ ∈ 0 ∀ ∈ 0

Since x0 and y0 are not local minimizers of f and g; respectively, we get that

x x ; x∗ 0 x [f < f(x )]; y y ; y∗ 0 y [g < g(y )]; h − 0 i ≤ ∀ ∈ 0 h − 0 i ≤ ∀ ∈ 0 ~ ~ i.e. x∗ @ f(x0) and y∗ @ g(y0). Now let us suppose that z∗ @∗M(z0): If x∗ ~ ∈ ∈ ∈ ∈ @ f(x ) @∗f(x ) there exists u [f < f(x )] such that u x ; x∗ = 0; then, as 0 \ 0 ∈ 0 h − 0 i u x ; x∗ + y y ; y∗ < 0 for each y [g < g(y )]; we get that y∗ @∗g(y ): h − 0 i h − 0 i ∈ 0 ∈ 0 The following result about performance functions is simple and useful. Proposition 2.5. Let W and X be locally convex spaces and let F : W X R; × → p(w) := inf F (w; x);S(w) := x X F (w; x) = p(w) : x X ∈ { ∈ | }

? ~ Then for @ = @∗;@ , for any w0 W such that p(w0) is ﬁnite, and for any x0 S(w0) w @?p w w ; @∈?F w ; x S w ∈w W one has 0∗ ( 0) iﬀ ( 0∗ 0) ( 0 0). When ( ) is non-empty for each , the same result∈ holds for @ν: ∈ ∈

Recall that if F is (convex) quasiconvex then p is (convex) quasiconvex. J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus 247

Proof. The result is essentially a consequence of the fact that w [p < p(w0)] iﬀ there exists x X such that (w; x) [F < F (w ; x )] and of the relation∈ ∈ ∈ 0 0 (w∗; 0); (w w ; x x ) = w∗; w w : h 0 − 0 − 0 i h 0 − 0i The last assertion is obvious.

We also observe that when (w∗; 0) @∗F (w ; x ) then x S(w ): 0 ∈ 0 0 0 ∈ 0 In the sequel, given another locally convex space Y , we denote by L(X;Y ) the class of continuous linear maps from X into Y and for A L(X;Y ), At stands for the transpose of A. ∈ Proposition 2.6. Let f = g A, where A L(X;Y ), and g : Y R is ﬁnite at ? ◦ ∈ ~ ν → y0 = Ax0. Then, if @ is one of the subdiﬀerentials @∗;@ ;@ , one has

At @?g(y ) @?f(x ): 0 ⊂ 0 If A is onto then ¡ t 1 ? ? (A )− @ f(x0) = @ g(Ax0): A R At X Y If is onto and ( ) is w∗-closed (in particular¡ if and are Banach spaces) one t ν ν t ~ has A (@ g(y0)) = @ f(x0); if moreover y0 is not a minimizer of g, then A (@ g(y0)) = ~ t @ f(x0) and A (@∗g(y0)) = @∗f(x0):

~ ~ Proof. Let us consider the case of @ . Given y∗ @ g(y ), for any x [f < f(x )] we ∈ 0 ∈ 0 have y := A(x) [g < g(y0)] hence y∗ A; x x0 = y∗; y y0 0 and x∗ := y∗ A ~ ∈ h ◦ − i h − i ≤ ◦ ∈ @ f(x0): t 1 ~ Suppose now that A is onto and y∗ (A )− (@ f(x0)). Let us show that y∗ belongs to ~ t ~∈ @ g(Ax0). Indeed, x∗ := A y∗ @ f(x0) and for each y Y such that g(y) < g(Ax0) there exists x X with y = Ax∈. So, ∈ ∈ t y Ax ; y∗ = Ax Ax ; y∗ = x x ;A y∗ 0: h − 0 i h − 0 i − 0 ≤ ~ Therefore y∗ @ g(Ax ). The other cases are similar.ª « ∈ 0 t ν 1 Suppose A is onto and R(A ) is w∗-closed; let x∗ @ f(x ). Given x A− (0) and ∈ 0 ∈ " 1; 1 we have f(x0 + "x) = f(x0) hence x∗; "x 0, and so x∗; x = 0. Thus ∈ {− } t h i ≤ h i x∗ N(A)⊥ = A (Y ∗) and there exists a continuous linear form y∗ on Y such that ∈ t 1 ν ν x∗ = y∗ A. It follows that y∗ (A )− (@ f(x )) = @ g(y ): ◦ ∈ 0 0 ~ When x0 is not a minimizer of f and x∗ @ f(x0), taking x X such that f(x) < f(x0) 1 ∈ ∈ one obtains that for any u A− (0) one has f(x + u) = g(Ax) = f(x) < f(x ) hence ∈ 0 x∗; x + u x0 0. It follows that x∗; u = 0 and x∗ = y∗ A for some y∗ Y ∗. As h − i ≤ ~ h i ◦ ∈ above one gets that y∗ @ g(y ) (and y∗ @∗g(y ) when x∗ @∗f(x )). ∈ 0 ∈ 0 ∈ 0 Another important chain rule is the following one. Proposition 2.7. Let g : X R and let ’ : R R be a nondecreasing ’ → ∪{∞}x X →’ t ∪{∞} t g x : function. Set ( ) = and consider 0 such that ( 0) R, where 0 := ( 0) Let f = ’ g. Then∞ ∞ ∈ ∈ ◦ ~ ~ ν ν @∗g(x ) @∗f(x );@ g(x ) @ f(x ) and @ g(x ) @ f(x ): 0 ⊂ 0 0 ⊂ 0 0 ⊃ 0 248 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus

If g(x0) is not a local minimizer of ’ the ﬁrst two inclusions are equalities while if g(x0) ν ν is not a local maximizer of ’ one has @ g(x0) = @ f(x0). If there is no local minimizer 1 ν ν ~ ~ of f in f − (f(x0)) then @ f(x0) = @ g(x0) = @ g(x0) = @ f(x0):

Proof. The inclusions are immediate since [f < f(x0)] [g < g(x0)] and [f f(x0)] [g g(x )]. When g(x ) is not a local minimizer (resp.⊂ maximizer) of ’ one≤ has [f⊃ < ≤ 0 0 f(x0)] = [g < g(x0)] (resp. [f f(x0)] = [g g(x0)]). The last assertion is a consequence ≤ ν ≤ ~ ν ~ of the other ones, and of the relations @ g(x0) @ g(x0), @ f(x0) = @ f(x0) when f satisﬁes the last assumption of Lemma 2:1. ⊂

The previous results can be used to compute the subdiﬀerential of the sublevel-convolution h := f3g. The following rule, which mimics the classical rule for the inﬁmal convolution, is a simple consequence of the formulas about sublevel sets. x ; y X x y z f x g y h z Proposition 2.8. Given 0 0 with 0 + 0 = 0, ( 0) = ( 0) = ( 0) R one has for h := f3g ∈ ∈

@~f(x ) @~g(y ) @~h(z ); 0 ∩ 0 ⊂ 0 ~ ~ @ f(x ) @∗g(y ) @∗f(x ) @ g(y ) @∗h(z ); 0 ∩ 0 ∪ 0 ∩ 0 ⊂ 0 @νf(x ) @νg(y ) @νh(z ); ¡ 0 ∩ 0¡ ⊃ 0 with equality in this last relation when for each z X there exist x; y X such that ∈ ∈ x + y = z, f(x) g(y) = h(z) i.e. the sublevel-convolution is exact. If x0 and y0 are not local minimizers∨ of f and g, respectively, equality holds in the ﬁrst and the second relations.

h < h z f < f x g < g y z @~f x @~g y Proof. Since [ ( 0)] = [ ( 0)] + [ ( 0)], for any 0∗ ( 0) ( 0) and any x [f < f(x )], y [g < g(y )] we have ∈ ∩ ∈ 0 ∈ 0

z∗; x + y z = z∗; x x + z∗; y y 0; h 0 − 0i h 0 − 0i h 0 − 0i ≤ ~ ~ ~ hence z∗ @ h(z ). Similarly, when z∗ @ f(x ) @∗g(y ) or z∗ @∗f(x ) @ g(y ) 0 ∈ 0 0 ∈ 0 ∩ 0 0 ∈ 0 ∩ 0 one gets z∗ @∗h(x ): 0 ∈ 0 x y f g z Conversely, when 0 and 0 are not local minimizers of and , respectively, given 0∗ ~ ∈ @ h(z0), for any x [f < f(x0)], taking a sequence (yn) in [g < g(y0)] with limit y0 we observe that ∈ z ; x x z ; x y x y ; 0∗ 0 = lim 0∗ + n 0 0 0 h − i n h − − i ≤ ~ ~ hence z∗ @ f(x ). Similarly, z∗ @ g(y ): An argument similar to the one in the last 0 ∈ 0 0 ∈ 0 part of the proof of Proposition 2.4 yields the case z∗ @∗h(z ): 0 ∈ 0 The assertion about @ν is a consequence of the inclusion [f f(x )] + [g g(y )] [h ≤ 0 ≤ 0 ⊂ ≤ h(z0)] which is an equality when the sublevel convolution is exact.

