MOLLIFIED DERIVATIVES AND SECOND-ORDER OPTIMALITY CONDITIONS

GIOVANNI P. CRESPI – DAVIDE LA TORRE – MATTEO ROCCA

Working Paper n.10.2003 – giugno

Dipartimento di Economia Politica e Aziendale Università degli Studi di Milano via Conservatorio, 7 20122 Milano tel. ++39/02/50321501 fax ++39/02/50321450

E Mail: [email protected]

In pubblicazione sul Journal of Nonlinear and Convex Analysis Mollified derivatives and second-order optimality conditions

Giovanni P. Crespi∗ Davide La Torre† Matteo Rocca‡

Abstract

The class of strongly semicontinuous functions is considered. For these func- tions the notion of mollified derivatives, introduced by Ermoliev, Norkin and Wets [8], is extended to the second order. By means of a generalized Tay- lor’s formula, second order necessary and sufficient conditions are proved for both unconstrained and constrained optimization. Finally a characterization of convex functions is given.

Keywords: Smooth approximations, Nonsmooth optimization, Strong semicontinuity .

1 Introduction

In this paper we extend to the second-order the approach introduced by Ermoliev, Norkin and Wets [8] to define generalized derivatives even for discontinuous func- tions, which often arise in applications (see [8] for references). A similar technique has been previously used by Craven in [7], but for the only class of locally Lipschitz functions. To deal with such problems a number of approaches have been proposed to develop a subdifferential calculus for nonsmooth and even discontinuous func- tions. Among the many possibilities, let us remember the notions due to Aubin [2],

∗Universit`aBocconi, I.M.Q., v.le Isonzo 25, 20137 Milano, Italia. e–mail: giovanni.crespi@uni- bocconi.it †Universit`adi Milano, Dipartimento di Economia Politica e Aziendale, via Conservatorio 7, 20122 Milano, Italia. e–mail: [email protected] ‡Universit`adell’Insubria, Dipartimento di Economia, via Ravasi 2, 21100 Varese, Italia. e–mail: [email protected]

1 Clarke [5], Ioffe [14], Michel and Penot [21], Rockafellar [23], in the context of Varia- tional Analysis. The previous approaches are based on the introduction of first-order generalized derivatives. Extensions to higher-order derivatives have been provided for instance by Hiriart-Hurruty, Strodiot and Hien Nguyen [13], Jeyakumar and Luc [15], Klatte and Tammer [16], Michel and Penot [20], Yang and Jeyakumar [31] , Yang [32]. Most of these higher-order approaches assume that the functions involved are of class C1,1, that is once differentiable with a locally Lipschitz gradient, or at least of class C1. Anyway, another possibility, concerning the differentiation of non- smooth functions dates back to the 30’s and is related to the names of Sobolev [27], who introduced the concept of “weak derivative” and later of Schwartz [26] who gen- eralized Sobolev’s approach with the “theory of distributions”. These tecniques are widely used in the theory of partial differential equations, in Mathematical Physics and in related problems, but they have not been applied to deal with optimization problems involving nonsmooth functions, until the work of Ermoliev, Norkin and Wets. The tools which allow to link the “modern” and the “ancient” approaches to Non- smooth Analysis are those of “mollifier” and of “mollified functions”. More specifi- cally, the approach followed by Ermoliev, Norkin and Wets appeals to some of the n results of the theory of distributions. They associate with a point x ∈ R a family of mollifiers (density functions) whose support tends toward x and converges to the

Dirac function. Given such a family, say {ψε, ε > 0}, one can define a family of n mollified functions associated to a function f : R → R as the convolution of f and

ψε (mollified functions will be denoted by fε). Hence a mollified function can be viewed as an averaged function. The mollified functions possess the same regularity 2 of the mollifiers ψε and hence, if they are at least of class C , one can define first and second-order generalized derivatives as the cluster points of all possible values of first and second-order derivatives of fε. For more details one can see [8]. We remember also that an approach based on similar techniques has beeen used to solve nonsmooth equations, with the introduction of smoothing functions and smoothing Newton methods [22].

2 In this paper, section 2 recalls the notions of mollifier, epi-convergence of a sequence of functions and some definitions introduced in [8]. Section 3 is devoted to the introduction of second-order derivatives by means of mollified functions; sections 4 and 5 deal, respectively, with second-order necessary and sufficient optimality conditions for unconstrained and constrained problems; finally, section 6 is devoted to a second–order characterization of convex functions.

2 Preliminaries

To follow the approach presented in [8] we first need to introduce the notion of mollifier (see e.g. [4]):

m Definition 1 A sequence of mollifiers is any sequence of functions {ψε : R → R+}, ε ↓ 0, such that:

n i) supp ψε := {x ∈ R | ψε(x) > 0} ⊆ ρεB, ρε ↓ 0, Z ii) ψε(x)dx = 1, n R m where B is the closed unit ball in R .

Although in the sequel we may consider general families of mollifiers, some ex- amples may be useful.

Example 1 Let ε be a positive number.

i) The functions:  1 ε  εm , max1,...,m |xi| ≤ 2 ψε(x) =  0, otherwise are called Steklov mollifiers.

ii) The functions:

 C  ε2   εm exp kxk2−ε2 , if kxk < ε ψε(x) =  0, if kxk ≥ ε R with C ∈ such that m ψε(x)dx = 1, are called standard mollifiers. R R

3 It is easy to check that the second family of functions is of class C∞.

m Definition 2 ([4]) Given a locally integrable function f : R → R and a sequence of bounded mollifiers, define the functions fε(x) through the convolution: Z fε(x) := f(x − z)ψε(z)dz. m R

The sequence fε(x) is said a sequence of mollified functions.

In the following all the functions considered will be assumed to be at least locally integrable.

m Remark 1 There is no loss of generality in considering f : R → R. The results m in this paper remain true also if f is defined on an open subset of R .

Some properties of the mollified functions can be considered classical.

m Theorem 1 ([4]) Let f ∈ C (R ). Then fε converges continuously to f, i.e. fε(xε) → f(x) for all xε → x. In fact fε converges uniformly to f on every compact subset of m R as ε ↓ 0.

The previous convergence property can be generalized.

m Definition 3 ([1], [25]) A sequence of functions {fn : R → R} epi–converges to m f : R → R at x, if:

i) lim infn→+∞ fn(xn) ≥ f(x) for all xn → x;

ii) limn→+∞ fn(xn) = f(x) for some sequence xn → x.

m The sequence {fn} epi–converges to f if this holds for all x ∈ R , in which case we write f = e − lim fn.

Remark 2 It can be easily checked that when f is the epi–limit of some sequence fn then f is lower semicontinuous. Moreover if fn converges continuously, then it also epi–converges.

4 m Definition 4 ([8]) A function f : R → R is said strongly lower semicontinuous

(s.l.s.c.) at x if it is lower semicontinuous at x and there exists a sequence xn → x with f continuous at xn (for all n) such that f(xn) → f(x). The function f is strongly lower semicontinuous if this holds at all x. The function f is said strongly upper semicontinuous (s.u.s.c.) at x if it is upper semicontinuous at x and there exists a sequence xn → x with f continuous at xn

(for all n) such that f(xn) → f(x). The function f is strongly lower semicontinuous if this holds at all x.

m Proposition 1 If f : R → R is s.l.s.c., then −f is s.u.s.c. .

Proof: It follows directly from the definitions. 2

m Theorem 2 ([8]) Let εn ↓ 0 as n → +∞. For any s.l.s.c. function f : R → R, and any associated sequence fn of mollified functions we have f = e − lim fn .

