UNIVERSITÀ DI PISA
DIPARTIMENTO DI STATISTICA E MATEMATICA APPLICATA ALL’ECONOMIA
Report n.185
Coercivity Concepts and Recession
Function in Constrained Problems
Riccardo CAMBINI - Laura CAROSI
Legally registered on July, 2000
Via Cosimo Ridolfi, 10 – 56124 PISA – Tel. (050) Centralino 945111 - Segr. Amm. 945233/231 - Segr. Stud. 945317 - Fax 945375
Cod. Fisc. 80003670504 – Partita IVA 00286820501
2
Cambini R. and L. Carosi
D
constrained problems, called f∞
, coinciding with the known one when the set D
is the whole space.
Beside that, coercivity of function f is the other key tool of our analysis.
In the literature there are several different concepts, which are often referred
to with the same name; sometimes they are defined by means of the recession
function, so that coercivity and recession function wrongly seem to be necessarily
related. In this paper we provide a generalization of these coercivity concepts
for constrained problem, trying to present them in a unified framework. A new
characterization of k-supercoercive function is given as well as the relationships
among coercivity concepts, recession functions and optimality conditions.
Furthermore we provide a new characterization of the existence of minimum
points which improves the one given by Auselender [2,3]. Some other conditions,
which can be easily verified in applicative problems, are stated and the condition
given by Auselender [4] for an asymptotically-linear region D is also improved.
Finally some necessary and sufficient conditions guaranteeing the unboundedness
of optimal solutions set are stated.
2. Preliminaries
In this paper we denote X = m and · is the usual associated norm on X;
our aim is to study conditions ensuring the existence of minimum points for the
following constrained problem:
min f (x)
PD :
x∈D⊆X
where f : X → ∪ {+∞} is a proper scalar function, that is dom(f ) = {x ∈
X : f (x) < +∞} = ∅, and D ⊆ X is a closed set. We also denote with SD =
{x0 ∈ D : f (x0 ) ≤ f (x) ∀x ∈ D} the set of minimum points for problem PD .
Note that the unconstrained problem associated with function f is also covered
just assuming D = X.
Let us now recall some concepts regarding to the notions of recession cone
and recession function which have a central role throughout the paper.
Given a nonempty set D ⊆ X, the asymptotic cone D∞ of D (often referred
to as recession cone) is defined by (1 ):
xn
D∞ = {y ∈ X : ∃{xn } ⊂ D, ∃{tn } ⊂ , tn → +∞,
→ y}
tn
xn
→ v, y = λv, λ ≥ 0}
= {0} ∪ {y ∈ X : ∃{xn } ⊂ D, xn → +∞,
xn 1 Remind that the asymptotic cone is a closed one and that a set D ⊆ m is bounded if
and only if D∞ = {0}. Remind also that when D is a closed convex set then the asymptotic
cone is convex and can be rewritten as follows:
D∞ = {y ∈ X : ∃x ∈ D such that x + λy ∈ D ∀λ > 0}
= {y ∈ X : x + λy ∈ D ∀x ∈ D ∀λ > 0}
Coercivity and Recession Function in Constrained Problems
3
By means of the asymptotic cone it is possible to define the recession function.
Let f : X → ∪ {+∞} be a proper function; the corresponding recession
function f∞ : X → ∪ {−∞, +∞} is defined by:
epi(f∞ ) = (epi(f ))∞
where, as usual, epi(f ) = {(x, µ) ∈ X × : f (x) ≤ µ}.
It is worth noticing (see Auselender [2,3]) that f∞ is positively homogeneous (i.e. 0 ∈ dom(f∞ ) and f∞ (λx) = λf∞ (x) ∀λ > 0) and that if f is lower
semicontinuous, that is lim inf x→x0 f (x) ≥ f (x0 ) ∀x0 ∈ X, then f∞ is lower
semicontinuous too. Note also that the following useful properties hold:
i) f∞ (0) = 0 or f∞ (0) = −∞ (2 ) ,
ii) f∞ (0) = 0 =⇒ f∞ (y) > −∞ ∀y = 0 (3 ) .
In studying the existence of minimum points for f a key role is played by the
behavior of f with respect to the directions y such that f∞ (y) = 0. With this
aim let us recall the following widely used notation:
ker(f∞ ) = {y ∈ X : f∞ (y) = 0}
Note that, by means of the positive homogeneity of f∞ , the set ker(f∞ ) comes
out to be a cone, with or without the origin (in the case f∞ (0) = −∞); if f
is lower semicontinuous then ker(f∞ ) is a cone with the origin, but it does not
necessarily contains lines.
In Baiocchi et Al. [6] the following equivalent expression for f∞ is given:
f (tn vn )
lim inf
f∞ (y) =
inf
: tn → +∞, vn → y
(1)
tn
{vn }⊂X,{tn }⊂ n→+∞
Let us now state a new equivalent formula for the recession function, which
will be useful in the rest of the paper. With this aim let us firstly introduce the
following definition.
Definition 1. Let us consider problem PD . Function f is said to be lower unbounded on compacts over D if ∃µ ∈ , 0 < µ < +∞, ∃{xn } ⊂ D, xn ≤ µ,
such that lim inf n→+∞ f (xn ) = −∞. Function f is said to be lower bounded on
compacts over D if it is not lower unbounded on compacts. In the case D = X
function f will be simply said to be respectively lower unbounded on compacts or
lower bounded on compacts.
Note that if at least one of the following conditions hold:
i) f is locally Lipschitz
ii) f is lower semicontinuous
By means of the definition epi(f∞ (0)) is a cone, so that f∞ (0) ∈ {−∞, 0, +∞}; being f
proper ∃y ∈ X such that f (y) < +∞ so that the halfline {(0, µ), µ ≥ 0} ⊂ (epi(f ))∞ and
hence f∞ (0) = +∞.
3 This property follows directly from the definition being (epi(f ))
∞ a closed cone.
2
4
Cambini R. and L. Carosi
then f is lower bounded on compacts; note also that these conditions are
sufficient but not necessary, as it is pointed out in the next examples.
Example 1. Let us consider the following scalar functions f : → :
√
i) f (x) = 3 x is lower bounded on compacts but it is not lipschitz in any
neighborhood
(−, ) of x0 = 0;
2
for x ≤ 0
x
ii) f (x) =
is lower bounded on compacts but it is not lower
x2 − 1 for x > 0
semicontinuous.