3. Subdifferential calculus for lower subdifferentials We devote the present section to the calculus with the lower subdifferential of Plastria < [10]. It is defined by x∗ @ f(x ) iff 0 ∈ 0

x∗; x x f(x) f(x ) x [f < f(x )]: h 0 − 0i ≤ − 0 ∀ ∈ 0 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus 249

We leave to the reader most of the task of writing the necessary modifications for the infradifferential of Gutiérrez[2] given by x∗ @≤f(x ) iff 0 ∈ 0

x∗; x x f(x) f(x ) x [f f(x )]: h 0 − 0i ≤ − 0 ∀ ∈ ≤ 0 Obviously, one has

< ν < @≤f(x ) @ f(x );@≤f(x ) @ f(x );@ f(x ) @∗f(x ): 0 ⊂ 0 0 ⊂ 0 0 ⊂ 0

Again, we start with the observation that these subdiﬀerentials are homotone. Now we consider the performance function p associated to a perturbation F : W X R by × → p(w) = infx X F (w; x): ∈ Lemma 3.1. p w F w ; x : w @

Proof. (see also [5]). (i) Let F (w; x) < F (w0; x0); it follows that p(w) < p(w0), whence

w w ; w∗ + x x ; 0 p(w) p(w ) F (w; x) F (w ; x ); h − 0 i h − 0 i ≤ − 0 ≤ − 0 0 < which shows that (w∗; 0) @ F (w ; x ). ∈ 0 0 (ii) Suppose that p(w ) < F (w ; x ). Then there exists x X such that F (w ; x) < 0 0 0 ∈ 0 F (w0; x0), whence

0 = w w ; w∗ + x x ; 0 F (w ; x) F (w ; x ) < 0; h 0 − 0 i h − 0 i ≤ 0 − 0 0 a contradiction. Therefore p(w0) = F (w0; x0). Let now w W be such that p(w) < p(w ). Then there exists (x ) X such that (F (w; x )) p∈(w); we may suppose that 0 n ⊂ n → F (w; xn) < F (w0; x0) = p(w0) for every n. Hence w w ; w x x ; F w; x F w ; x n : 0 ∗ + n 0 0 ( n) ( 0 0) N h − i h − i ≤ − ∀ ∈ < Taking the limit we get w w ; w∗ p(w) p(w ). Therefore w∗ @ p(w ). h − 0 i ≤ − 0 ∈ 0 Let us turn to composition with a continuous linear map. g Y A L X;Y x X g Ax Proposition 3.2. Let : R, ( ) and 0 be such that ( 0) R. Then, for f := g A → ∈ ∈ ∈ ◦ @

t Moreover, if R(A ) is w∗-closed (in particular if X and Y are Banach spaces) and Ax0 is not a minimizer of g, then

< t < @ f(x0) = A (@ g(Ax0)) : 250 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus

Proof. The ﬁrst assertion is contained in [10] Th. 3.5. Let us recall its simple proof. Let < y∗ @ g(Ax ) and f(x) < f(x ). Then g(Ax) < g(Ax ), and so ∈ 0 0 0 t x x ;A y∗ = Ax Ax ; y∗ g(Ax) g(Ax ) = f(x) f(x ): − 0 h − 0 i ≤ − 0 − 0 t < Therefore Aª y∗ @ f(x«): ∈ 0 t 1 < t < Suppose now that A is onto and y∗ (A )− (@ f(x0)). Then x∗ := A y∗ @ f(x0). Let y Y be such that g(y) < g(Ax ). There∈ exists x X such that y = Ax.∈ So, ∈ 0 ∈ t y Ax ; y∗ = Ax Ax ; y∗ = x x ;A y∗ f(x) f(x ) = g(y) g(Ax ): h − 0 i h − 0 i − 0 ≤ − 0 − 0 < Therefore y∗ @ g(Ax ): ª « ∈ 0 t Suppose now that R(A ) is w∗-closed and Ax0 is not a minimizer for g. It follows that t < R(A ) = (ker A)⊥. Let x∗ @ f(x0) @∗f(x0). By the proof of Proposition 2.6 x∗ t t ∈ ⊂ < ∈ R(A ). Thus x∗ = A y∗ for some y∗ Y ∗. By (3.1) we have that y∗ @ g(Ax0), and so t < ∈ ∈ x∗ A (@ g(Ax )). ∈ 0 < Remark 3.3. The same conclusions hold for @ replaced by @≤ in the preceding statement, even without asking that Ax0 is not a minimizer of f in the last part, with the same proof.

The following result is a special case of what precedes, taking for A a projection. It can also be proved directly. Corollary 3.4. Let f : X R and let f : X Y R be deﬁned by f(x; y) = f(x). If x ; y X Y f x → x × → f ( 0 0) with ( 0) R and if 0 is not a minimizer of , then ∈ × ∈ e e @ inf g and t0 is not a local minimizer ’ g x g x > ’ t t of . If either (a) ( 0) 0 or (b) ( 0) 0 and ( ) = 0 for every R then ≤ ∈ − @<’(t ) @

Proof. (i) Let us denote ’ g by f. Suppose ﬁrst that t0 is a minimizer of ’. Then x0 is a minimizer of f, and so relation◦ (3.2) is obvious. < < Suppose now that t is not a minimizer of ’, and let t∗ @ ’(t ) and x∗ @ g(x ). 0 ∈ 0 ∈ 0 Since t is not a minimizer and ’ is nondecreasing, t∗ > 0. Consider x X such that 0 ∈ f(x) < f(x0). Then g(x) < g(x0), whence

t∗(g(x) g(x )) ’(g(x)) ’(g(x )); x x ; x∗ g(x) g(x ): − 0 ≤ − 0 h − 0 i ≤ − 0 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus 251

Multiplying the second relation by t∗ > 0, we obtain that x x0; t∗x∗ f(x) f(x0). < h − i ≤ − Therefore t∗x∗ @ f(x ): ∈ 0 < (ii) Let now u∗ @ f(x ). As t is not a local minimizer of ’, we have that [f < f(x )] = ∈ 0 0 0 [g < g(x0)]. Thus

x x ; u∗ f(x) f(x ) < 0 x [ g < g(x )[: (3.4) h − 0 i ≤ − 0 ∀ ∈ 0

(a) Let t 0. Taking x [g < 0] and s > 0, we have 2x +sx [g < g(x )] = [ f < f(x ] 0 ≤ ∈ 0 ∈ 0 0 hence x0 + sx; u∗ 0. Therefore x0; u∗ = lims 0 x0 + sx; u∗ 0. Let t < t0. From → (3.4) weh obtain thati ≤ h i h i ≤

x; u∗ x ; u∗ + ’(t) ’(t ) x [g t]: h i ≤ h 0 i − 0 ∀ ∈ ≤ As inf g < t < 0, applying [8] Prop. 5.2 with its notation, we get

sup w; u∗ w [g t] = t‹ (u∗) x ; u∗ + ’(t) ’(t ) < 0: (3.5) {h i | ∈ ≤ } ∂g(0) ≤ h 0 i − 0 ‹ u > = @g @g w Thus := ∂g(0)( ∗) 0. Since 0 (0) and (0) is ∗-closed, is ﬁnite. Using again 1 ∈ [8] Prop. 5.2 we get w∗ := − u∗ @g(0). By (3.5) we obtain that ∈

t x ; w∗ + ’(t) ’(t ) < x ; w∗ t < t : ≤ h 0 i − 0 h 0 i ∀ 0

Dividing by , taking the limit as t t0 and using the inclusion w∗ @g(0), we get → < ∈ < t0 = x0; w∗ = g(x0) = t0, whence w∗ @g(x0) @ g(x0) and @ ’(t0). The conclusionh followsi in this case. ∈ ⊂ ∈ (b) Suppose now that t > 0. Thus, for each t ]0; t [, using again [8] Prop. 5.2, we have 0 ∈ 0