Remark 3 It can be seen that, according to Remark 2, Theorem 1 follows from Theorem 2.

m Proposition 2 Let εn ↓ 0 as n → +∞. For any s.u.s.c. function f : R → R and m any associated sequence fn of mollified functions, we have for any x ∈ R :

i) lim supn→+∞ fn (xn) ≤ f(x) for any sequence xn → x;

ii) limn→+∞ fn (xn) = f(x) for some sequence xn → x.

Proof: It is immediate from definition 4 and Proposition 1 2

The following Proposition plays a crucial role in the sequel.

k Proposition 3 ([26, 27]) Whenever the mollifiers ψε are of class C , so are the associated mollified functions fε.

By means of mollified functions it is possible to define generalized directional derivatives for a nonsmooth function f, which, under suitable regularity of f, coin- cide with Clarke’s generalized derivative. Such an approach has been deepened by several authors (see e.g. [7, 8]) in the first–order case.

5 m Definition 5 ([8]) Let f : R → R, n ↓ 0 as n → +∞ and consider the sequence 1 {fεn } of mollified functions with associated mollifiers ψεn ∈ C . The upper mollified m derivative of f at x in the direction d ∈ R , with respect to (w.r.t.) the mollifiers sequence ψn is defined as:

> D ψf(x; d) := sup lim sup ∇fεn (xn) d, xn→x n→+∞ where the supremum is taken over all possible sequences xn tending to x.

Similarly, we might introduce the following:

m Definition 6 Let f : R → R, n ↓ 0 as n → +∞ and consider the sequence 1 {fεn } of mollified functions with associated mollifiers ψεn ∈ C . The lower mollified m derivative of f at x in the direction d ∈ R , w.r.t. the mollifiers sequence ψn is defined as: > D ψf(x; d) := inf lim inf ∇fεn (xn) d, xn→x n→+∞ where the infimum is taken over all possible sequences xn tending to x.

m Let us recall that, for any sequence an ∈ R , Limsupn→∞an denotes the set of all cluster points of an. Under the setting of Definition 5, in [8] it has been defined also a generalized gradient w.r.t. the mollifiers sequence ψn , in the following way:

∂ψf(x) := {Limsupn→∞∇fεn (xn), xn → x}

i.e. the set of cluster points of all possible sequences {∇fn (xn)} such that xn → x. Clearly (see e.g. [8]) for the above mentioned upper mollified derivative it holds:

> D ψf(x; d) ≥ sup L d, L∈∂ψf(x) > D ψf(x; d) ≤ inf L d. L∈∂ψf(x)

This generalized gradient has been used in [8] to prove first–order necessary optimal- ity conditions for nonsmooth optimization. The equivalence with the well–known no- tions of Nonsmooth Analysis is contained in the following Proposition. We recall that 0 f(x0+td)−f(x0) fC (x; d) = lim supx0→x, t↓0 t , denotes Clarke’s generalized derivative of f

6 m > 0 m at the point x in the direction d, while ∂C f(x) = {v ∈ R |v d ≤ fC (x; d), ∀d ∈ R } denotes Clarke’s generalized gradient.

m Proposition 4 ([8]) i) Let f : R → R be locally integrable. Then for every 1 0 choice of sequences εn ↓ 0 and ψεn ∈ C , we have D ψf(x; d) ≤ fC (x; d). If f 0 is also continuous, then D ψf(x; d) = fC (x; d).

ii) If f is lower semicontinuous and locally integrable, then conv ∂ψf(x) ⊆ ∂C f(x) m (here conv A stands for the convex hull of the set A ⊆ R ). If, in addition, f

is locally Lipschitz, then conv ∂ψf(x) = ∂C f(x).

Remark 4 ¿From the previous proposition and the well–known properties of Clarke’s 1 generalized gradient, we deduce that, if f and ψεn ∈ C , then ∂ψf(x) = {∇f(x)} and > D ψf(x; d) = D ψf(x; d) = ∇f(x) d. More generally, it is easy to see that, whenever 1 f(x) = g(x) + h(x), with g of class C and h = 0 a.e., then ∂ψf(x) = {∇g(x)} and > D ψf(x; d) = D ψf(x; d) = ∇g(x) d.

The following example shows that the inequalities and inclusions of Proposition 4 can be strict.

Example 2 We consider the following function f : R → R:  p  − |x|, x = ±1/n, n = 1, 2,...  f(x) =   −x, elsewhere

0 0 It can be easily verified that f is s.l.s.c.. We get fC (0; −1) = fC (0; 1) = +∞, thus

∂C f(0) = R, while D ψf(0; d) = D ψf(0; d) = −d and ∂ψf(0) = −1.

Warga [28, 29, 30] defines subgradients of continuous functions using a construc- tion similar to that of [8]. In this last paper comparison results between mollified derivatives and Warga’s notion are given. For the aim of our paper, we will need to point out the following proposition (contained in [8]) of which we give an alternative proof.

7 m m Proposition 5 Let f : R → R and x ∈ R . Then, for every choice of sequences 1 εn ↓ 0 and ψεn ∈ C we have:

m i) D ψf(·; d) is upper semicontinuous (u.s.c.) at x for all d ∈ R ;

m ii) D ψf(·; d) is lower semicontinuous (l.s.c.) at x for all d ∈ R .

Proof: We can prove only i), since ii) follows with the same reasoning. m Assume d ∈ R is fixed. First we note that the upper semicontinuity is obviuous if

D ψf(x; d) = +∞. Otherwise, for all K > D ψf(x; d), there exists a neighborhood

U of x and an integer n0 so that:

0 > 0 ∇fn (x ) d < K, ∀n > n0, ∀x ∈ U.

Therefore, for each x0 ∈ U, we have:

0 > D ψf(x ; d) = sup lim sup ∇fn (xn) d ≤ K, 0 xn→x n→+∞ which shows that D ψf(·; d) is u.s.c. . 2

Furthermore, we point out the following property:

1 Proposition 6 Whenever the choice of sequences εn ↓ 0 and ψεn ∈ C , D ψf(x; ·) and D ψf(x; ·) are positively homogeneous functions.

Furthermore, if D ψf(x; ·) 6= −∞ (D ψf(x; ·) 6= +∞ respectively), then it is subad- ditive (resp. superadditive) and hence convex (resp. concave) as a function of the direction d.

Proof: The positive homogeneity is trivial. Concerning the second part of the The- m orem, we have, ∀d1, d2 ∈ R :

> D ψf(x; d1 + d2) = sup lim sup ∇f(xn) (d1 + d2) ≤ xn→x n→+∞ > > ≤ sup lim sup ∇f(xn) d1 + sup lim sup ∇f(xn) d2 = xn→x n→+∞ xn→x n→+∞ = D ψf(x; d1) + D ψf(x; d2), and hence D ψf(x; ·) is subadditive. Convexity follows considering positive homogenei- ty and subadditivity. The proof for D ψf(x; ·) is analogous. 2

8 3 Second–order mollified derivatives

As suggested in [8], by requiring some more regularity of the mollifiers, it is possible to construct also second–order generalized derivatives.

m Definition 7 Let f : R → R, n ↓ 0 as n → +∞ and consider the sequence of 2 mollified functions {fεn }, obtained from a family of mollifiers ψεn ∈ C . We define m the second–order upper mollified derivative of f at x in the directions d and v ∈ R , w.r.t. to the mollifiers sequence {ψn }, as:

2 > D ψf(x; d, v) := sup lim sup d Hfεn (xn)v, xn→x n→+∞

2 where Hfεn (x) is the Hessian matrix of the function fεn ∈ C at the point x and the supremum is taken over all possible sequences xn tending to x.

m Definition 8 Let f : R → R, n ↓ 0 and consider the sequence of mollified func- 2 tions {fn }, obtained from a family of mollifiers ψn ∈ C . We define the second– m order lower mollified derivative of f at x in the directions d and v ∈ R , w.r.t. the mollifiers sequence {ψn }, as:

2 > D ψf(x; d, v) := inf lim inf d Hfεn (xn)v, xn→x n→+∞ where the infimum is taken over all possible sequences xn tending to x.