By means of the following notation:
lim inf f (xn ) : xn → +∞
lim inf f (x) = inf
x→+∞
{xn }⊂X
n→+∞
we are now able to state the following characterization of the recession function.
Theorem 1. Let f : X → ∪ {+∞} be a proper function, then
i) f∞ (0) = 0 if and only if f is lower bounded on compacts and
f (x)
> −∞,
x→+∞ x
lim inf
f∞ (0) = −∞ otherwise;
ii) for any y = 0 it results
f (x)
x
y
:
→
f∞ (y) = y lim inf
x
y
x→+∞ x
Proof. i) ⇐) Suppose on the contrary that f∞ (0) = −∞; then, by means of the
definition, (0, −1) ∈ (epi(f ))∞ so that, being (0, −1) = 1:
∃{(xn , µn )} ⊂ epi(f ), (xn , µn ) → +∞,
(xn , µn )
→ (0, −1).
(xn , µn )
Let d = limn→+∞ xxnn (this limit exists eventually substituting {xn } with a
n
subsequence); we then have (xnµ,µ
→ −1, so that µn < 0 definitively, and:
n )
xn
xn
xn →0
=
xn
µn (xn , µn )
,
)
(
xn xn so that
µn
xn f (xn )
xn (x)
lim inf x→+∞ fx
→ −∞; being f (xn ) ≤ µn < 0 it then follows that
−∞ which implies limn→+∞ xn < +∞ provided that
→
>
(xn )
→ −∞ only if limn→+∞ xn = 0, being f lower bounded on
−∞; hence fx
n
compacts. Since (xn , µn ) → +∞, we have µn → −∞ and hence f (xn ) → −∞,
since f (xn ) ≤ µn , which contradicts the lower boundedness on compacts of f .
Coercivity and Recession Function in Constrained Problems
5
⇒) Let us suppose on the contrary that f is not lower bounded on compacts
(x)
= −∞, then the following condition holds:
or that lim inf x→+∞ fx
∃{xn } ⊂ X, f (xn ) → −∞, such that
f (xn )
→ −∞
xn (xn ,f (xn ))
and d = limn→+∞
Let (y, β) = limn→+∞ (x
n ,f (xn ))
we have that f (xn ) < 0 definitively and:
xn
xn .
Being
xn
xn f (xn )
xn
( x
,
)
n xn f (xn )
|f (xn )|
xn f (xn )
xn
( x
,
)
n |f (xn )| |f (xn )|
y = limn→+∞
xn
(xn ,f (xn ))
= limn→+∞
β = limn→+∞
f (xn )
(xn ,f (xn ))
= limn→+∞
f (xn )
xn → −∞
=0
= −1
Being {(xn , f (xn ))} ⊂ epi(f ) and (xn , f (xn )) → +∞ we have that (0, −1) ∈
(epi(f ))∞ ; hence {(0, µ) : µ < 0} ⊂ (epi(f ))∞ = epi(f∞ ) implying f∞ (0) = −∞
which is a contradiction.
ii) We prove that if y = 0 then f∞ (y) given by (1) is equivalent to the
following:
xn
y
f (xn )
: xn → +∞,
→
f∞ (y) = y inf m lim inf
n→+∞ xn xn y
{xn }⊂
Let xn = tn vn ; being tn → +∞ we can assume tn > 0 ∀n so that xn = tn vn .
Firstly note that condition vn → y is equivalent to the two conditions vn =
xn y
vn
xn
tn → y and vn = xn → y ; observe also that the couple of conditions
xn → y are equivalent to the two following ones:
tn
xn = tn → y. Formula (1) can then be rewritten as:
tn → +∞ and vn =
xn → +∞ and vn f∞ (y) =
inf
{vn },{xn }⊂m
lim inf
n→+∞
It results also:
y =
xn
f (xn )
y
vn : xn → +∞, vn → y ,
→
xn xn y
inf m lim inf vn : vn → y
{vn }⊂
n→+∞
so that the thesis is proved.
3. Coercivity concepts
In the literature, the existence of minimum points for unconstrained problems
is strictly related to the coercivity property of the objective function. Several
definitions of coercivity can be found in the literature and different concepts are
often referred to with the same name.
In this section we aim at defining some coercivity concepts related to constrained problems, generalizing the ones given in the literature and trying to
study them in a unified framework. From now on we use the following notation:
lim inf f (xn ) : xn → +∞
lim inf f (x) = inf
x→+∞D
{xn }⊂D
n→+∞
6
Cambini R. and L. Carosi
Definition 2. Let us consider problem PD . Function f is said to be (4 ):
i) semicoercive on D if ∃x0 ∈ D such that lim inf x→+∞D f (x) ≥ f (x0 )
ii) coercive on D if lim inf x→+∞D f (x) = +∞
(x)
>0
iii) strictly coercive on D if lim inf x→+∞D fx
iv) supercoercive on D if lim inf x→+∞D
f (x)
x
= +∞
v) k-supercoercive on D, with k ≥ 1, if lim inf x→+∞D
f (x)
xk
>0
In the case D = X, function f will be simply said to be semicoercive, coercive,
strictly coercive, supercoercive or k-supercoercive.
Directly from the given definitions, we have that on the set D a coercive
function is also semicoercive, a strictly coercive function is also coercive, a supercoercive function is also strictly coercive, a k-supercoercive function is also
supercoercive for any k > 1, a strictly coercive function is nothing but a ksupercoercive function with k = 1.
Let us now point out that these introduced concepts are not sufficient to
guarantee neither the existence of minimum points for problem PD nor the lower
boundedness on compacts of the function, as it is shown in the next Example 2
where function f is k-supercoercive but not lower bounded on compacts.
Example 2. Let us consider the following function f : → ∪ {+∞}:
f (x) =
k
|x| + log(x2 ) for x = 0
+∞
for x = 0
with k > 1. Function f is k-supercoercive on X = even if SD = ∅ since
inf x∈ f (x) = limx→0 f (x) = −∞.
The following results provide a characterization of k-supercoercive and strictly
coercive functions.