0 sup w; u∗ w [g t] = t« (u∗) x ; u∗ + ’(t) ’(t ) < : ≤ {h i | ∈ ≤ } ∂g(0) ≤ h 0 i − 0 ∞ u @g u @g +@g g x ; u Therefore, ∗ R+ (0). If ∗ 0 (0) = 0 (0) = (dom )−, then 0 ∗ 0 ’∈1 t ’ t < ∈ · « uh > i ≤ and so 0 ( 2 0) ( 0) 0, a contradiction. Therefore := ∂g(0)( ∗) 0 and 1 ≤ − w∗ := − u∗ @g(0). Thus ∈

t x ; w∗ + ’(t) ’(t ) < x ; w∗ t ]0; t [: ≤ h 0 i − 0 h 0 i ∀ ∈ 0 < As above, it follows that x ; w∗ = g(x ) = t , whence w∗ @g(x ) @ g(x ) and h 0 i 0 0 ∈ 0 ⊂ 0 (t t0) ’(t) ’(t0), for each t ]0; t0[. Taking x = 0 in (3.4), we see that the preceding− ≤ inequality− is also valid for t =∈ 0, while for t < 0 the inequality is obvious. Thus @<’(t ) again. ∈ 0 Example 3.6. Let ’ : R R and g : R R be deﬁned by ’(t) := t3, g(x) = max(x; 1). Then one has @→<’(0) = but @<→(’ g)(0) = [1; [. This example shows the necessity− of restrictive assumptions∅ such as sublinearity◦ of g:∞

Let us observe that for any function g and any x0 X such that t0 := g(x0) = inf g @<’ t ∈ X ∈ R, relation (3.3) holds if and only if ( 0) = . When dim = 1, relation (3.3) 6 ∅ holds whenever t0 = 0 and g is sublinear. One may ask whether the assumptions of the statement (ii) are crucial.6 252 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus

Example 3.7. Let ’; ’; ’; g : R R and g : R2 R be deﬁned by: ’(t) = max(t; 0), ’(t) := min(1; 2’(t)), g(r; s) = r and→ → e e 2t if t 0, 1 x if x 0, ’(t) = ≤ g(x) = 2 ≤ (t if t > 0, (x if x > 0. e e x ; t g x ’ @<’ t @ 1. ◦ k · k k 0k The necessity of the condition ’(t) = 0 for t 0 in case (b) is shown by the function ’ g and x = 1: @vector space and g(x) = x for every x X: k k ∈ X ’ Corollary 3.8. Let be a normed space and : R R be increasing on R+ with ’ t t f X f→x ∪{∞}’ x x X ( ) = 0 for every R_. Consider : R, ( ) = ( ) and 0 such that f x : ∈ → k k ∈ ( 0) R Then ∈ < < < @ f(x0) = @ ’( x0 ) @ (x0) k k · k·k < = x∗ X∗ x ; x∗ = x x∗ ; x∗ @ ’( x ) : { ∈ | h 0 i k 0k k k k k ∈ k 0k }

Proof. The conclusion is obvious for x0 = 0, both sides of the relation being X∗, while for x = 0 the conclusion follows from assertion (ii)(b) of the preceding proposition. 0 6

One may ask if there are other situations when (3.3) holds. Such a case occurs when ’ and g are convex functions. To be more speciﬁc, we have: Proposition 3.9. Let ’ : R R be nondecreasing and convex, g : X R → ∪{∞} → ∪{∞} be convex, g(X) int(dom ’) = and x0 dom f which is not a minimizer of f := ’ g. Then (3.3) holds.∩ 6 ∅ ∈ ◦

Proof. Let t0 = g(x0). Using [15, Th. 2.7.5], we have that @f(x0) = @(g)(x0) @’(t ) . Since x is not a minimizer of f, t is not a minimizer of ’∪{and x is not| ∈ a 0 } 0 0 0 minimizer of g. Therefore @’(t0) ]0; [. It follows that @f(x0) = @’(t0) @g(x0). Using ⊂ ∞ k X · x k now [5] Prop. 10, which says that for a convex function : R and 0 dom < → ∪{∞} ∈ which is not a minimizer of k, we have @ k(x0) = [1; [ @k(x0), we obtain that (3.3) holds. ∞ ·

Note that with the preceding data, when x dom f is a minimizer of f, then either x 0 ∈ 0 is a minimizer of g (and (3.3) holds if g(x0) int(dom ’)), or g(x0) is a minimizer of ’; in this case it may happen that (3.3) is not∈ satisﬁed. The fact that in (3.2) one does not have equality in general (even if one of the functions is convex) is shown by the following example. J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus 253

Example 3.10. Let ’; : R [0; [, ’(t) = tp, (t) = tq for t 0, ’(t) = (t) = 0 for → ∞ ≥ < p 1 t < 0, where 0 < q < 1 p < . Then for every t > 0 we have that @ ’(t) = [pt − ; [, < q 1 ≤ ∞ ∞ @ (t) = [t − ; [. Taking q = 1=p with p > 1 we have that ∞ @<(’ )(t) = [1; [ = @<( ’)(t); ◦ ∞ ◦ @<’((t)) @<(t) = [p; [ = @<(’(t)) @<’(t): · ∞ · f X x f In the sequel, for : R and 0 dom , we consider that → ∪{∞} ∈ 0 @

< < Proof. (i) Let x∗ @ f(x0), y∗ @ g(x0), ]0; 1[ and z∗ = x∗ + (1 )y∗. Consider x X such that h∈(x) < h(x ). It∈ follows that ∈f(x) < f(x ) and g(x) <− g(x ). Hence ∈ 0 0 0

x x ; x∗ f(x) f(x ) h(x) h(x ); h − 0 i ≤ − 0 ≤ − 0 x x ; y∗ g(x) g(x ) h(x) h(x ): h − 0 i ≤ − 0 ≤ − 0 Multiplying the ﬁrst relation by and the second one by 1 , then adding them, we get < − x x0; z∗ h(x) h(x0). Therefore z∗ @ h(x0). The cases = 0; 1 follow as in the proofh − of (ii).i ≤ − ∈ < (ii) Let x∗ N([f < r0]; x0) and y∗ @ g(x0): Consider x X such that h(x) < h(x0). It follows that∈ x [f < r ] and g(x)∈< g(x ). Hence ∈ ∈ 0 0

x x ; x∗ 0; x x ; y∗ g(x) g(x ) h(x) h(x ): h − 0 i ≤ h − 0 i ≤ − 0 ≤ − 0 < Adding both relations we get that x∗ + y∗ @ h(x ): ∈ 0

< < Note that when @ f(x0) and @ g(y0) are nonempty, the left hand side of relation (3.7) is the convex hull of @

Example 3.13. Let › ]0; 1[, – [1; [ and f; g : R R; ∈ ∈ ∞ → – if x ] ; –]; – if x ] ; 0]; − ∈ − ∞ − −η γ2 ∈ − ∞ f(x) = x if x ] –; 0]; g(x) = + x – if x ]0;›];  ∈ −  γ − ∈ x2 if x ]0; [; (1 + ›)x › if x ]›; [:  ∈ ∞  − ∈ ∞ Then   – if x ] ; –]; x− if x ∈ ] −–; ∞0]−; f g x  x2 x ∈ −;› ; ( )( ) =  if ]0 ] ∨  › x › x ∈ ›; ;  (1 + ) if ] 1] x2 − if x ∈ ]1; ]:  ∈ ∞  We have f(1) = g(1) = 1, @