The following proposition summarizes some basic properties of second–order mol- lified derivatives.

m m Proposition 7 Let f : R → R and x ∈ R .

i) If λ > 0, then:

2 2 D ψλf(x; d) = λD ψf(x; d); 2 2 D ψλf(x; d) = λD ψf(x; d).

Moreover, if λ < 0 we get:

2 2 D ψλf(x; d) = λD ψf(x; d).

9 2 2 ii) The maps (d, v) → D ψf(x; d, v) and (d, v) → D ψf(x; d, v) are symmetric (that 2 2 2 2 m is D ψf(x; d, v) = D ψf(x; v, d) and D ψf(x; d, v) = D ψf(x; v, d) ∀d, v ∈ R ).

2 2 iii) The functions D ψf(x; d, ·) and D ψf(x; d, ·) are positively homogeneous, for m any fixed d ∈ R .

m 2 2 iv) Whenever d ∈ R , if D ψf(x; d, ·) 6= −∞ (D ψf(x; d, ·) 6= +∞ resp.), then it is sublinear (superlinear).

2 2 m v) D ψf(x; d, −v) = −D ψf(x; d, v), ∀d, v ∈ R .

m 2 vi) Whenever x ∈ R , D ψf(·; d, v) is upper semicontinuous (u.s.c.) at x for every m d, v ∈ R .

m 2 vii) Whenever x ∈ R , D ψf(·; d, v) is lower semicontinuous (l.s.c.) at x for every m d, v ∈ R .

2 2 viii) If f(x) = g(x) + h(x), with g of class C and h = 0 a.e., then D ψf(x; d, v) = 2 > D ψf(x; d, v) = d Hg(x)v.

Proof: i), ii), iii) and viii) are obvious from the definitions. The proof of iv) is similar to that of Proposition 6. To prove v), observe that we have:

2 > D ψf(x; d, −v) = sup lim sup −d Hfn (xn)v = xn→x n→+∞ > = sup − lim inf d Hfn (xn)v = xn→x n→+∞ > = − inf lim inf d Hfn (xn)v = xn→x n→+∞ 2 = −D ψf(x; d, v).

The proofs of vi) and vii) are analogous to that of Proposition 5. 2

In the following we will set for simplicity:

2 2 D ψf(x; d) := D ψf(x; d, d) and: 2 2 D ψf(x; d) := D ψf(x; d, d).

10 Remark 5 One of the main advantages of considering mollified derivatives is that we need not to go through a first–order approximation to get the second–order derivative. Practically we derive both first and second–order generalized derivatives as the limit of two indipendent well defined sequences of “numbers”.

Using these notions of derivatives, we shall introduce a Taylor’s formula for strongly semicontinuous functions:

m Theorem 3 (Mean value theorem and Taylor’s formula) Let f : R → R be m a s.l.s.c. (resp. s.u.s.c.) function and let n ↓ 0, t > 0, d and x ∈ R .

1 i) If ψn ∈ C is a sequence of mollifiers, there exists a point ξ ∈ [x, x + td] such that: f(x + td) − f(x) ≤ tD ψf(ξ; d)

(f(x + td) − f(x) ≥ tD ψf(ξ; d))

2 ii) If ψn ∈ C is a sequence of mollifiers, there exists ξ ∈ [x, x + td] such that:

t2 2 f(x + td) − f(x) ≤ tD ψf(x; d) + 2 D ψf(ξ; d), t2 2 (f(x + td) − f(x) ≥ tD ψf(x; d) + 2 D ψf(ξ; d))

assuming that the righthand sides are well defined, i.e. it does not happen the expression +∞ − ∞.

Proof: We prove only the second part. The proof of the first part is similar.

For any xn → x, we can easily write Taylor’s formula for each mollified function: t2 f (x + td) − f (x ) = t∇f (x )>d + d>Hf (ξ )d n n n n n n 2 n n where ξn ∈ (xn, xn + td). Without loss of generality, we can think that ξn → ξ ∈

[x, x + td]. Now, we consider the lim sup as n → +∞ and the definition of D ψf(x; d) 2 and D ψf(x; d) to get:

lim sup fn (xn + td) − lim sup fn (xn) ≤ lim sup[fn (xn + td) − fn (xn)] ≤ n→+∞ n→+∞ n→+∞ 2 2 > t > t 2 ≤ lim sup[t∇fn (xn) d + d Hfn (ξn)d] ≤ tD ψf(x; d) + D ψf(ξ; d). n→+∞ 2 2

11 By the strong lower semicontinuity assumed on f, there exists a sequence yn → x such that:

lim f (yn) = f(x). n→+∞ n Thus, recalling Theorem 2, we have, considering in particular this sequence:

lim sup fn (yn + td) − lim sup fn (yn) = lim sup fn (yn + td) − lim fn (yn) ≥ n→+∞ n→+∞ n→+∞ n→+∞ lim inf f (yn + td) − lim f (yn) ≥ f(x + td) − f(x), n→+∞ n n→+∞ n from which the thesis follows. The other formula follows in a similar way, recalling Proposition 2 instead of Theo- rem 2. 2

It should be clear that, for both semicontinuity of the generalized derivatives and Taylor’s formula, we need some conditions to avoid “triviality” of the derivatives, such as local Lipschitziannes of f so that, as already seen, the first–order mollified derivative is finite, since it coincides with Clarke’s derivative.

Remark 6 One has to observe that the previous second–order derivatives may be infinity. Furthermore they are dependent on the specific family of mollifiers which we choose and also on the sequence n. However the results presented above and those in the sequel hold true for any mollifiers sequence (provided the mollifiers are 2 at least of class C ) and for any choice of n.

2 Now we wish to prove that for a suitable class of functions, D ψf(x; d, v) and 2 D ψf(x, v, d) are finite, independent on εn and on the choice of the mollifiers sequence 2 ψεn ∈ C and coincide with the second–order derivative in Clarke’s sense introduced in [6] and [13]. Before proving this result we recall the definition and the theorem that follow.

m 1,1 m Definition 9 A function f : R → R is said to be of class C at x0 ∈ R when it is differentiable in a neighborhood of x0 and its gradient is locally Lipschitz at x0.

m 1,1 Theorem 4 ([18]) Let f : R → R be a bounded function. Then f is of class C m at x0 ∈ R if and only if there exist a neighborhood U of x0, a right neighborhood

12 V of 0 ∈ R and a constant M ≥ 0 such that:

f(x + 2td) − 2f(x + td) + f(x) ≤ M, t2 1 m ∀x ∈ U, t ∈ V and d ∈ S (the unit sphere in R ).

m 1,1 m Theorem 5 i) A continuous function f : R → R is of class C at x0 ∈ R

if and only if there exist a neighborhood U of x0, and a constant M ≥ 0 such 2 that, for every choice of sequences εn ↓ 0 and ψεn ∈ C it holds:

2 2 −M ≤ D ψf(x; d) ≤ D ψf(x; d) ≤ M,

for every x ∈ U and d ∈ S1.

m 1 m ii) if f : R → R is a function of class C in a neighborhood of x0 ∈ R , then: 0 > 0 > 2 ∇f(x + tv) d − ∇f(x ) d D ψf(x; d, v) = lim sup , x0→x,t↓0 t

2 for any choice of the sequence n ↓ 0 and of the mollifiers ψn ∈ C .