Theorem 2. Let us consider problem PD . Then:
i) f is k-supercoercive on D if and only if the following condition holds:
k
∃a, b, c ∈ , a > 0, such that f (x) ≥ a x + b ∀x ∈ D, x > c
ii) f is k-supercoercive on D and lower bounded on compacts if and only if the
following condition holds:
k
∃a, b ∈ , a > 0, such that f (x) ≥ a x + b ∀x ∈ D
4
Note that, in the case D = X, in [8] the supercoercive functions are referred to as 1coercive, while in [2, 3] the strictly coercive functions are called coercive.
Coercivity and Recession Function in Constrained Problems
7
Proof. i) ⇐) From the hypothesis it is:
f (x)
lim inf
x→+∞D
k
x
≥ lim inf
x→+∞D
a+
b
k
x
=a>0
so that f is k-supercoercive.
i) ⇒) Suppose on the contrary that ∀a, b, c ∈ , a > 0, ∃x ∈ D, x > c,
k
such that f (x) < a x + b. Choose a = n1 , b = 0 and c = n, n = 1, 2, 3, . . . ;
f (xn )
then ∃xn ∈ D, xn > n, such that x
< n1 . This implies:
k
n
lim inf
x→+∞D
f (x)
x
k
≤ lim
n→+∞
f (xn )
k
xn ≤0
which is a contradiction.
ii) ⇐) Analogously to i) we have that f is k-supercoercive; f is also lower
k
bounded, since f (x) ≥ a x + b ≥ b ∀x ∈ D.
ii) ⇒) Suppose on the contrary that ∀a, b ∈ , a > 0, ∃x ∈ D such that
k
f (x) < a x + b. Choose a = n1 and b = −n; then ∃xn ∈ D such that f (xn ) <
k
1
n xn − n. If the sequence {xn } is bounded then f (xn ) → −∞ so that f is
lower unbounded on compacts, which is a contradiction. If the sequence {xn } is
unbounded then:
n
f (x)
f (xn )
1
−
lim inf
≤ lim
< lim
≤0
k
n→+∞ x k
n→+∞
n
x→+∞D xk
x
n
n
which is a contradiction.
Since a strictly coercive function is nothing but a k-supercoercive function
with k = 1, the following corollary trivially holds.
Corollary 1. Let us consider problem PD . Then:
i) f is strictly coercive on D if and only if the following condition holds:
∃a, b, c ∈ , a > 0, such that f (x) ≥ a x + b ∀x ∈ D, x > c
ii) f is strictly coercive on D and lower bounded on compacts if and only if the
following condition holds:
∃a, b ∈ , a > 0, such that f (x) ≥ a x + b ∀x ∈ D
Let us now introduce the following definition of recession function for constrained problems.
Definition 3. Let us consider problem PD . We say recession function over D
D
the function f∞
: D∞ → ∪ {−∞, +∞} defined as follows:
8
Cambini R. and L. Carosi
D
i) f∞
(0) = 0 if and only if f is lower bounded on compacts over D and:
lim inf
x→+∞D
f (x)
> −∞,
x
D
(0) = −∞ otherwise;
f∞
ii) for any y = 0 it is:
D
f∞
(y)
= y
lim inf
x→+∞D
f (x)
x
y
:
→
x x
y
Analogously to the set ker(f∞ ) we will use the following notation:
D
D
) = {y ∈ D∞ : f∞
(y) = 0}.
ker(f∞
Let us note that, directly from the definition (5 ), it is:
D
(y) ∀y ∈ D∞
f∞ (y) ≤ f∞
and that the equality may not hold, as it is shown in the next example.
Example 3. Let us consider X = 2 , the function f : 2 → such that f (x, y) =
xy, and let D = D∞ = 2+ = {(x, y) ∈ 2 : x ≥ 0, y ≥ 0}. It results f∞ (y) =
D
D
f∞
(y) ∀y ∈ Int(D∞ ) while −∞ = f∞ (y) < f∞
(y) = 0 ∀y ∈ Fr(D∞ ).
It is now possible to study the relationships existing among the coercivity
concepts and the recession function. With this aim it is worth noticing that the
coercivity concepts are not based on the recession function associated with f ,
but just on the behavior of f when x → +∞; actually in the literature the
coercivity concepts are often defined by means of the recession function (6 ), so
that they wrongly seem to be necessarily related. Note that in the case D = X
statement ii) of the following theorem has already been proved by Zǎlinescu
[11]; for the sake of completeness we provide an independent proof of the whole
theorem.
Theorem 3. Let us consider problem PD .
D
(y) ≥ 0 ∀y ∈ D∞ , y = 0
i) if f is semicoercive on D then f∞
D
ii) f is strictly coercive on D if and only if f∞
(y) > 0 ∀y ∈ D∞ , y = 0
D
iii) f is supercoercive on D if and only if f∞ (y) = +∞ ∀y ∈ D∞ , y = 0
5
Note also that the following implication holds:
g(x) =
f (x) for x ∈ D
+∞ for x ∈
/D
=⇒
g∞ (y) =
D (y) for x ∈ D
f∞
∞
+∞ for x ∈
/ D∞
6 In [2–4], for example, in the case D = X a function f is said to be coercive (strictly coercive
with our notation) if f∞ (y) > 0 ∀y = 0 and is said to be weakly coercive if f∞ (y) ≥ 0 ∀y = 0
and f is constant on each line with direction y ∈ ker(f∞ ).
Coercivity and Recession Function in Constrained Problems
9
Proof. i) By means of the semicoercivity of f it results:
lim inf
x→+∞D
f (x)
f (x0 )
≥ lim inf
=0
x
x→+∞D x
D
(y) ≥ 0 ∀y ∈ D∞ , y = 0.
so that f∞
ii) ⇒) The thesis follows directly from the definitions.
⇐) From Theorem 2 it is sufficient to prove that ∃a, b, c ∈ , a > 0, such that
f (x) ≥ a x + b ∀x ∈ D, x > c. Suppose on the contrary that ∀a, b, c ∈ ,
a > 0, ∃x ∈ D, x > c, such that f (x) < a x + b. Choose a = n1 , b = 0 and
(xn )
< n1 . Defining
c = n, n = 1, 2, 3, . . . ; then ∃xn ∈ D, xn > n, such that fx
n
(xn )
D
y = limn→+∞ xxnn ∈ D∞ we then have f∞
(y) ≤ limn→+∞ fx
≤ 0 which is a
n
contradiction.
iii) ⇒) The thesis follows directly from the definitions.