(ii) If f(x0) < g(y0); then < < < N([f < r ]; x ) @ g(y ) @ M(x ; y ) X∗ @ g(y ): 0 0 × 0 ⊂ 0 0 ⊂ × 0 Moreover, if y0 is not a local minimizer of g then @

Proof. The conclusions of (i) and the first part of (ii) are obvious if (x0; y0) is a minimizer of M. In the contrary case, let f; g : X Y R be given by f(x; y) = f(x) and × → g(x; y) = g(y). Of course, M = f g. By Proposition 3.2 and Corollary 3.4 we have ∨ < < e e < e < that @ f(x0; y0) = @ f(x0) 0 in case (i) and @ g(x0; y0) = 0 @ g(y0) in cases × { }e { } × (i)e and (ii). As [f < r ] = [f < r ] Ye , the conclusion (i) and the first inclusion of the 0 0 × first parte of (ii) follow from Proposition 3.11; the seconde inclusion in (ii) is immediate: < taking y [g < ge(y0)] and x = x0 in the definition of (x∗; y∗) @ M(x0; y0) we get that < ∈ ∈ y∗ @ g(y ). ∈ 0 Suppose now that we are in case (ii) and y0 is not a local minimizer of g. There exists < a net (yi)i I [g < g(y0)] such that (yi) y0. Then for any (x∗; y∗) @ M(x0; y0) we ∈ <⊂ → ∈ have y∗ @ g(y ) and ∈ 0 x x ; x∗ + y y ; y∗ < 0 x [f < r ]; i I: h − 0 i h i − 0 i ∀ ∈ 0 ∈ J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdifferential calculus 255

Passing to the limit for i I, we obtain that x x ; x∗ 0 for every x [f < r ], ∈ h − 0 i ≤ ∈ 0 whence x∗ N([f < r ]; x ). ∈ 0 0 Remark 3.15. By Proposition 3.14 (ii), if

< One can say that the above example is not very conclusive because @ f(x0) = or inf f = inf g. In fact one observes that taking m := max inf f; inf g (supposed to∅ be 6 { } a real number), we have that M(x; y) = fm(x) gm(y); where fm(x) = m f(x) and g (y) = m g(y). In the next example inf f = inf∨g and @

Proof. Taking (xn)n 1 [f < f(x0)] with (xn) x0 and y < y0, we get that y∗ 0, ≥ ⊂ → ≥ while taking x [f < f(x )] and (y ) y , we get that x∗ N([f < f(x )]; x ). Suppose ∈ 0 n % 0 ∈ 0 0 that s∗y∗ [0; 1[ and take x [f < f(x0)]. By assumption, there exists y ] ; y0[ ∈ ∈ e ∈ − ∞ such that g(y) t with t := f(x): Then we have g (t) y: Moreover, as g(y0) < g(y0) for ≥ e e e ≤ y0 < y0; we have g (t0) = y0: Thus, as g (t) < g (t0); we obtain

e e s∗(f(x) f(x )) = s∗(t t ) g (t) g (t ) y y : − 0 − 0 ≤ − 0 ≤ − 0 Hence

x x ; x∗ + y∗s∗(f(x) f(x )) x x ; x∗ + y∗(y y ) f(x) f(x ) h − 0 i − 0 ≤ h − 0 i − 0 ≤ − 0 1 < and so u∗ := (1 s∗y∗)− x∗ @ f(x0); x being arbitrary in [f < f(x0)]. When g is the identity mapping,− ge is also∈ the identity mapping and @

Example 3.16 shows that (3.9) does not hold in general for g(t) = t p with p = 1: | | 6 We take now g as a composition of an increasing function with the norm. Proposition 3.19. Let ’ : [0; [ [0; [ be an increasing function with ’(0) = 0, f X Y ∞ → ∞ x X y Y x X : R and a normed vector space. Consider 0 , 0 , ∗ ∗ and → ∈ < ∈ ∈ y∗ Y ∗ such that y0; y∗ = y0 y∗ and (x∗; y∗ ) @ (f ’)(x0; y0 ). Then ∈ < h i k k · k k k k ∈ ∨ k k (x∗; y∗) @ (f ’ )(x ; y ). ∈ ∨ ◦ k·k 0 0 Proof. Let (x; y) X Y be such that f(x) ’( y ) < f(x ) ’( y ). Then ∈ × ∨ k k 0 ∨ k 0k

x x ; x∗ + y y ; y∗ x x ; x∗ + y∗ ( y y ) h − 0 i h − 0 i ≤ h − 0 i k k k k − k 0k f(x) ’( y ) f(x ) ’( y ); ≤ ∨ k k − 0 ∨ k 0k < i.e. (x∗; y∗) @ (f ’ )(x ; y ). ∈ ∨ ◦ k·k 0 0 The converse is also true under some additional hypotheses. Proposition 3.20. Let ’ : [0; [ [0; [ be an increasing function with ’(0) = 0, f X f ∞ → ∞Y x X; : R be such that inf = 0 and a normed space. Consider also 0 → ∈ y0 Y , x∗ X∗ and y∗ Y ∗ 0 such that x0 is not a local minimizer for f and ’ is ∈ ∈ ∈ \{ } < right-continuous at y0 . If (x∗; y∗) @ (f ’ )(x0; y0) and ›0 := f(x0) ’( y0 ), then k k ∈ ∨ ◦ k·k ∨ k k

< f(x ) ’( y ); (x∗; y∗ ) @ (f ’)(x ; y ); 0 ≤ k 0k k k ∈ ∨ 0 k 0k y ; y∗ = y y∗ ; x∗ N ([f < › ]; x ) : h 0 i k 0k · k k ∈ 0 0 < < Moreover, if f(x ) < ’( y ) then y∗ @ ’( y ) and so y∗ @ (’ )(y ). 0 k 0k k k ∈ k 0k ∈ ◦ k · k 0 Proof. Since ’ is right-continuous at t := y , g := ’ is upper semi-continuous at 0 k 0k ◦ k·k y0. Supposing that g(y0) < f(x0), by Proposition 3.14, we get the contradiction y∗ = 0 because N([g < f(x )]; y ) = 0 . Therefore f(x ) ’( y ). In particular y = 0 (since 0 0 { } 0 ≤ k 0k 0 6 f(x0) > inf f = ’(0)). We have that

x x ; x∗ + y y ; y∗ f(x) ’( y ) f(x ) ’( y ) h − 0 i h − 0 i ≤ ∨ k k − 0 ∨ k 0k for every (x; y) X Y such that f(x) ’( y ) < › ; such (x; y) exist in our conditions. ∈ × ∨ k k 0 Considering t [0; t0[, x X with f(x) ’(t) < ›0, and arbitrary y Y with y = t from the above∈ relation we∈ get ∨ ∈ k k

x x ; x∗ + t y∗ y ; y∗ f(x) ’(t) f(x ) ’(t ): (3.10) h − 0 i k k − h 0 i ≤ ∨ − 0 ∨ 0

Since x0 is not a local minimizer of f, there exists (xn)n 1 [f < f(x0)] such that ≥ (x ) x . From (3.10) we obtain that ⊂ n → 0

x x ; x∗ + t y∗ y ; y∗ < 0 t [0; t [: h n − 0 i k k − h 0 i ∀ ∈ 0 Taking the limit for n , we obtain that → ∞

t y∗ y ; y∗ y y∗ t [0; t [: k k ≤ h 0 i ≤ k 0k · k k ∀ ∈ 0 258 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus

Taking t t we get y ; y∗ = y y∗ . Now, from (3.10), we obtain that ↑ 0 h 0 i k 0k · k k x x ; x∗ + y∗ (t t ) f(x) ’(t) f(x ) ’(t ) (3.11) h − 0 i k k − 0 ≤ ∨ − 0 ∨ 0 < whenever f(x) ’(t) < f(x ) ’(t ), i.e. (x∗; y∗ ) @ (f ’)(x ; y ). Fixing ∨ 0 ∨ 0 k k ∈ ∨ 0 k 0k x [f < › ] and taking t t in (3.11), we obtain that x∗ N([f < › ]; x ): ∈ 0 ↑ 0 ∈ 0 0 < If f(x ) < ’(t ), in (3.11) we may take x = x , obtaining that y∗ @ ’(t ). 0 0 0 k k ∈ 0 Corollary 3.21. Let Y be a normed space, f : X R be such that inf f = 0 and → ∪ {∞} M(x; y) = f(x) y : Suppose that x0 X is not a local minimizer of f and y0 Y is such that f(x ) =∨ kyk . Then (3.9) holds∈ at (x ; y ): ∈ 0 k 0k 0 0 g Y g y y : x ; y @

The following lemma is an important step toward the case of the sublevel-convolution. Lemma 3.22. Let f; g : X R and let M;H : X X R , be given by M x; y f x g y H→ w;∪ x {∞}M x; w x × → r ∪ {∞}H w ; x ( ) = ( ) ( ) and ( ) = ( ). Suppose that 0 := ( 0 0) R. Then ∨ − ∈ < < (w∗; x∗) @ H(w ; x ) (w∗ + x∗; w∗) @ M(x ; w x ): ∈ 0 0 ⇔ ∈ 0 0 − 0 In particular, for y := w x , one has 0 0 − 0 < < (w∗; 0) @ H(w ; x ) (w∗; w∗) @ M(x ; y ): ∈ 0 0 ⇔ ∈ 0 0 f x g y C @

If f(x0) < g(y0) then

< < (y∗; x∗ y∗) (x∗; y∗) N([f < r ]; x ) @ g(y ) @ H(x ; y ): { − | ∈ 0 0 × 0 } ⊂ 0 0 Proof. Let A L(X X;X X) be given by A(w; x) = (x; w x). One gets easily t ∈ × × − that A (w∗; x∗) = (x∗; w∗ x∗). As H = M A and A is an isomorphism, the conclusion follows from Propositions− 3.2 and 3.14. ◦

It is known (see [11], [15, Cor. 2.7.6]) that for proper convex functions f; g : X R , x dom f, y dom g, w = x + y and r = (f3g)(w ), one has → ∪{∞} 0 ∈ 0 ∈ 0 0 0 0 0 @f(x )3@g(y ) if f(x ) = g(y ) = r ; @(f3g)(w ) = 0 0 0 0 0 (3.12) 0 N(dom f; x ) @g(y ) if f(x ) < g(y ) = r ; º 0 ∩ 0 0 0 0 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus 259

where, for subsets A and B of X

A3B := A (1 )B (0 A B) (A 0 B) ; (3.13)  ∩ −  ∪ · ∩ ∪ ∩ · λ ]0,1[ ∈[   here, as in [8], 0 A = 0+A with 0+ = 0 , unless A is a subdiﬀerential in which case we use the convention· (3.6). ∅ { } Recall that when A; B are closed convex sets such that PA PB = for P =]0; [, one has (see [8]) ∩ 6 ∅ ∞

A3B = cl A (1 )B = cl (A#B) :  ∩ −  λ ]0,1[ ∈[   From formula (3.12) one obtains that for f; g, x0; y0; w0, r0 as above one has

A natural question is whether (3.14) is valid for arbitrary (quasi-convex) functions. In the next result we shall see that the inclusion always holds, while the converse one is true in very special cases. With respect to this⊃ point, some remarks are in order. Given f; g : X R , it is obvious that inf f3g = inf f inf g. Moreover, for → ∪ {∞} ∨ m := inf f inf g, fm := f m; gm := g m we have f3g = fm3gm: Moreover, for every « ∨ x ∨f f∨x « @< f « x @ « and consider x∗ @ f(x ). If 0 0 ∈ 0 (f «)(x) < (f «)(x0) = f(x0), f(x) < f(x0). Thus x x0; x∗ f(x) f(x0) ∨ ∨ < h − i ≤ − ≤ (f «)(x) (f «)(x0). Therefore x∗ @ (f «)(x0). Thus, it may be convenient to suppose∨ that− inf∨f = inf g when considering∈ f3g∨, especially when one wants to prove the inclusion @<(f3g)(w ) @

Proof. (i) Suppose that f(x0) > g(y0). Since x0 is not a local minimizer of f, there exists a net (xi)i I [f < f(x0)] such that (xi) x0. Of course, because (w0 xi) y0 ∈ ⊂ → − → and g is u.s.c. at y0, there exists i0 I such that (w0 xi)i i [g < f(x0)]: We get the · 0 contradiction h(w ) f(x ) g(w ∈ x ) < f(x ) = −h(w ): ⊂ 0 ≤ i0 ∨ 0 − i0 0 0 (ii), (iii) Consider H : X X R , given by H(w; x) = f(x) g(w x). We observe that h is the performance× → function∪ {∞} associated to H: ∨ − 260 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus

< < < Let w∗ @ f(x )3@ g(y ) in case (ii) and w∗ N([f < g(y )]; x ) @ g(y ) in case (iii); ∈ 0 0 ∈ 0 0 ∩ 0 it is obvious that (w∗; w∗) C (deﬁned at Lemma 3.22) in the ﬁrst case and (w∗; w∗) < ∈ ∈ N([f < g(y0)]; x0) @ g(y0) in the second one. In both cases, by Lemma 3.22, (w∗; 0) < × < ∈ @ H(w ; x ). By Lemma 3.1 we have that h(w ) = H(w ; x ) and w∗ @ h(w ). 0 0 0 0 0 ∈ 0 When g has the form ’ , a more special result can be given. ◦ k·k Proposition 3.24. Let X be a normed vector space and h = f3(’ ), where ’ : R ’ t t f X ◦ k·k → R is increasing on R+, with ( ) = 0 for 0 and : R is such that ≤ → ∪ {∞} inf f = 0. Consider also x dom f, y ; w X such that w = x + y and w∗ X∗: 0 ∈ 0 0 ∈ 0 0 0 ∈ (i) If < (w∗; w∗ ) @ (f ’)(x ; y ); y ; w∗ = y w∗ ; (3.15) k k ∈ ∨ 0 k 0k h 0 i k 0k · k k < then h(w ) = f(x ) ’( y ) and w∗ @ h(w ): 0 0 ∨ k 0k ∈ 0 (ii) If h(w0) = f(x0) ’( y0 ), x0 is not a local minimizer of f and ’ is right-continuous ∨ k k < at y then f(x ) ’( y ): Moreover, if w∗ @ h(w ) then w∗ = 0, w∗ k 0k 0 ≤ k 0k ∈ 0 6 ∈ N ([f < h(w0)]; x0) and (3.15) holds. (iii) Under the conditions of (ii), if f(x ) < ’( y ) then 0 k 0k @

< Proof. (i) By Proposition 3.19 we obtain that (w∗; w∗) @ M(x0; y0), where g := ’ ∈ < ◦ , M(x; y) = f(x) g(y). Therefore, by Lemma 3.22, (w∗; 0) @ H(w0; x0), where Hk·k(w; x) = f(x) g(w∨ x). Since h is the performance function of∈H, using Lemma 3.1, ∨ − < we obtain that h(w ) = H(w ; x ) = f(x ) ’( y ) and w∗ @ h(w ): 0 0 0 0 ∨ k 0k ∈ 0 (ii) The ﬁrst part follows from Proposition 3.23 (i). Since x0 is not a local minimizer of f, we have that h(w0) f(x0) > inf f = inf h = 0. Hence w0 is not a minimizer of h. < ≥ Let w∗ @ h(w0); it follows that w∗ = 0. Since h is the performance function of H ∈ 6 < deﬁned above we have, by Lemmas 3.1 and 3.22, that (w∗; w∗) @ (f ’ )(x0; y0). By Proposition 3.20 the conclusion follows. ∈ ∨ ◦ k·k < (iii) Suppose that f(x ) < ’( y ) (= h(w )) and take w∗ @ h(w ). Of course, the 0 k 0k 0 ∈ 0 conclusion of part (ii) holds; moreover, from Proposition 3.20, we have that w∗ < < < k k ∈ @ ’( y0 ). It follows that w∗ N([f < h(w0)]; x0) @ ’( y0 ) @ (y0). As the last equalityk k of (3.16) is given by Corollary∈ 3.8, the conclusion∩ k followsk · fromk·k Proposition 3.23 (iii).