Proof:

1,1 i) Necessity. Let f be a function of class C at x0 and let ψε We prove that there > exists a constant M ≥ 0 and a positive number ε0 such that |d Hfε(x)d| ≤ M, m 1 ∀x ∈ R , ∀d ∈ S and ∀ε ∈ (0, ε0). It is easy to see that for every ε > 0 we have:

> fε(x + 2td) − 2fε(x + td) + fε(x) d Hfε(x)d = lim = t↓0 t2 Z f(x − z + 2td) − 2f(x − z + td) + f(x) lim 2 ψε(z)dz. t↓0 m t R Recalling theorem 4, we have that for x and z in suitable neighborhoods U of 0 m 1 x0 and U of 0 ∈ R respectively, d ∈ S and t > 0 ”small enough”, it holds:

f(x − z + 2td) − 2f(x − z + td) + f(x) ≤ M, t2 for some constant M ≥ 0. Hence, remembering Definition 1, it follows the

existence of a number ε0 > 0 such that: Z f(x − z + 2td) − 2f(x − z + td) + f(x) > d Hfε(x)d ≤ lim 2 ψε(z)dz ≤ M, t↓0 m t R

13 0 1 for every x ∈ U, z ∈ U , d ∈ S and ε ∈ (0, ε0). Now the thesis follows recalling 2 2 the definitions of D ψf(x; d) and D ψf(x; d).

Sufficiency. Let U1 be a neighborhood of x0 and V a right neighborhood of 1 0 ∈ R such that U1 ⊂ U and x + 2td ∈ U for every x ∈ U, t ∈ V and d ∈ S .

We can write, for such x, t and d and for every couple of sequences εn ↓ 0 and 2 ψεn ∈ C :

> 2 > fεn (x + 2td) = fεn (x) + 2t∇fεn (x) d + 2t d Hfεn (ξn)d,

where ξn ∈ (x, x + 2td), ∀n and: t2 f (x + td) = f (x) + t∇f (x)>d + d>Hf (ξ0 )d, εn εn εn 2 εn n 0 where ξn ∈ (x, x + td), ∀n. Hence we have: f (x + 2td) − 2f (x + td) + f (x) εn εn εn = 2d>Hf (ξ )d − d>Hf (ξ0 )d. t2 εn n εn n

Sending n to +∞, without loss of generality we can assume that ξn → ξ ∈ 0 0 [x, x+2td] and ξn → ξ ∈ [x, x+td]and recalling Theorem 1 and the definitions 2 2 of D ψf(x; d) and D ψf(x; d) we have:

2 2 0 > > 0 2D ψf(ξ; d) − D ψf(ξ ; d) ≤ lim inf 2d Hfεn (ξn)d − lim sup d Hfεn (ξn)d ≤ n→+∞ n→+∞

f(x + 2td) − 2f(x + td) + f(x) > ≤ 2 ≤ lim sup 2d Hfεn (ξn)d− t n→+∞ > 0 2 2 0 − lim inf d Hfε (ξ )d ≤ 2D f(ξ; d) − D f(ξ ; d). n→+∞ n n ψ ψ 2 2 Hence, since D ψf(x; d) and D ψf(x; d) are bounded by a constant M for x ∈ U and d ∈ S1, we obtain: f(x + 2td) − 2f(x + td) + f(x) −3M ≤ ≤ 3M. t2 Now the proof is complete recalling theorem 4.

1 ii) If f is of class C in a neighborhood of x0, we obtain for i = 1, ··· , m: Z   ∂fε ∂f ∂f (x) = (x − z)ψε(z)dz = (x). ∂x m ∂x ∂x i R i i ε The proof is complete recalling Proposition 4.10 in [8].

2

14 Remark 7 Point ii) of the previous theorem implies that when f is a function of 1,1 2 class C at x0, D ψf(x0; d, v) coincides with the derivative introduced in [6] and [13], > that is Clarke’s generalized derivative of ∇f(·) d at x0 in the direction v. Hence, 2 2 in this case D ψf(x0; d, v) is finite. (This last property holds also for D ψf(x0; d, v) 2 since it coincides with −D ψf(x0; d, −v)).

4 Optimality Conditions

In this section we give second–order necessary and sufficient optimality conditions. We begin considering the following problem:

P1) minx∈K f(x)

m where K ⊆ R .

Definition 10 The radial tangent cone of the set K at x is given by:

m R(K, x) := {d ∈ R | ∀α > 0 : ∃t ∈ (0, α) , x + td ∈ K} =

m {d ∈ R | ∃tn ↓ 0 : x + tnd ∈ K} .

Definition 11 The set:

m T (K, x0) := {d ∈ R | ∃dn → d, ∃tn ↓ 0 : x0 + tndn ∈ K} is called the Bouligand tangent cone of the set K at x.

Clearly we have the following inclusion:

R(K, x) ⊆ T (K, x).

Definition 12 The second–order radial tangent set of K at x in the direction d is:

 t2  R2(K, x, d) = w ∈ m | ∃t ↓ 0 : x + t d + n w ∈ K . R n n 2

Clearly R2(K, x, 0) = R(K, x).

15 m 2 Theorem 6 Let f : R → R be s.l.s.c., let εn ↓ 0 and ψεn ∈ C and assume m 2 that D ψf(x0; d) is finite for every d ∈ R and D ψf(x0; d, w)y is finite for every d m d, w ∈ R . If x0 ∈ K is a local solution of problem P1), then:

i) D ψf(x0; d) ≥ 0, ∀d ∈ R(K, x0),

m 2 ii) If d ∈ R is such that D ψf(x0; d) = 0, then D ψf(x0; w) + D ψf(x0; d) ≥ 0, 2 ∀w ∈ R (K, x0, d).

Proof: i) Let d ∈ R(K, x0). Then there exists a sequence tn ↓ 0 such that x0 + tnd ∈

K. Since x0 is optimal, applying the mean value theorem, we obtain:

0 ≤ f(x0 + tnd) − f(x0) ≤ D ψf(ξn; d), where for every n, ξn ∈ [x0, x0 + tnd]. Hence, from the upper semicontinuity of

D ψf(x0; d), we have:

0 0 ≤ lim sup D ψf(ξn; d) ≤ lim sup D ψf(x ; d) ≤ D ψf(x0; d). 0 n→+∞ x →x0 m 2 ii) Let d ∈ R be such that D ψf(x0; d) = 0 and let w ∈ R (K, x0, d). Hence there 2 tn exists a sequence tn ↓ 0 such that x0 + tnd + 2 w ∈ K and we have, for n ”large enough”: t2 t t2 t 0 ≤ f(x + t d + n w) − f(x ) ≤ t D f(x ; d + n w) + n D 2 f(ξ ; d + n w), 0 n 2 0 n ψ 0 2 2 ψ n 2 2 tn where ξn ∈ [x0, x0 + tnd + 2 w]. Recalling the sublinearity of D ψf(x0; ·) and of 2 2 2 D ψf(x0; d, ·), the simmetry of D ψf(x0, d, v) and that by definition D ψf(x0; d) = 2 D ψf(x0; d, d), we have: t2 t2 0 ≤ f(x + t d + n w) − f(x ) ≤ t D f(x ; d) + n D f(x ; w)+ 0 n 2 0 n ψ 0 2 ψ 0 t2 t4 t3 + n D 2 f(ξ ; d) + n D 2 f(ξ ; w) + n D 2 f(ξ ; d, w) = 2 ψ n 8 ψ n 2 ψ n t2 t2 t4 t3 n D f(x ; w) + n D 2 f(ξ ; d) + n D 2 f(ξ ; w) + n D 2 f(ξ ; d, w). 2 ψ 0 2 ψ n 8 ψ n 2 ψ n Now observe that from the finiteness assumption on the derivatives and their upper semicontinuity it follows that, for every ε > 0, there exists a positive integer n0 such that, for every n > n0 it holds:

2 2 D ψf(ξn; d) ≤ D ψf(x0; d) + ε;

16 2 2 D ψf(ξn; w) ≤ D ψf(x0; w) + ε;

2 2 D ψf(ξn; d, w) ≤ D ψf(x0; d, w) + ε.