⇐) Suppose on the contrary that:
f (x)
f (xn )
lim inf
= inf
: xn → +∞ < +∞
lim inf
x→+∞D x
{xn }⊂D n→+∞ xn (xn )
then ∃{xn } ⊂ D such that xn → +∞ and lim inf n→+∞ fx
< +∞; den
D
(y) < +∞ and this is a
noting with y = limn→+∞ xxnn ∈ D∞ it results f∞
contradiction.
D
Remark 1. Note that having f∞
(y) = +∞ ∀y ∈ D∞ , y = 0, that is having
D
(0) = 0, as it is shown in
a supercoercive function f , does not imply that f∞
Example 2 where f is k-supercoercive and lower unbounded on compacts so
D
(0) = −∞.
that f∞
In the next section we are going to deep on conditions characterizing the
existence of minimum points for a proper function f .
As a preliminary result, let us point out the behavior of coercivity with
respect of the existence of minimum points.
Theorem 4. Let us consider problem PD as well as the following properties:
D
(y) > 0 ∀y ∈ D∞ , y = 0
i) f is lower semicontinuous and f∞
ii) f is strictly coercive on D and lower semicontinuous
iii) f is semicoercive on D and lower semicontinuous
iv) SD = ∅
D
v) f is semicoercive on D and f∞
(0) = 0
vi) f is semicoercive on D and lower bounded on compacts over D
vii) inf x∈D f (x) > −∞
D
viii) f is lower bounded on compacts over D and f∞
(y) ≥ 0 ∀y ∈ D∞ , y = 0
D
ix) f∞ (y) ≥ 0 ∀y ∈ D∞
Then the following sequence of implications holds:
i) ⇔ ii) ⇒ iii) ⇒ iv) ⇒ v) ⇔ vi) ⇒ vii) ⇒ viii) ⇔ ix)
10
Cambini R. and L. Carosi
Proof. i) ⇔ ii) Follows directly from ii) of Theorem 3.
ii) ⇒ iii) From definition 2 trivially follows that any strictly coercive function
is also semicoercive. Hence the result follows directly from i) of Theorem 3.
iii) ⇒ iv) Let {xn } ⊂ D be a sequence such that f (xn ) → inf x∈D f (x). If
{xn } is unbounded (xn → +∞) then by means of the semicoercivity on D of
f we have that ∃x0 ∈ D such that f (x0 ) ≤ lim inf x→+∞D f (x) ≤ inf x∈D f (x)
so that x0 ∈ SD and hence SD = ∅. If {xn } is bounded, that is ∃µ > 0 such that
xn ≤ µ, then it is possible to extract a converging subsequence {xj } ⊆ {xn }
such that, being D closed, xj → x ∈ D; from the lower semicontinuity of f we
then have inf x∈D f (x) = lim inf j→+∞ f (xj ) ≥ f (x) so that x ∈ SD and hence
SD = ∅.
iv) ⇒ v) The semicoercivity of f follows directly from the definition choosing
x0 among the points of SD ; being SD = ∅ then f is also lower bounded so that
D
f∞
(0) = 0 by means of i) of Definition 3.
v) ⇒ vi) Follows directly from i) of Definition 3.
vi) ⇒ vii) Suppose by contradiction that inf x∈D f (x) = −∞ and let {xn } ⊂
D be a sequence such that f (xn ) → inf x∈D f (x) = −∞. Being f semicoercive
on D the sequence {xn } is bounded and this contradicts the fact that f is lower
bounded on compacts over D.
vii) ⇒ viii) Function f has a finite infimum on D, hence it is trivially lower
bounded on compacts over D; suppose now on the contrary that ∃y ∈ D∞ ,
D
(y) < 0. Let {xn } ⊂ D, xn → +∞, be the sequence such
y = 0, such that f∞
that:
f (xn )
n→+∞ xn D
f∞
(y) = y lim
with
xn
y
→
xn y
It then results:
f (xn )
f D (y)
lim xn = ∞
lim xn = −∞
n→+∞ xn n→+∞
y n→+∞
lim f (xn ) = lim
n→+∞
and this is a contradiction.
D
viii) ⇒ ix) Suppose on the contrary that f∞
(0) < 0; being f lower bounded
on compacts over D it follows from i) of Definition 3 that:
lim inf
x→+∞D
f (x)
= −∞
x
D
so that ∃y ∈ D∞ , y = 0, such that f∞
(y) = −∞ and this is a contradiction.
D
(0) = 0 implies that
ix) ⇒ viii) Just note that, by means of the definition, f∞
f is lower bounded on compacts over D.
vi) ⇒ v) By means of the previously proved implications, we have that vi)
D
implies ix) so that f∞
(0) = 0.
The following examples point out that the implications given in the previous
theorem are proper.
Coercivity and Recession Function in Constrained Problems
11
Example 4. Let us consider the following functions f : → ∪ {+∞} and let
D = X = :
i) f (x) = 0 is a continuous semicoercive function which is not strictly coercive;
hence inTheorem 4 iii) ii).
0 for x < 0
ii) f (x) =
is semicoercive and SD = {x : x < 0} = ∅ even if
x + 1 for x ≥ 0
f is notlower semicontinuous; hence in Theorem 4 iv) iii).
x2 for x ≤ 0
D
is supercoercive with f∞
iii) f (x) =
(0) = f∞ (0) = 0 but it is
2
x − 1 for x > 0
not lower semicontinuous and SD = ∅ with inf x∈ f (x) = −1 > −∞; hence
in Theorem 4 v) iv). Note that this example shows the importance of the
lower semicontinuity of f in order to have minimum points, providing that:
D
f is supercoercive, f∞
(0) = 0 and inf f (x) > −∞ ⇒ SD = ∅.