In the case ’(t) = t we obtain a more precise result using Corollary 3.21 and the preceding proposition. Corollary 3.25. Let X be a normed vector space and h = f3 , where f : X R is such that inf f = 0. Consider also x ; y ; w X such that wk·k = x + y . If →x is 0 0 0 ∈ 0 0 0 0 not a local minimizer of f and h(w0) = f(x0) y0 then f(x0) y0 . Moreover, if f(x ) = y then ∨ k k ≤ k k 0 k 0k @

Proof. The ﬁrst part follows by Proposition 3.24 (ii). < Suppose that h(w0) = f(x0) = y0 and take w∗ @ h(w0). Using again Proposition k k ∈ < < 3.24 (ii) and Corollary 3.21, we obtain that w∗ @ f(x0)3@ (y0). Therefore the inclusion holds in (3.17). The converse inclusion∈ is true, byk·k Proposition 3.23, even ⊂ without requiring that x0 is not a local minimizer of f.

4. A new class of functions The lower subdifferential and the subdifferentials of normal type are both important ν ~ but have distinct features. For instance, we observed that @ f(x0) and @ f(x0) always < contain 0 and @∗f(x0) is nonempty whenever f is u.s.c. while @ f(x0) is often empty. Moreover, we noticed that the inclusions one gets in the calculus rules sometimes are in opposite directions. Thus it is of interest to find conditions under which these two types of subdifferential are related. We first observe that, taking into account the convention (3.6), the relation

@~f x @

Since the inclusion @ f x @proper convex function : R belongs to ( 0) if 0 is not a f f x X: → C minimizer of and P(dom 0) = −

Proof. Let x∗ @∗f(x ). Since x is not a minimizer of f, one has x∗ = 0 and [f f(x )] ∈ 0 0 6 ≤ 0 is contained in the closure of [f < f(x )], so that x∗ N([f f(x )]; x ) 0 . By [8] 0 ∈ ≤ 0 0 \{ } Prop. 5.4 it follows that x∗ [0; [@f(x0). Let 0 be such x∗ @f(x0) with the ∈ ∞ ≥x ∈ f x X convention of [8] reproduced in relation (3.6). Since ∗ = 0 and P(dom 0) = , so 6 − that N(dom f; x0) = 0 , as observed in [8] Remark 5.1, we cannot have = 0. Thus { } < x∗ ]0; [@f(x ) =]0; [([1; [@f(x )) =]0; [@ f(x ) by [5] Prop. 10. ∈ ∞ 0 ∞ ∞ 0 ∞ 0 Example 4.2. The convex function with domain [ 1; 1] given by f(x) = √1 x2 does x x − f x X − − not belong to ( 0) for 0 = 1. Here the assumption P(dom 0) = is not satisﬁed. We also note thatC the assumption @

Although not all quasiconvex functions belong to the class (x0); as the preceding example (or the example x x3) shows, the stability properties describedC below enrich this class. 7→ Proposition 4.4. Given locally convex spaces X;Y , x0 X, A L(X;Y ) and g (y0), t ∈ ∈ ∈ C where y0 := Ax0, let f := g A. If A is onto, if R(A ) is w∗-closed (in particular if X and Y are Banach spaces) and◦ if y is not a minimizer of g, then f (x ). 0 ∈ C 0 t Proof. By Proposition 2.6 we have @∗f(x0) = A (@∗g(y0)) and by Proposition 3.2 we @

Proof. Let z∗ := (x∗; y∗) @∗M(z0). Using Proposition 2.4 we may suppose x∗ @~f x @

and the incident (or adjacent, or upper epi-) derivative given by f x tu f x f i x; v ( + ) ( ); v X; ( ) := sup lim sup inf t − U (v) t 0 u U ∈ ∈N & ∈ where (v) is the family of neighborhoods of v. These derivatives are such that their N i epigraphs are the contingent cone T (Ef ; xf ) and the incident cone T (Ef ; xf ) to the epigraph Ef of f at xf := (x; f(x)) given respectively by

1 i 1 T (Ef ; xf ) := lim sup t− (Ef xf );T (Ef ; xf ) := lim inf t− (Ef xf ): t 0 − t 0 − & & A pleasant situation occurs when these two derivatives coincide; then f is said to be epi- differentiable at x. Such a condition is less stringent than requiring that f 0(x; ) coincides with the upper derivative · f x tu f x f ] x; v ( + ) ( ) ( ) := lim sup t − (t,u) (0 ,v) → + i ] since f 0(x; ) f (x; ) f (x; ): When f is directionally steady at x in the sense that for · ≤ · ≤ · each v X 0 one has ∈ \{ } f(x + tu) f(x + tv) lim − = 0; (t,u) (0 ,v) t → + then one has f i(x; v) = f ](x; v) v X 0 ∀ ∈ \{ } i and f 0(x; ) and f (x; ) coincide with their radial counterparts on X 0 . Observe that f is directionally· steady· at x whenever f is directionally Hadamard\{ differentiable} at x; ] i.e. when f 0(x; ) = f (x; ), or when X is a normed space and f is Lipschitzian on a neighborhood of· x or, more· generally, when f is directionally Lipschitzian at x in the sense: for any v X 0 there exist a neighborhood V of v and " > 0, c > 0 such that ∈ \{ } f(x + tv0) f(x + tv00) ct v0 v00 for any v0; v00 V , t [0;"]: | − | ≤ k − k ∈ ∈ In the sequel we say that f is calm at x if f 0(x; 0) = 0; in fact it would be enough to i i suppose f is incidently calm in the sense f (x; 0) = 0 (note that f (x; 0) and f 0(x; 0) are either 0 or ). −∞ For f finite at x we set

< i < i @≺f(x) := @ f 0(x; )(0);@≺ f(x) := @ f (x; )(0): · · If the first (resp. second) set is nonempty, f is calm (resp. incidently calm) at x. The sub- i differential @≺ f(x) is more important than @≺f(x) because it coincides with the Fenchel subdifferential at 0 of the function p associated to p := f i(x; ) via a variant of the < · Crouzeix’s decomposition of p described in [5]. Here p< is defined by p<(v) := p(v) 1 i whenever v cl Dp 0 , + else, where Dp := p− (] ; 0[). Note that @≺ f(x) is nonempty whenever∈ ∪f { is} quasiconvex∞ and p does not take− ∞ the value (or p(0) = 0, −∞ i.e. f is (incidently) calm at x0) because p< is l.s.c. and sublinear (see [5]). Let us start with calculus rules related to order. We first observe that both subdifferentials are homotone, so that easy consequences can be derived. 264 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdifferential calculus

Proposition 5.1. Let f and g be quasiconvex functions ﬁnite and calm at x0, with f(x ) = g(x ), and let h := f g. Suppose hi(x ; u) < 0 for some u X and g is 0 0 ∨ 0 ∈ directionally steady at x0. Then

i i i @≺ h(x ) = co @≺ f(x ) @≺ g(x ) : (5.1) 0 0 ∪ 0 ¡ The proof relies on the following lemma.