Hence for n ” large enough” it holds:

2 tn f(x0 + tnd + 2 w) − f(x0) 0 ≤ 2 ≤ tn 2 t2 ≤ D f(x ; w) + D 2 f(x ; d) + ε + n [D 2 f(x ; w) + ε] + t [D 2 f(x ; d, w) + ε]. ψ 0 ψ 0 4 ψ 0 n ψ 0 Sending n to +∞ we obtain, since ε is arbitrary:

2 D ψf(x0; w) + D ψf(x0; d) ≥ 0 and the theorem is proved. 2

Corollary 1 Under the assumptions of Theorem 6, a necessary condition for x0 to be a local solution of Problem P1) is that:

i) D ψf(x0; d) ≥ 0, ∀d ∈ R(K, x0);

2 ii) D ψf(x0; d) ≥ 0, ∀d ∈ R(K, x0) suchthat D ψf(x0; d) = 0.

Proof: It follows from the previous theorem, observing that if d ∈ R(K, x0), then 2 0 ∈ R (K, x0, d). 2

Remark 8 In Example 2, the point x = 0 is not a minimizer and does not satisfy condition i) of the previuos Theorem. On the contrary x = 0 fulfils necessary 0 optimality conditions expressed through Clarke generalize derivative (fC f(0; d) ≥ 0, ∀d).

m Remark 9 When K is an open subset of R , then from corollary 1 one obtains that the following conditions are necessary for x0 to be an unconstrained local minimizer of f:

m i) D ψf(x0; d) ≥ 0, ∀d ∈ R ;

17 2 m ii) D ψf(x0; d) ≥ 0, ∀d ∈ R suchthat D ψf(x0; d) = 0.

m Theorem 7 Let f : R → R be s.u.s.c., x0 ∈ K and assume that for some choice of 2 sequences εn ↓ 0 and ψεn ∈ C , one of the following conditions holds ∀d ∈ T (K, x0)∩ S1:

i) if D ψf(x0; d) > 0, then there exist a real number α(d) > 0 and a neighborhood of the direction d, say U(d) such that:

0 0 0 1 D ψf(x0 + td ; d ) > 0, ∀t ∈ (0, α(d)), ∀d ∈ U(d) ∩ S ;

ii) if D ψf(x0; d) = 0, then there exist a real number α(d) > 0 and a neighborhood of the direction d, say U(d), such that, for each t ∈ (0, α(d)) and for each 0 1 0 2 0 0 d ∈ U(d) ∩ S we have D ψf(x0; d ) ≥ 0 and D ψf(x0 + td ; d ) > 0.

Then x0 is a local solution of problem P1).

Proof: Ab assurdo, let assume there exists a feasible sequence xn → x0 such that f(xn) < f(x0). It can be easily written, without loss of generality xn = x0 + tndn, 1 1 dn ∈ S , dn → d ∈ S , tn ↓ 0, and hence d ∈ T (K, x0).

i) If D ψf(x0; d) > 0, then, as n → +∞:

0 ≥ f(x0 + tndn) − f(x0) ≥ tnD ψf(ξn; dn),

with ξn ∈ [x0, x0 + tndn], which is trivially a contradiction.

ii) If D ψf(x0; d) = 0, then:

t2 0 ≥ f(x + t d ) − f(x ) ≥ t D f(x ; d ) + D 2 f(ξ ; d ), 0 n n 0 n ψ 0 n 2 ψ n n

with ξn ∈ [x0, x0 + tndn], which is again a contradiction.

2

Remark 10 Conditions similar to those of the previous theorem have been proved in [32] for functions of class C1,1.

18 Now we deal with the following constrained optimization problem:

P2) min f(x) s.t.

gi(x) ≤ 0, i = 1, . . . , r

m where f, gi : R → R. We will define the set of active constraints at a point x0 as the index set I(x0): {i = 1, . . . , r : gi(x0) = 0} and the feasible set as m Γ := {x ∈ R : gi(x) ≤ 0, i = 1, . . . , r}. Concerning this problem, we will first investigate first–order conditions expressed by means of mollified derivatives.

Lemma 1 (Generalized Abadie Lemma) Let f and gi be s.l.s.c. for i ∈ I(x0), gi be u.s.c. for i∈ / I(x0) and assume that x0 ∈ Γ is a local solution of problem P2). 2 m Then, whatever the choice of sequences εn ↓ 0 and ψεn ∈ C , @d ∈ R such that:   D ψf(x0; d) < 0

 D ψgi(x0; d) < 0, i ∈ I(x0)

m Proof: Since x0 is a local solution of P2), we can easily check that, ∀d ∈ R , @α(d) > 0 such that ∀t ∈ (0, α(d)):   D ψf(x0 + td; d) < 0

 D ψgi(x0 + td; d) < 0, i ∈ I(x0)

Indeed, if for some d such an α(d) would exist, from Theorem 3 we would get, ∀t ∈ (0, α(d)):

f(x0 + td) < f(x0) and

gi(x0 + td) < 0, i ∈ I(x0).

Since gi, i∈ / I(x0) are u.s.c. we obtain also, for t “small enough”, gi(x0 + td) <

0, i∈ / I(x0). This fact contradicts that x0 ∈ Γ is a local solution of P2). m Hence, for any fixed d ∈ R one can find a sequence tn ↓ 0 such that for all n it holds or D ψf(x0 + tnd; d) ≥ 0 either D ψgi(x0 + tnd; d) ≥ 0, for some fixed i ∈ I(x0). Recalling that the first–order upper mollified derivative is u.s.c. we obtain that either D ψf(x0; d) ≥ 0 or D ψgi(x0; d) ≥ 0 and hence we get the thesis. 2

19 Theorem 8 (Generalized F. John Conditions) Let f, gi, i ∈ I(x0) be s.l.s.c., 2 gi, i∈ / I(x0) be u.s.c. and let εn ↓ 0 and ψεn ∈ C . Assume that x0 ∈ Γ is a local solution of problem P2) and that D ψf(x0; ·) and D ψgi(x0; ·), i ∈ I(x0) are finite.

Then there exist scalars τ ≥ 0, λi ≥ 0, i ∈ I(x0), not all zero, such that:

τD f(x ; d) + P λ D g (x ; d) ≥ 0, ∀d ∈ m. (1) ψ 0 i∈I(x0) i ψ i 0 R

Proof: ¿From the previous Lemma we know that the system:   D ψf(x0; d) < 0

 D ψgi(x0; d) < 0 i ∈ I(x0) has no solution. Since the first–order upper mollified derivatives are convex (Propo- sition 6), from a well known Theorem of the alternative ([3] Theorem 7.1.2), we obtain the thesis. 2

Remark 11 Of course a relevant question is which conditions would ensure τ > 0 (or equivalently τ = 1) in formula (1). It can be easily seen that this is the case if the following generalized Slater–type constraint qualification condition holds:

m ∃ d ∈ R such that D ψgi(x0; d) < 0, i ∈ I(x0).