1
x2 for x = 0 is lower bounded on compacts with finite infimum,
iv) f (x) =
+∞ for x = 0
even if it
is
hence in Theorem 4 vii) vi).
not semicoercive;
2
− log(x ) for x = 0
v) f (x) =
is lower semicontinuous, hence lower bounded
+∞ for x = 0
D
(y) = f∞ (y) = 0 ∀y ∈ , even if f is not semicoercive
on compacts, and f∞
and inf x∈ f (x) = −∞; hence in Theorem 4 viii) vii). Moreover, the
D
(y) ≥ 0 ∀y ∈ m is a
converse of i) in Theorem 3 does not hold and f∞
necessary but not sufficient condition for the existence of a finite infimum.
4. Existence of Optimal Solutions
Theorem 4 underlines the importance of the recession function and coercivity
properties for optimality conditions. It comes immediately out that strictly coercivity on D together with the lower semincontinuity is a sufficient optimality
D
condition while f∞
(y) ≥ 0 for every y ∈ D∞ is a necessary one. When the feasible region D is the whole space, these conditions coincide with the standard
ones existing in the literature, but is worthy noting that in general we cannot
D
with f∞ as it is shown by Example 3 where the function f (x, y) has
replace f∞
D
(y) ≥ 0 ∀y ∈ D∞ while f∞ (y) = −∞
minimum points over D = 2+ and f∞
D
∀y ∈ Fr(D∞ ). Analogously to the unconstrained case, condition f∞
(y) ≥ 0 is
necessary but not sufficient for the existence of the minimum points and this
suggests to investigate the behavior of f along the direction y = 0 such that
D
f∞
(y) = 0.
The goal of this section is to state results characterizing the existence of
minimum points analyzing the behavior of the objective function f along the
D
). In particular we will generalize in different ways the
directions y ∈ ker(f∞
following result.
12
Cambini R. and L. Carosi
Proposition 1 (Auslender [2]). Let us consider problem PD with D = X and
suppose function f to be lower semicontinuous. SD = ∅ if and only if both the
two following conditions hold:
i) f∞ (d) ≥ 0 ∀d ∈ m ,
ii) ∀{xn } ⊂ m , such that xn → +∞, xxnn → d ∈ ker(f∞ ) and the sequence
{f (xn )} is bounded above, there exist ρn ∈]0, 1], zn → z, with d − z < 1,
such that for n sufficiently large:
f (xn − xn ρn zn ) ≤ f (xn )
The generalization of the previous Proposition 1 is based on the following
lemma (Attouch [1] gives a result similar to statements i) and ii) in reflexive
Banach spaces, while Zǎlinescu [11] proves an analogous theorem under stronger
assumptions).
Lemma 1. Let us consider problem PD and suppose function f to be lower semiD
(y) ≥ 0 ∀y ∈ D∞ and define, for any integer
continuous. Suppose also that f∞
n ≥ 1, the following auxiliary function:
2
gn (x) = f (x) + n x , n = 1, 2, . . .
where n > 0 ∀n and n → 0+ . Then the following properties hold:
i) ∃xn ∈ argmin{gn (x), x ∈ D} for any integer n ≥ 1;
ii) limn→+∞ f (xn ) = inf x∈D f (x)
If the sequence {n } is also strictly decreasing then:
iii) xn ≤ xn+1 for any integer n ≥ 1;
iv) the sequence {f (xn )} is decreasing;
D
(y) ≥ 0 ∀y ∈ D∞ it comes out that function gn is superProof. i) Being f∞
coercive on D for any integer n and hence for Theorem 4 there exists xn ∈
argmin{gn (x), x ∈ D}.
ii) Let z be any element of D. By means of the definition, for any integer n
we have:
2
f (xn ) ≤ f (xn ) + n xn = gn (xn ) ≤ gn (z) = f (z) + n z
2
so that:
2
lim f (xn ) ≤ lim f (z) + n z = f (z) ∀z ∈ D
n→+∞
n→+∞
and hence limn→+∞ f (xn ) = inf x∈D f (x).
iii) Let us consider, for any n, the following inequalities:
2
2
f (xn ) + n xn = gn (xn ) ≤ gn (xn+1 ) = f (xn+1 ) + n xn+1 2
2
f (xn+1 ) + n+1 xn+1 = gn+1 (xn+1 ) ≤ gn+1 (xn ) = f (xn ) + n+1 xn Coercivity and Recession Function in Constrained Problems
13
adding these inequalities it yields:
2
2
2
2
n xn + n+1 xn+1 ≤ n xn+1 + n+1 xn so that:
2
2
(n − n+1 ) xn ≤ (n − n+1 ) xn+1 2
Being the sequence {n } strictly decreasing it is (n − n+1 ) > 0 so that xn ≤
2
xn+1 and the thesis is proved.
2
2
iv) By means of the definition of gn+1 (x) and being xn ≤ xn+1 , we
have that for any n:
2
f (xn+1 ) + n+1 xn+1 = gn+1 (xn+1 ) ≤ gn+1 (xn ) =
2
2
= f (xn ) + n+1 xn ≤ f (xn ) + n+1 xn+1 so that f (xn+1 ) ≤ f (xn ) and the thesis is proved.
By means of Lemma 1, we are able to state the following characterization for
the existence of optimal points.
Theorem 5. Let us consider problem PD and suppose function f to be lower
semicontinuous. SD = ∅ if and only if both the two following conditions hold:
D
(y) ≥ 0 ∀y ∈ D∞ ,
i) f∞
D
ii) ∀{xn } ⊂ D, such that xn → +∞, xxnn → y ∈ ker(f∞
) and the sequence
{f (xn )} is strictly decreasing, there exist {ρn } ⊂]0, 1], {zn } ⊂ X, {vn } ⊂ X,
with vn = 1 and (xn (vn − ρn zn )) ∈ D, such that for n sufficiently large:
vn − zn < 1
and
f (xn (vn − ρn zn )) ≤ f (xn )
Proof. ⇒) Condition i) follows directly from Theorem 4; as regards to condition
n −y0
ii) let y0 ∈ SD = ∅ and assume ρn = 1, vn = xxnn and zn = xx
; it results
n
(xn (vn − ρn zn )) = y0 ∈ D, vn = 1 and, for n sufficiently large, vn − zn =
y0 xn < 1, so that the thesis is proved since f (y0 ) ≤ f (xn ) being y0 ∈ SD .