Lemma 5.2. Suppose h := f g where f and g are finite and calm at x0, with f(x0) = ∨ i i i g(x0) and g is directionally steady at x0. Then h (x0; v) = f (x0; v) g (x0; v) for each v X: ∨ ∈ i i i i Proof. The inequalities h (x0; v) f (x0; v), h (x0; v) g (x0; v) are obvious, so that hi(x ; v) f i(x ; v) gi(x ; v). As≥hi(x ; 0) 0, equality≥ holds for v = 0 when f and g 0 ≥ 0 ∨ 0 0 ≤ are finite and calm at x0. On the other hand, it is not difficult to prove that

hi(x ; v) f i(x ; v) g](x ; v): 0 ≤ 0 ∨ 0 Thus, when g](x ; v) = gi(x ; v) for v X 0 , equality holds. 0 0 ∈ \{ } i i Proof of Proposition 5.1. Let us ﬁrst observe that, setting p := f (x0; ), q := g (x0; ), i · · r := h (x0; ); so that r = p q by the preceding lemma, one has cl Dr = cl Dp cl Dq where 1 · ∨ ∩ Dp := p− (] ; 0[), with a similar notation with q and r. In fact, given u Dr = Dp Dq, for any v cl−∞D cl D and any t ]0; 1[ one has p((1 t)v+tu) < 0 as p ∈ is sublinear;∩ ∈ p ∩ q ∈ − |cl Dp similarly, q((1 t)v+tu) < 0, so that (1 t)v+tu Dr and v = limt 0((1 t)v+tu) cl Dr. & Extending p (resp.− q; r) by + outside− of cl D ∈ 0 (resp. cl D 0 ,− resp. cl D∈ 0 ) ∞ p ∪ { } q ∪ { } r ∪ { } into a sublinear function p< (resp. q<, r< ) we get

r (v) = p (v) q (v) for each v X: < < ∨ < ∈ Therefore, by a well-known rule of convex subdiﬀerential calculus,

i i i @≺ h(x ) = @r (0) = co (@p (0) @q (0)) = co @≺ f(x ) @≺ g(x ) : 0 < < ∪ < 0 ∪ 0 ¡

The case of a supremum of two functions of independent variables does not require the assumptions of the general case. Proposition 5.3. Let f : X R , g : Y R and M : X Y R , M x; y f x g y → ∪{∞}x ; y X →Y ∪{∞} M x ; y × : → f∪{∞}x < ( ) = ( ) ( ). Consider ( 0 0) such that ( 0 0) R If ( 0) ∨ ∈ × i ∈ i g(y0), if f is u.s.c. at x0 and if g is l.s.c. and calm at y0 then @≺ M(x0; y0) = 0 @≺ g(y0) i i i { }× when 0 = @≺ g(y ) and @≺ M(x ; y ) = X∗ Y ∗ when 0 @≺ g(y ). If f(x ) = g(y ) then ∈ 0 0 0 × ∈ 0 0 0

i i i @≺ M(x ; y ) @≺ f(x ) (1 )@≺ g(y ) : (5.2) 0 0 ⊃ 0 × − 0 λ [0,1] ∈[ ¡

Equality holds in the preceding relation if f and g are quasiconvex and calm at x0 and y0 i i i i respectively and if either 0 = @≺ f(x ), 0 = @≺ g(y ) or 0 @≺ f(x ), 0 @≺ g(y ). ∈ 0 ∈ 0 ∈ 0 ∈ 0 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus 265

Proof. Under the assumptions of the ﬁrst assertion we have M(u; v) = g(v) for (u; v) in i i a neighborhood of (x0; y0). Thus M (x0; y0; x; y) = g (y0; y) for any x X; y Y . The assertion then follows from Corollary 3.4. ∈ ∈ i i Now let us suppose f(x0) = g(y0). We ﬁrst observe that M (x0; y0; x; y) = f (x0; x) i ∨ g (y0; y) for any x X; y Y . In fact, for any t > 0 and any neighborhoods U, V of x and y respectively∈ we have∈

inf rt(u; v) = inf pt(u) qt(v) = inf pt(u) inf qt(v) ; (u,v) U V (u,v) U V ∨ u U ∨ v V ∈ × ∈ × ² ∈ ³ ² ∈ ³ with 1 r (u; v) := t− (M(x + tu; y + tv) M(x ; y )) ; t 0 0 − 0 0 1 1 pt(u) := t− (f(x0 + tu) f(x0)) and qt(v) := t− (g(y0 + tv) g(y0)), hence, by the con- tinuity of the operation− (r; s) r s, we have − 7→ ∨

lim sup inf rt(u; v) = lim sup inf pt(u) sup inf qt(v) ε 0+

= lim sup inf pt(u) lim sup inf qt(v) ε 0+ u U ∨ ε 0+ v V ² → 0

@r (0; 0) = co ((@p (0) 0 ) ( 0 @q (0))) : < < × { } ∪ { } × < i i i i Since 0 @≺ f(x0) (resp. 0 @≺ g(y0)) means that f (x0; ) 0 (resp. g (y0; ) 0), both ∈ ∈ i · ≥ i · ≥ sides of relation (5.2) are X∗ Y ∗ when (0; 0) @≺ f(x ) @≺ g(y ). × ∈ 0 × 0 Let us turn to performance functions: Proposition 5.4. Let p be the performance function associated with a perturbation F : W X p w F w; x p w F w ; x : w R by ( ) = infx X ( ). Suppose that ( 0) = ( 0 0) R If ∗ i× → i ∈ i ∈ ∈ @≺ p(w0) then (w∗; 0) @≺ F (w0; x0). Moreover, when p (w0;:) = q(:), where q(w) := i ∈ i i infx X F (w0; x0; w; x), one has w∗ @≺ p(w0) if (w∗; 0) @≺ F (w0; x0): ∈ ∈ ∈ The proof depends on the following lemma of independent interest which also gives a suﬃcient condition ensuring the assumption of the last assertion. This condition is a variant of [4] Proposition 3.5. Lemma 5.5. Let F , p, w , x be as above. Then for each w W one has 0 0 ∈ i i p (w0; w) q(w) := inf F (w0; x0; w; x): x X ≤ ∈ 266 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus

If W and X are normed spaces, if F is continuous, diﬀerentiable with respect to its ﬁrst variable with a partial derivative DW F jointly continuous in (w; x) and if for any sequences (un) w0, ("n) 0+ there exists some (xn) x0 such that F (un; xn) p(un) + "n then pi(w →; ) = q: → → ≤ 0 ·

This last condition is satisﬁed if x0 is the unique minimizer of F (w0; ) and if there exist W w r > p w · x X w a neighborhood 0 of 0 and a real number supw W ( ) such that ∈ 0 { ∈ | ∃ ∈ W0;F (w; x) < r is relatively compact. When X is a ﬁnite dimensional space, this last condition amounts} to a coercivity condition.

w W x X p w F w ; x Proof. Let 0 and 0 such that ( 0) = ( 0 0) R. For any neighborhoods U and V of w ∈W and x ∈X, respectively, we have for any∈t > 0 that ∈ ∈

F (w + tw0; x + tx0) F (w ; x ) p(w + tw0) p(w ) inf 0 0 − 0 0 inf 0 − 0 ; w U, x V t w U t 0∈ 0∈ ≥ 0∈ whence F i(w ; x ; w; x) pi(w ; w). Hence q pi(w ; ): 0 0 ≥ 0 ≥ 0 · i In order to prove the last assertion, let w W and let r > p (w0; w): For any sequence (t ) 0 there exists a sequence (w ) w∈ such that n → + n → 1 r > lim sup (p(w0 + tnwn) p(w0)): n tn −

Taking un := w0 + tnwn; ﬁn := p(w0) + tnr p(w0 + tnwn) > 0 for n large enough, n − "n := min(ﬁn; 2− ) and picking (xn) x0 such that F (un; xn) p(un) + "n, using the mean value theorem we get → ≤

F (w0 + tnwn; xn) F (w0; xn) DW F (w0; x0)w lim sup − ≤ n tn p(w + t w ) + ﬁ F (w ; x ) lim sup 0 n n n − 0 n ≤ n tn p(w ) + t r F (w ; x ) lim sup 0 n − 0 0 = r; ≤ n tn

i so that DW F (w0; x0)w p (w0; w). On the other hand, for any sequence (tn) 0+ one has ≤ → F w t w; x F w ; x i ( 0 + n 0) ( 0 0) F (w0; x0; w; 0) lim sup − = DW F (w0; x0)w: ≤ n tn Therefore, pi(w ; w) F i(w ; x ; w; 0) D F (w ; x )w pi(w ; w) and equality holds. 0 ≤ 0 0 ≤ W 0 0 ≤ 0

i i i Proof of Proposition 5.4. As p (w0; ) q and p (w0; 0) = F (w0; x0; 0; 0) when i · ≤ i @≺ p(w0) is non empty, applying Lemma 3.1 we get that for any w∗ @≺ p(w0) = < i < < i i ∈ @ p (w0; )(0) @ q(0) we have (w∗; 0) @ F (w0; x0; ; )(0; 0) = @≺ F (w0; x0). More- · i ⊂ ∈ · <· i over, when p (w0; ) = q, any w∗ such that (w∗; 0) @ F (w0; x0; ; )(0; 0) belongs to i < i · i ∈ · · @≺ q(0) = @ p (w ; )(0) = @≺ p(w ): 0 · 0 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus 267