Now we prove necessary and sufficient second–order optimality conditions for problem P2).

Theorem 9 Let f, gi, i ∈ I(x0) be s.l.s.c., gi, i∈ / I(x0) be u.s.c. and assume that 2 x0 ∈ Γ is a local solution of problem P2). Moreover let εn ↓ 0, ψεn ∈ C and assume that D ψf(x0; ·) and D ψgi(x0; ·), i ∈ I(x0), are finite.

Then, if τ ≥ 0, λi ≥ 0, i ∈ I(x0) satisfy (1), the following condition holds:

if d ∈ R(Γ(λ), x ) is such that τD f(x ; d) + P λ D g (x ; d) = 0 0 ψ 0 i∈I(x0) i ψ i 0 (2) then τD 2 f(x ; d) + P λ D 2 g (x ; d) ≥ 0 ψ 0 i∈I(x0) i ψ i 0 where Γ(λ) = {x ∈ Γ | P λ g (x) = 0}. i∈I(x0) i i

20 Proof: Let d ∈ R(Γ(λ), x ) be such that τD f(x ; d) + P λ D g (x ; d) = 0 0 ψ 0 i∈I(x0) i ψ i 0 and observe that, since D ψf(x0; ·) and D ψgi(x0; ·), i ∈ I(x0) are finite, we can write, for t > 0:

t2 2 f(x0 + td) − f(x0) ≤ tD ψf(x0; d) + 2 D ψf(ξ; d) t2 2 gi(x0 + td) − gi(x0) ≤ tD ψgi(x0; d) + 2 D ψgi(ξi; d), i ∈ I(x0) where ξ, ξi ∈ [x0, x0 + td]. Hence we have: X X τf(x0 + td) + λigi(x0 + td) − τf(x0) − λigi(x0) ≤

i∈I(x0) i∈I(x0)   t2 X ≤ τD 2 f(ξ; d) + λ D 2 g (ξ ; d) . 2  ψ i ψ i i  i∈I(x0) For t “small enough”, the lefthandside is nonnegative and hence, using the upper semicontinuity of second–order mollified derivatives:

2 X 2 0 ≤ lim sup[τD ψf(ξ; d) + λiD ψgi(ξi; d)] ≤ t↓0 i∈I(x0) 2 X 2 ≤ τ lim sup D ψf(ξ; d) + λi lim sup D ψgi(ξi; d) ≤ t↓0 t↓0 i∈I(x0) 2 X 2 ≤ τD ψf(x0; d) + λiD ψgi(x0; d), i∈I(x0) and so we get the thesis. 2

Theorem 10 Let f, gi, i ∈ I(x0) be s.u.s.c. and x0 ∈ Γ. Moreover, assume that for 2 some choice of sequences εn ↓ 0 and ψεn ∈ C there exist scalars λi ≥ 0, i ∈ I(x0) 1 such that ∀d ∈ T (Γ, x0) ∩ S one of the following conditions holds:

i) If D f(x ; d) + P λ D g (x ; d) > 0, then there exist a real α(d) > 0 ψ 0 i∈I(x0) i ψ i 0 and a neighborhood of the direction d, U(d), so that:

D f(x + td0; d0) + P λ D g (x + td0; d0) > 0 ∀t ∈ (0, α(d)), ∀d0 ∈ U(d). ψ 0 i∈I(x0) i ψ i 0

ii) If D f(x ; d) + P λ D g (x ; d) = 0, then there exist a real α(d) > 0 ψ 0 i∈I(x0) i ψ i 0 and a neighborhood of the direction d, U(d), so that:

D 2 f(x + td0; d0) + P λ D 2 g (x + td0; d0) > 0 ∀t ∈ (0, α(d)), ∀d0 ∈ U(d). ψ 0 i∈I(x0) i ψ i 0

21 Then x0 is a (strict) local solution of P2).

Proof: By contradiction assume there exists a feasible sequence xn → x0 so that 1 f(xn) − f(x0) ≤ 0. We shall write xn = x0 + tndn for some dn → d ∈ T (Γ, x0) ∩ S .

i) If D f(x ; d) + P λ D g (x ; d) > 0, then we would have: ψ 0 i∈I(x0) i ψ i 0

f(x0 + tndn) − f(x0) ≥ tnD ψf(ξn; dn)

and i gi(x0 + tndn) − gi(x0) ≥ tnD ψgi(ξn; dn), i ∈ I(x0),

i where ξn, ξn ∈ [x0, x0 + tndn]. Using multipliers λi, we get: X X 0 ≥ f(x0 + tndn) + λigi(x0 + tndn) − f(x0) − λigi(x0) ≥

i∈I(x0) i∈I(x0) X i ≥ tnD ψf(ξn; dn) + tn λiD ψgi(ξn; dn) i∈I(x0) which contradict the hypothesis, for n large enough.

ii) If D f(x ; d) + P λ D g (x ; d) = 0, then we shall write: ψ 0 i∈I(x0) i ψ i 0

t2 f(x + t d ) − f(x ) ≥ t D f(x ; d ) + n D 2 f(ξ ; d ) 0 n n 0 n ψ 0 n 2 ψ n n

and t2 g (x + t d ) − g (x ) ≥ t D g (x ; d ) + n D 2 g (ξi ; d ), i ∈ I(x ), i 0 n n i 0 n ψ i 0 n 2 ψ i n n 0

i where ξn, ξn ∈ [x0, x0 + tndn]. Using multipliers λi and the assumption, we get:

X X 0 ≥ f(x0 + tndn) + λigi(x0 + tndn) − f(x0) − λigi(x0) ≥

i∈I(x0) i∈I(x0) t2 t2 X ≥ n D 2 f(ξ ; d ) + n λ D 2 g (ξi ; d ) 2 ψ n n 2 i ψ i n n i∈I(x0) which contradict again the hypothesis, for n large enough.

2

22 5 Characterization of convex functions

In this section we give a characterization of convex functions by means of second– order mollified derivatives. The following result is classical:

m Lemma 2 ([34]) Let f : R → R be a continuous function. Then f is convex if and only if: f(x + td) − 2f(x) + f(x − td) ≥ 0, t2 m ∀x, d ∈ R , ∀t ∈ R.

m Lemma 3 ([9]) Let f : R → R be a continuos function. Then f is convex if and only if the mollified functions fε, obtained from a sequence of mollifiers ψε, are convex for every ε > 0.

m 2 Theorem 11 Let f : R → R be a continuous function and let εn ↓ 0 and ψεn ∈ C . A necessary and sufficient condition for f to be convex is that:

2 m m D ψf(x; d) ≥ 0, ∀x ∈ R , ∀d ∈ R .