2
⇐) Let gn = f (x) + n x , n = 1, 2, . . . , with {n } strictly decreasing; by
means of condition i) and Lemma 1 we have that ∃xn ∈ argmin{gn (x), x ∈ D}
∀n and that {f (xn )} is decreasing with limn→+∞ f (xn ) = inf x∈D f (x). Let us
now prove that if the sequence {xn } is unbounded then we can not extract
a strictly decreasing subsequence of {f (xn )}. Suppose by contradiction that
xn → +∞ and that {f (xn )} is strictly decreasing (substituting {xn } with a
subsequence if necessary); let y = limn→+∞ xxnn , being f (xn ) < f (x1 ) we have
(xn )
≤ limn→+∞ fx(xn1) = 0 so that, by means of the definition, it
limn→+∞ fx
n
D
(y) ≤ 0 and hence, for i), y ∈ ker(f∞ ). By means of ii) there exist
results f∞
ρn ∈]0, 1], zn , vn ∈ X, with vn = 1 and (xn (vn − ρn zn )) ∈ D, such that
14
Cambini R. and L. Carosi
for n sufficiently large vn − zn < 1 and f (xn (vn − ρn zn )) ≤ f (xn ); it then
results:
1
2
xn = gn (xn ) ≤ gn (xn (vn − ρn zn ))
n
1
2
= f (xn (vn − ρn zn )) + xn (vn − ρn zn )
n
1
2
2
≤ f (xn ) + xn vn − ρn zn n
f (xn ) +
and hence, being 0 < ρn ≤ 1, vn = 1, and vn − zn < 1:
1 ≤ vn − ρn zn = vn − ρn vn + ρn vn − ρn zn = vn (1 − ρn ) + ρn (vn − zn ) ≤ vn (1 − ρn ) + ρn vn − zn = 1 + ρn (vn − zn − 1) < 1
which is impossible.
We then have that {f (xn )} does not admit any strictly decreasing subsequence regardless the sequence corresponding {xn } is bounded or unbounded.
In the latter case ∃q such that f (xn ) = f (xq ) ∀n ≥ q, xq ∈ D, and hence
inf x∈D f (x) = limn→+∞ f (xn ) = f (xq ) so that the infimum is reached as a
minimum. If otherwise {xn } is bounded then it is possible to extract a subsequence x̃n → y0 ∈ D such that, by means of the lower semicontinuity of f ,
inf x∈D f (x) = limn→+∞ f (x̃n ) = f (y0 ), implying that the infimum is reached as
a minimum. The proof is then complete.
Remark 2. It is worth noticing the differences existing between Theorem 5 and
the known Proposition 1:
i) in Proposition 1 the unconstrained case is considered, while in Theorem 5 we
study a constrained problem, covering the unconstrained case just assuming
D = X;
ii) in Proposition 1 vn is fixed to be equal to xxnn , while in Theorem 5 vn may
be any vector with vn = 1;
iii) in Proposition 1 it is required that vn − zn → v − z < 1, while in Theorem 5 we simply assume that definitively vn − zn < 1, thus we cover also
the case vn − zn → v − z = 1;
iv) in Proposition 1 all the bounded above sequences {f (xn )} are considered,
while in Theorem 5 just the strictly decreasing ones.
The following useful corollary can also be stated, providing a practical condition to determine the existence of optimal points. Note that this result can not
be derived from Proposition 1.
Coercivity and Recession Function in Constrained Problems
15
Corollary 2. Let us consider problem PD and suppose function f to be lower
semicontinuous. SD = ∅ if and only if both the two following conditions hold:
D
(y) ≥ 0 ∀y ∈ D∞ ,
i) f∞
D
ii) ∀{xn } ⊂ D, such that xn → +∞, xxnn → y ∈ ker(f∞
) and the sequence
{f (xn )} is strictly decreasing, there exists {yn } ⊂ D such that for n sufficiently large:
yn < xn and
f (yn ) ≤ f (xn )
Proof. ⇒) Condition i) follows directly from Theorem 4; as regards to condition
ii) we just have to choose y0 ∈ SD = ∅ and assume yn = y0 ∀n.
n −yn
; it results {ρn } ⊂]0, 1],
⇐) Assume ρn = 1, vn = xxnn and zn = xx
n
vn = 1, (xn (vn − ρn zn )) = yn ∈ D so that, by means of the hypothesis, for
n sufficiently large it is:
vn − zn =
yn <1
xn and f (xn (vn − ρn zn )) = f (yn ) ≤ f (xn )
so that the thesis follows directly from Theorem 5.
Another useful result can be stated when the set D is asymptotically-linear.
Remind that a nonempty closed set D ⊆ X is said to be an asymptotically-linear
set if ∀ρ > 0, ∀{xn } ⊂ D such that xn → +∞ and xxnn → y, for n sufficiently
large it is (xn − ρy) ∈ D (see Auselender [4]).
Note that directly from the definition it follows that the finite intersection
or the finite union of asymptotically-linear sets is still an asymptotically-linear
set. Note also that, as it has been proved in Auselender [4], an asymptotically
polyhedral set D is also asymptotically-linear (7 ).
The following sufficient condition holds.
Corollary 3. Let us consider problem PD and suppose function f to be lower
semicontinuous and the set D ⊆ X to be asymptotically-linear. If the two following conditions hold:
D
i) f∞
(y) ≥ 0 ∀y ∈ D∞ ,
D
ii) ∀{xn } ⊂ D, such that xn → +∞, xxnn → y ∈ ker(f∞
) and the sequence
{f (xn )} is strictly decreasing, there exists ρ > 0 such that for n sufficiently
large f (xn − ρy) ≤ f (xn ),
then the optimal set SD is nonempty.
Remind, see [4], that a set D ⊂ X is said to be a simple asymptotically polyhedral set if
∃µ ≥ 0 for which the set Dµ := D ∩ {x : x ≥ µ} admits the decomposition Dµ = K + M
with K compact and M a polyhedral cone. D is said to be asymptotically polyhedral if it is
the intersection of a finite number of sets, each of them being the union of a finite number of
simple asymptotically polyhedral sets.