Now let us consider the case of the composition with a nondecreasing function. More refined assumptions could be used, but we prefer to give a simple statement. Proposition 5.6. Let g : X R and let ’ : R R be a nondecreasing → ∪{∞} → ∪{∞} function. Set ’( ) = and consider x0 X such that ’ is finite and differentiable at ∞ ∞ i ∈ t = g(x ) and g0(x ; ) and g (x ; ) are finite. 0 0 0 · 0 · i Then, if ’0(t ) = 0 one has @≺(’ g)(x ) = @≺ (’ g)(x ) = X∗ while if ’0(t ) = 0 0 ◦ 0 ◦ 0 0 6 i i @≺(’ g)(x ) = ’0(t ) @≺g(x );@≺ (’ g)(x ) = ’0(t ) @≺ g(x ): ◦ 0 0 · 0 ◦ 0 0 · 0

Proof. Let us set f := ’ g. When ’0(t0) = 0, one easily sees that f 0(x0; ) = 0 and i ◦ i · f (x0; ) = 0, so that 0 is a minimizer of f 0(x0; ) and f (x0; ) hence @≺f(x0) = X∗, i · · · @≺ f(x0) = X∗. Suppose now that ’0(t0) = 0, so that ’0(t0) > 0: Then f 0(x0; ) = i i 6 · ’0(t0) g0(x0; ) and f (x0; ) = ’0(t0) g (x0; ). The result follows from the relation @<(rh)(0)· = r@· · 0, h arbitrary.· ·

Let us turn to composition with a continuous linear map. g Y A L X;Y x X g y Proposition 5.7. Let : R, ( ) and 0 be such that ( 0) R for y := Ax . Then, for f := g →A; ∈ ∈ ∈ 0 0 ◦ t i t i @≺f(x ) A (@≺g(y )) ;@≺ f(x ) A @≺ g(y ) : 0 ⊃ 0 0 ⊃ 0 i If A is onto, if X and Y are Banach spaces and if inf g0(y0; ) < 0, respectively¡ inf g (y0; ) < 0, then · · t i t i @≺f(x0) = A (@≺g(y0)) ;@≺ f(x0) = A @≺ g(y0) :

¡ 1 Proof. As the epigraph Ef of f and the epigraph Eg of g are related by Ef = B− (Eg), B L X ;Y B x; r Ax; r T E ; x where ( R R) is given by ( ) = ( ), we have ( f f ) 1 ∈ × × ⊂ B− (T (Eg; yg)) for xf := (x0; f(x0)), yg := (y0; g(y0)). It follows that

f 0(x ; u) g0(y ; Au) u X; 0 ≥ 0 ∀ ∈ and similarly

f i(x ; u) gi(y ; Au) u X: 0 ≥ 0 ∀ ∈ Therefore f i(x ; ) > when gi(x ; ) > and 0 · −∞ 0 · −∞ < t < t @≺f(x ) = @ f 0(x ; )(0) A (@ g0(y )(0)) = A (@≺g(y )) ; (5.3) 0 0 · ⊃ 0,· 0 and similarly

i < i t < i t i @≺ f(x ) = @ f (x ; )(0) A @ g (y )(0) = A @≺ g(y ) : 0 0 · ⊃ 0,· 0

¡ ¡ 1 When A is onto and X and Y are Banach spaces one has T (Ef ; xf ) = B− (T (Eg; yg)) and f 0(x0; u) = g0(y0; Au) for each u X (see [3] for example). If 0 is not a minimizer of ∈ i g0(y0; ) we can apply Proposition 3.2 and we get equality in (5.3). The case of @≺ f(x0) is similar.·

Let us ﬁnish with the sublevel-convolution. 268 J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdiﬀerential calculus f; g X h f g x f y g Proposition 5.8. Let : R , = 3 and 0 dom , 0 dom , w = x + y with h(w ) = f(x ) →g(y ∪): {∞} ∈ ∈ 0 0 0 0 0 ∨ 0 (i) If g(y0) < f(x0) and g is u.s.c. at y0 then w0 is a local maximizer of h and x0 is a i local minimizer of f, so that @≺ f(x0) = X∗: (ii) If f(x0) = g(y0), if f and g are quasiconvex and calm at x0 and y0 respectively and if i i i i i (0; 0) @≺ f(x0) @≺ g(y0) or if 0 = @≺ f(x0), 0 = @≺ g(y0) then @≺ (f3g)(w0) i ∈ i × ∈ ∈ ⊂ @≺ f(x0)3@≺ g(y0):

Proof. (i) Let V be a neighborhood of 0 such that g(y + v) < f(x ) for each v V . 0 0 ∈ Then, for v V we have h(w0 + v) f(x0) g(y0 + v) = f(x0) = h(w0). It follows that i ∈ ≤ ∨ h (w0; ) 0. Moreover, for v V we have f(x0 v) f(x0) as otherwise we would · ≤ ∈ − ≥ i have f(x0 v) g(y0 + v) < f(x0) = h(w0), a contradiction. Thus f (x0; ) 0 and i − ∨ · ≥ @≺ f(x0) = X∗ (ii) Consider H : X X R , given by H(w; x) = f(x) g(w x). We observe again that h is the performance× → ∪ function {∞} associated with H and H∨(w; x)− = M(x; w x) = M A(w; x), where A(w; x) = (x; w x) and M(x; y) = f(x) g(y). As A is an isomorphism− ◦ t − ∨ and A (u∗; v∗) = (v∗; u∗ v∗) we have − i i (w∗; x∗) @≺ H(w ; x ) (w∗; w∗ + x∗) @≺ M(x ; w x ): ∈ 0 0 ⇔ ∈ 0 0 − 0 In particular, i i (w∗; 0) @≺ H(w ; x ) (w∗; w∗) @≺ M(x ; y ): ∈ 0 0 ⇔ ∈ 0 0 i i i Let w∗ @≺ h(w0). Then (w∗; 0) @≺ H(w0; x0) hence (w∗; w∗) @≺ M(x0; y0). Taking ∈ ∈ i ∈ i Proposition 5.3 into account we can find [0; 1], x∗ @≺ f(x0), y∗ @≺ g(y0) such that i ∈ i ∈ ∈ w∗ = (1 )x∗ = y∗. Thus w∗ @≺ f(x )3@≺ g(y ). − ∈ 0 0 References [1] D. Azé,M. Volle: A stability result in quasi-convex programming, J. Optim. Th. Appl. 67(1) (1990), 175–184. [2] J. M. Gutiérrez: Infragradientes y direcciones de decrecimiento, Rev. Real Acad. C. Ex., Fis. y Nat. Madrid 78 (1984), 523–532. [3] J.-P. Penot: On regularity conditions in mathematical programming, Math. Progr. Study 19 (1982), 167–199. [4] J.-P. Penot: Central and peripheral results in the study of marginal and performance functions, in: Mathematical Programming with Data Perturbations, A.V. Fiacco, ed. Lecture Notes in Pure and Applied Maths 195, Dekker, New York (1998) 305–337. [5] J.-P. Penot: Are generalized derivatives useful for generalized convex functions?, in: Gener- alized Convexity and Generalized Monotonicity, J.-P. Crouzeix et al. eds., Kluwer, Dordrecht (1998) 3–59. [6] J.-P. Penot: What is quasiconvex analysis?, Optimization (2000). [7] J.-P. Penot, M. Volle: Inversion of real-valued functions and applications, Math. Methods Oper. Res. 34 (1990) 117–141. [8] J.-P. Penot, C. Z˘alinescu:Harmonic sum and duality, J. Convex Analysis 7 (2000) 95–113. [9] J.-P. Penot, C. Z˘alinescu:Regularization of quasiconvex functions, in preparation. J.-P. Penot, C. Z˘alinescu/ Elements of quasiconvex subdifferential calculus 269

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