Proof: Necessity. By definition:

2 > D ψf(x; d) = inf lim inf d Hfεn (xn)d. xn→x n→+∞

Recalling the previous lemma, from the convexity of the functions fε, we have:

> m d Hfεn (x)d ≥ 0, ∀x, d ∈ R , ∀n and the necessity follows. Sufficiency. We can write for every n:

f (x + td) − 2f (x) + f (x − td) εn εn εn = t2 1  t2 t2  = t∇f (x)>d + d>Hf (ξ )d − t∇f (x)d + d>Hf (ξ0 )d = t2 εn 2 εn n εn 2 εn n

1 h i = d>Hf (ξ )d + d>Hf (ξ0 )d , 2 εn n εn n

23 0 where ξn ∈ (x, x + td), ξn ∈ (x, x − td). As n → +∞ we can assume that ξn → ξ ∈ 0 0 [x, x + td] and ξn → ξ ∈ [x, x − td] and recalling Theorem 1 we obtain:

f(x + td) − 2f(x) + f(x − td) 1 h > > 0 i = lim d Hfε (ξn)d + d Hfε (ξ )d ≥ t2 n→+∞ 2 n n n

> > 0 lim inf d Hfε (ξn)d + lim inf d Hfε (ξ )d ≥ n→+∞ n n→+∞ n n 1 1 ≥ D 2 f(ξ; d) + D 2 f(ξ0; d) ≥ 0. 2 ψ 2 ψ 2

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Papers già pubblicati/Papers already published

94.01 – D. CHECCHI, La moderazione salariale negli anni 80 in Italia. Alcune ipotesi interpretative basate sul comportamento dei sindacati 94.02 – G. BARBA NAVARETTI, What Determines Intra-Industry Gaps in Technology? A Simple Theoretical Framework for the Analysis of Technological Capabilities in Developing Countries 94.03 – G. MARZI, Production, Prices and Wage-Profit Curves:An Evaluation of the Empirical Results 94.04 – D. CHECCHI, Capital Controls and Conflict of Interests 94.05 – I. VALSECCHI, Job Modelling and Incentive Design: a Preliminary Study 94.06 – M. FLORIO, Cost Benefit Analysis: a Research Agenda 94.07 – A. D’ISANTO, La scissione di società e le altre operazioni straordinarie: natura, presupposti economici e problematiche realizzative 94.08 – G. PIZZUTTO, Esistenza dell’ equilibrio economico generale: approcci alternativi 94.09 – M. FLORIO, Cost Benefit Analysis of Infrastructures in the Context of the EU Regional Policy 94.10 – D. CHECCHI - A. ICHINO - A. RUSTICHINI, Social Mobility and Efficiency - A Re-examination of the Problem of Intergenerational Mobility in Italy 94.11 – D. CHECCHI - G. RAMPA - .L. RAMPA, Fluttuazioni cicliche di medio termine nell’economia italiana del dopoguerra 95.01 – G. BARBA NAVARETTI, Promoting the Strong or Supporting the Weak? Technological Gaps and Segmented Labour Markets in Sub-Saharan African Industry 95.02 – D. CHECCHI, I sistemi di assicurazione contro la disoccupazione: un'analisi comparata 95.03 – I. VALSECCHI, Job Design and Maximum Joint Surplus 95.04 – M. FLORIO, Large Firms, Entrepreneurship and Regional Policy: "Growth Poles" in the Mezzogiorno over Forty Years 95.05 – V. CERASI - S. DALTUNG, The Optimal Size of a Bank: Costs and Benefits of Diversification 95.06 – M. BERTOLDI, Il miracolo economico dei quattro dragoni: mito o realtà? 95.07 – P. CEOLIN, Innovazione tecnologica ed alta velocità ferroviaria: un'analisi 95.08 – G. BOGNETTI, La teoria della finanza a Milano nella seconda metà del Settecento: il pensiero di Pietro Verri 95.09 – M. FLORIO, Tax Neutrality in the King-Fullerton Framework, Investment Externalities, and Growth 95.10 – D. CHECCHI, La mobilità sociale: alcuni problemi interpretativi e alcune misure sul caso italiano 95.11 – G. BRUNELLO - D. CHECCHI , Does Imitation help? Forty Years of Wage Determination in the Italian Private Sector 95.12 – G. PIZZUTTO, La domanda di lavoro in condizioni di incertezza 95.13 – G. BARBA NAVARETTI - A. BIGANO, R&D Inter-firm Agreements in Developing Countries. Where? Why? How? 95.14 – G. BOGNETTI - R. FAZIOLI, Lo sviluppo di una regolazione europea nei grandi servizi pubblici a rete 96.01 – A. SPRANZI, Il ratto dal serraglio di W. A. Mozart. Una lettura non autorizzata 96.02 – G. BARBA NAVARETTI - I. SOLOAGA - W. TAKACS, Bargains Rejected? Developing Country Trade Policy on Used Equipment 96.03 – D. CHECCHI - G. CORNEO, Social Custom and Strategic Effects in Trade Union Membership: Italy 1951-1993 96.04 – V. CERASI, An Empirical Analysis of Banking Concentration 96.05 – M. FLORIO, Il disegno dei servizi pubblici locali dal socialismo municipale alla teoria degli incentivi 96.06 – G. PIZZUTTO, Piecewise Deterministic Markov Processes and Investment Theory under Uncertainty: Preliminary Notes 96.07 – I. VALSECCHI, Job Assignment and Promotion 96.08 – D. CHECCHI, L'efficacia del sistema scolastico in prospettiva storica 97.01 – I. VALSECCHI, Promotion and Hierarchy: A Review 97.02 – D. CHECCHI, Disuguaglianza e crescita. Materiali didattici 97.03 – M. SALVATI, Una rivoluzione copernicana: l'ingresso nell'Unione Economica e Monetaria 97.04 – V. CERASI - B. CHIZZOLINI - M. IVALDI, The Impact of Deregulation on Branching and Entry Costs in the Banking Industry 97.05 – P.L. PORTA, Turning to Adam Smith 97.06 – M. FLORIO, On Cross-Country Comparability of Government Statistics:OECD National Accounts 1960-94 97.07 – F. DONZELLI, Pareto's Mechanical Dream 98.01 – V. CERASI - S. DALTUNG, Close-Relationships between Banks and Firms: Is it Good or Bad? 98.02 – M. FLORIO - R. LUCCHETTI - F. QUAGLIA, Grandi e piccole imprese nel Centro-Nord e nel Mezzogiorno: un modello empirico dell'impatto occupazionale nel lungo periodo 98.03 – V. CERASI – B. CHIZZOLINI – M. IVALDI, Branching and Competitiveness across Regions in the Italian Banking Industry 98.04 – M. FLORIO – A. GIUNTA, Planning Contracts in Southern Italy, 1986-1997: a Prelimary Evaluation 98.05 – M. FLORIO – I. VALSECCHI, Planning Agreements in the Mezzogiorno: a Principle Agent Analysis 98.06 – S. COLAUTTI, Indicatori di dotazione infrastrutturale: un confronto tra Milano e alcune città europee 98.07 – G. PIZZUTTO, La teoria fiscale dei prezzi in un’economia aperta 98.08 – M. FLORIO, Economic Theory, Russia and the fading “Washington Consensus” 99.01 – A. VERNIZZI – A. SABA, Alcuni effetti della riforma della legislazione fiscale italiana nei confronti delle famiglie con reddito da lavoro dipendente 99.02 – C. MICHELINI, Equivalence Scales and Consumption Inequality: A Study of Household Consumption Patterns in Italy 99.03 – S.M. IACUS, Efficient Estimation of Dynamical Systems 99.04 – G. BOGNETTI, Nuove forme di gestione dei servizi pubblici 99.05 – G.M. BERNAREGGI, Milano e la finanza pubblica negli anni 90: attualità e prospettive 99.06 – M. FLORIO, An International Comparison of the Financial and Economic Rate of Return of Development 99.07 – M. FLORIO, La valutazione delle politiche di sviluppo locale 99.08 – I. VALSECCHI, Organisational Design: Decision Rules, Operating Costs and Delay 99.09 – G. PIZZUTTO, Arbitraggio e mercati finanziari nel breve periodo. Un’introduzione