7
16
Cambini R. and L. Carosi
Proof. We just have to verify condition ii) of Corollary 2. Assume yn = (xn −ρy);
being D asymptotically-linear then for n sufficiently large it is yn ∈ D, by means
of the hypothesis it is also f (yn ) ≤ f (xn ). We are left to prove that for n
sufficiently large it is xn − ρy < xn ; with this aim let us firstly note that:
lim y T xn = lim y T
n→+∞
n→+∞
xn
2
lim xn = y lim xn = +∞;
n→+∞
xn n→+∞
2
then for n sufficiently large it is y T xn > 12 ρ y and hence:
2
2
2
2
xn − ρy = xn − 2ρy T xn + ρ2 y < xn so that definitively xn − ρy < xn and the thesis is proved.
It is now worth to compare Corollary 3 with the following result.
Proposition 2 (Auslender [4]). Let us consider problem PD and suppose function f to be lower semicontinuous and the set D ⊆ X to be asymptotically-linear.
If the two following conditions hold:
i) inf x∈D f (x) > −∞ and f∞ (y) ≥ 0 ∀y ∈ D∞ , y = 0,
ii) f ∈ F (8 ), that is to say that ∀{xn } ⊂ X, such that xn → +∞ and xxnn →
y ∈ ker(f∞ ), ∀ρ > 0, for n sufficiently large it results f (xn − ρy) ≤ f (xn ),
then the optimal set SD is nonempty.
Note that Corollary 3 is more general that the result by Auslender since:
D
i) in Proposition 2 f∞ (y) ≥ 0 is required, while in Corollary 3 f∞
(y) ≥ 0, with
D
f∞ (y) ≥ f∞ (y), is just needed;
ii) as it has been proved in Theorem 4, condition i) of Proposition 2 implies
condition i) of Corollary 3, while the converse is not true;
iii) in condition ii) of Proposition 2 all the sequences ∀{xn } ⊂ X are considered,
while in ii) of Corollary 3 just the feasible sequences such that {f (xn )} is
strictly decreasing are used;
iv) in condition ii) of Proposition 2 all ρ > 0 are considered, while in Corollary
3 just one of them is used.
Directly from Corollary 3 follows the next result which generalizes the sufficient condition, given in Auselender [2,3], related to asymptotically polyhedral
sets and based on the hypothesis that the objective function f is constant along
the directions y ∈ ker(f∞ ) (in other words, based on the hypothesis that f is
weakly coercive).
8
Note that this condition is equivalent to Definition 7 given in [4], page 51. To prove this
note that from one side it represent the particular case of Def.7 with n = f (xn ); on the other
side it implies Def.7 since Sf (α) ⊆ Sf (β) ∀α ≤ β so that for any n if xn ∈ Sf (n ) then
f (xn ) ≤ n and xn − ρy ∈ Sf (f (xn )) ⊆ Sf (n ).
Coercivity and Recession Function in Constrained Problems
17
Corollary 4. Let us consider problem PD and suppose function f to be lower
semicontinuous and the set D ⊆ X to be asymptotically-linear. If function f is
D
) then:
nondecreasing along the directions y ∈ ker(f∞
SD = ∅
⇐⇒
D
f∞
(y) ≥ 0 ∀y ∈ D∞
We conclude this section looking for conditions related to the boundedness
or unboundedness of the set SD of minimum points.
Theorem 6. Let us consider problem PD and suppose function f to be lower
semicontinuous and the set SD to be nonempty. Then:
D
D
i) if SD is unbounded then f∞
(y) ≥ 0 ∀y ∈ D∞ and ker(f∞
) = {0}.
Suppose also D to be a closed convex set and f to be constant along the
D
); then:
directions y ∈ ker(f∞
D
D
(y) ≥ 0 ∀y ∈ D∞ and ker(f∞
) = {0}.
ii) SD is unbounded if and only if f∞
D
Proof. i) By means of Theorem 4 we just have to prove that ker(f∞
) =
{0}.
xn
Let {xn } ⊂ SD such that xn → +∞ and let xn → y ∈ D∞ ; it results
(xn )
D
D
(y) ≤ limn→+∞ fx
= 0 and hence y ∈ ker(f∞
).
0 ≤ f∞
n
D
ii) ⇐) Let y ∈ ker(f∞
), y = 0, and let x0 ∈ SD ; being D closed and convex
D
then x0 + λy ∈ D ∀λ > 0, being f constant along the directions y ∈ ker(f∞
),
y = 0, we have also f (x0 ) = f (x0 + λy) ∀λ > 0 and hence x0 + λy ∈ SD ∀λ > 0.
The next result follows directly from Theorems 4 and 6.
Corollary 5. Let us consider problem PD and suppose function f to be lower
semicontinuous. Then:
D
(y) > 0 ∀y ∈ D∞ , y = 0, then SD is nonempty and bounded.
i) if f∞
Suppose also D to be a closed convex set and f to be constant along the
D
); then:
directions y ∈ ker(f∞
D
ii) SD is nonempty and bounded if and only if f∞
(y) > 0 ∀y ∈ D∞ , y = 0.
The following examples point out the importance in Theorems 6 and 5 of the
D
convexity of D and the constant behavior of f along the directions y ∈ ker(f∞
).
Example 5. Let us consider the following functions f : 2 → and sets D ⊂ 2 :
i) f (x, y) = y and D = {x ≥ 0, y ≥ 1} ∪ {x ≥ 0, y ≥ x}; D is nonconvex and
D
);
SD = {(0, 0)T } is bounded even if y = (1, 0)T ∈ ker(f∞
2
−x
2
ii) f (x, y) = y −
and D = + ; D is convex, f is not constant along the
T
D
direction y = (1, 0) ∈ ker(f∞
) and SD = {(0, 0)T } is bounded.
The following theorem is useful when the set D has empty interior, or when
it is known ”a priori” that there are no minimum points in the interior of D,
such as in many mathematical programming problems.