00.01 – D. LA TORRE – M. ROCCA, A.e. Convex Functions on Rn 00.02 – S. M. IACUS – YU A. KUTOYANTS, Semiparametric Hypotheses Testing for Dynamical Systems with Small Noise 00.03 – S. FEDELI – M. SANTONI, Endogenous Institutions in Bureaucratic Compliance Games 00.04 – D. LA TORRE – M. ROCCA, Integral Representation of Functions: New Proofs of Classical Results 00.05 – D. LA TORRE – M. ROCCA, An Optimization Problem in IFS Theory with Distribution Functions 00.06 – M. SANTONI, Specific excise taxation in a unionised differentiated duopoly 00.07 – H. GRAVELLE – G. MASIERO, Quality incentives under a capitation regime: the role of patient expectations 00.08 – E. MARELLI – G. PORRO, Flexibility and innovation in regional labour markets: the case of Lombardy 00.09 – A. MAURI, La finanza informale nelle economie in via di sviluppo 00.10 – D. CHECCHI, Time series evidence on union densities in European countries 00.11 – D. CHECCHI, Does educational achievement help to explain income inequality? 00.12 – G. BOESSO – A. VERNIZZI, Carichi di famiglia nell’Imposta sui Redditi delle Persone Fisiche in Italia e in Europa: alcune proposte per l’Italia 01.01 G. NICOLINI, A method to define strata boundaries 01.02 – S. M. IACUS, Statistical analysis of the inhomogeneous telegrapher’s process 01.03 – M. SANTONIi, Discriminatory procurement policy with cash limits can lower imports: an example 01.04 – D. LA TORRE, L’uso dell’ottimizzazione non lineare nella procedura di compressione di immagini con IFS 01.05 – G. MASIERO, Patient movements and practice attractiveness 01.06 – S. M. IACUS, Statistic analysis of stochastic resonance with ergodic diffusion noise 01.07 – B. ANTONIOLI – G. BOGNETTI, Modelli di offerta dei servizi pubblici locali in Europa 01.08 – M. FLORIO, The welfare impact of a privatisation: the British Telecom case-history 01.09 – G. P. CRESPI, The effect of economic policy in oligopoly. A variational inequality approach. 01.10 – G. BONO – D. CHECCHI, La disuguaglianza a Milano negli anni ’90 01.11 – D. LA TORRE, On the notion of entropy and optimization problems 01.12 – M. FLORIO – A. GIUNTA, L’esperienza dei contratti di programma: una valutazione a metà percorso 01.13 – M. FLORIO – S. COLAUTTI, A logistic growth law for government expenditures: an explanatory analysis 01.14 – L. ZANDERIGHI, Town Center Management: uno strumento innovativo per la valorizzazione del centro storico e del commercio urbano 01.15 – A. MAFFIOLETTI – M. SANTONI, Do trade union leaders violate subjective expected utility? Some insights from experimental data 01.16 – D. LA TORRE, An inverse problem for stochastic growth models with iterated function systems 01.17 – D. LA TORRE – M. ROCCA, Some remarks on second-order generalized derivatives for C1,1 functions 01.18 – A. BUCCI, Human capital and technology in growth 01.19 – R. BRAU – M. FLORIO, Privatisation as price reforms: an analysis of consumers’ welfare change in the UK 01.20 – A. SPRANZI, Impresa e consumerismo: la comunicazione consumeristica 01.21 – G. BERTOLA – D. CHECCHI, Sorting and private education in Italy 01.22 – G. BOESSO, Analisi della performance ed external reporting: bilanci e dati aziendali on-line in Italia 01.23 – G. BOGNETTI, Il processo di privatizzazione nell’attuale contesto internazionale 02.01 – D. CHECCHI – J. VISSER, Pattern persistence in european trade union density 02.02 – G. P. CRESPI – D. LA TORRE – M. ROCCA, Second order optimality conditions for differentiable functions 02.03 – S. M. IACUS – D. LA TORRE, Approximating distribution functions by iterated function systems 02.04 – A. BUCCI – D. CHECCHI, Crescita e disuguaglianza nei redditi a livello mondiale 02.05 – A. BUCCI, Potere di mercato ed innovazione tecnologica nei recenti modelli di crescita endogena con concorrenza imperfetta 02.06 – A. BUCCI, When Romer meets Lucas: on human capital, imperfect competition and growth 02.07 – S. M. IACUS – D. LA TORRE, On fractal distribution function estimation and applications 02.08 – P. GIRARDELLO – O. NICOLIS – G. TONDINI, Comparing conditional variance models: theory and empirical evidence 02.09 – L. CAMPIGLIO, Issues in the measurement of price indices: a new measure of inflation 02.10 – D. LA TORRE – M. ROCCA, A characterization of Ck,1 functions 02.11 – D. LA TORRE – M. ROCCA, Approximating continuous functions by iterated function systems and optimization problems 02.12 – D. LA TORRE – M. ROCCA, A survey on C1,1 functions: theory, numerical methods and applications 02.13 – D. LA TORRE – M. ROCCA, C1,1 functions and optimality conditions 02.14 – D. CHECCHI, Formazione e percorsi lavorativi dei laureati dell’Università degli Studi di Milano 02.15 – D. CHECCHI – V. DARDANONI, Mobility comparisons: Does using different measures matter? 02.16 – D. CHECCHI – C. LUCIFORA, Unions and Labour Market Institutions in Europe 02.17 – G. BOESSO, Forms of voluntary disclosure: reccomendations and business practices in Europe and U.S. 02.18 – A. MAURI – C.G. BAICU, Storia della banca in Romania – Parte Prima - 02.19 – D. LA TORRE – C. VERCELLIS, C1,1approximations of generalized support vector machines 02.20 – D. LA TORRE, On generalized derivatives for C1,1 vector functions and optimality conditions 02.21 – D. LA TORRE, Necessary optimality conditions for nonsmooth optimization problems 02.22 – D. LA TORRE, Solving cardinality constrained portfolio optimization problems by C 1,1 approximations 02.23 – M. FLORIO – K. MANZONI, The abnormal returns of UK privatisations: from underpricing to outperformance 02.24 – M. FLORIO, A state without ownership: the welfare impact of British privatisations 1979-1997 02.25 – S.M.IACUS – D. LA TORRE, Nonparametric estimation of distribution and density functions in presence of missing data: an IFS approach 02.26 – S.M. IACUS – G. PORRO, Il lavoro interinale in Italia: uno sguardo all’offerta 02.27 –G.P.CRESPI – D. LA TORRE, M. ROCCA, Second-order optimality conditions for nonsmooth multiobjective 02.28– D. CHECCHI –T. JAPPELLI, School Choice and Quality 03.01– D. CHECCHI, The Italian educational system family background and social stratification 03.02 – G. NICOLINI, – D. MARASIN, I Campionamento per popolazioni rare ed elusive: la matrice dei profili 03.03 – S. COMI, Intergenerational mobility in Europe: evidence from ECHP 03.04 – A. MAURI, Origins and early development of banking in Ethiopia 03.05 – A. ALBERICI, Strategie bancarie e tecnologia 03.06 – D. LA TORRE – M. ROCCA, On C^(1,1) constrained optimization problems 03.07 – M. BRATTI – A. BUCCI, Effetti di complementarietà, accumulazione di capitale umano e crescita economica: teoria e risultati empirici 03.08 – R. MacCULLOCH – S. PEZZINI, The role of freedom, growth and religion in the taste for revolution 03.09 – L. PILOTTI –N. RIGHETTO, Web strategy and intelligent software agents in decision process for networks knowledge based 03.10 – G. P. CRESPI –D. LA TORRE – M. ROCCA, Mollified derivatives and second-order optimality conditions