18
Cambini R. and L. Carosi
Theorem 7. Let us consider problem PD and suppose function f to be lower
semicontinuous, nondecreasing along the directions y ∈ D∞ and constant along
D
the directions y ∈ ker(f∞
), suppose also D + D∞ , with D D + D∞ , to be a
closed convex set. If at least one of the following conditions hold:
i) the set of minimum points for f over (D + D∞ ) \ D is bounded,
ii) D + D∞ is strictly convex and the set of minimum points for f over Int(D +
D∞ ) \ D is bounded,
D
(y) > 0 ∀y ∈ D∞ , y = 0.
then SD is nonempty and bounded if and only if f∞
Proof. By means of Theorem 5 we just have to prove that if SD is nonempty
D
D
(y) > 0 ∀y ∈ D∞ , y = 0. For Theorem 4 f∞
(y) ≥ 0
and bounded then f∞
D
∀y ∈ D∞ ; suppose now by contradiction that ∃y ∈ ker(f∞ ), y = 0 and let
x0 ∈ SD . Being f nondecreasing along the directions y ∈ D∞ we have that
f (x0 ) ≤ f (x0 + λy) ∀λ > 0, ∀y ∈ D∞ , so that x0 is a minimum point for f
over D + D∞ ; since D + D∞ is a closed convex set and (D + D∞ )∞ = D∞ then
D
x0 + λy ∈ D + D∞ ∀λ > 0; being f constant along the directions y ∈ ker(f∞
)
we then have f (x0 ) = f (x0 + λy) ∀λ > 0 and hence all the points x0 + λy,
∀λ > 0, are minimum points for f over D + D∞ ; note also that being SD
bounded ∃λ > 0 such that x0 + λy ∈ (D + D∞ ) \ D, ∀λ > λ > 0. The set of
this minimum points is unbounded, which contradicts condition i). If ii) holds
then, being D + D∞ strictly convex, x0 + λy ∈ Int(D + D∞ ), ∀λ > 0, and hence
x0 + λy ∈ Int(D + D∞ ) \ D, ∀λ > λ > 0, which is again a contradiction being
this set of minimum points unbounded.
Acknowledgements. The authors wish to thank J.P.Penot for his helpful and valuable comments and suggestions given during his short visit in Pisa. The paper has been discussed and
developed jointly by the authors. In particular, sections 1,2 and 3 have been handled by Laura
Carosi while section 4 is due to Riccardo Cambini. The paper has been partially supported by
M.U.R.S.T.
References
1. H. Attouch, “Viscosity solutions of minimization problems”, SIAM J. Optim., Vol.6, no.3,
pp.769-806, 1996.
2. A. Auslender, “Noncoercive Optimization Problems”, Math. Oper. Res., Vol.21, no.4,
pp.769-782, 1996.
3. A. Auslender, “How to deal with the unbounded in optimization: Theory and Algorithms”,
Math. Program., Vol.79, pp.3-18, 1997.
4. A. Auslender, “Existence of optimal solutions and duality results under weak conditions”,
Math. Program., Vol.88, pp.45-59, 2000.
5. M.Avriel, W.E. Diewert, S. Schaible, and I. Zang, Generalized Concavity, Mathematical
Concepts and Methods in Science and Engineering, vol.36, Plenum Press, New York, 1988.
6. C. Baiocchi, G. Buttazzo, F. Gastaldi and F. Tomarelli, “General existence theorems
for unilateral problems in continuum mechanics”, Arch. Ration. Mech. Anal., Vol.100,
pp.149-189, 1988.
7. G. Buttazzo and F. Tomarelli, “Nonlinear Neuman problems”, Adv. Math., Vol.89, pp.126142, 1991.
8. J. B. Hiriart-Hurruti, Conditions for global optimality, R. Horst and P.M. Pardalos (eds.),
Handbook of Global Optimization, Kluwer Academic Publishers, 1995, 1-26.
Coercivity and Recession Function in Constrained Problems
19
9. A. Mas Colell, M. D. Whinston, and J.R. Green , Microeconomic Theory, Oxford University Press, New York, NY, 1995.
10. R.T. Rockafellar, Convex Analisys, Princeton University Press, Princeton NJ, 1970
11. C. Zǎlinescu, Stability for a class of nonlinear optimization, in Proceedings of the summer
school in nonsmooth optimization and related topics - Erice, F.H. Clarke, V. F. Demyanov
and F. Giannessi (eds.), Plenum Press, New York, 1989, 437-458.
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Report n.25 - Attrazione ed entropia.(V.Bruno), 1989
Report n.26 - Invexity in Nonsmooth Programming. (G.Giorgi - S.Mititelu), 1989
Report n.27 -Lineamenti econometrici dell'evoluzione del reddito nazionale in relazione ad altri fenomeni
economici. (V.Bruno), 1989
Report n.28 - Equivalence in Linear Fractional Programming. (A.Cambini - L.Martein), 1989
Report n.29 - Centralità e potenziale demografico per l'analisi dei comportamenti demografici: il caso
della Toscana. (O.Barsotti - M.Bottai - M.Costa), 1990
Report n.30 - A sequential method for a bicriteria problem arising in portfolio selection theory.
(A.E.Marchi), 1990
Report n.31 - Mobilità locale e pianificazione territoriale. (M.Bottai), 1990
Report n.32 - Solving a Quadratic Fractional Program by means of a Complementarity Approach.
(A.Marchi), 1990
Report n.33 - Sulla relazione tra un problema bicriteria e un problema frazionario. (A.Marchi), 1990
Report n.34 - Variabili latenti e "self-selection" nella valutazione dei processi formativi.(E.Gori), 1991
Report n.35 - About an Interactive Model for Sexual Populations. (P.Manfredi - E.Salinelli), 1991
Report n.36 - Alcuni aspetti matematici del modello di Sraffa a produzione semplice.(G.Giorgi), 1991
Report n.37 - Parametric Linear Fractional Programming for an Unbounded Feasible Region.
(A.Cambini-S.Schaible-C.Sodini), 1991
Report n.38 - International migration to Northern Mediterranean countries. The cases of Greece, Spain
and Italy. (I.Emke-Poulopoulos, V.Gozàlves Pérez, L.Lecchini, O.Barsotti),1991
Report n.39 - A LP Code Implementation.(G.Gasparotto), 1991
Report n.40 - Un problema di programmazione quadratica nella costituzione di capitale.
(R.Cambini), 1991
Report n.41 - Stime ed errori campionari nell'indagine ISTAT sulle forze di lavoro.(G.Ghilardi), 1991
Report n.42 - Alcuni valori medi, variabilità paretiana ed entropia.(V.Bruno), 1991
Report n.43 - Gli effetti del trascinamento dei prezzi sulle misure dell'inflazione: aspetti metodologici.
(G.Boletto), 1991
Report n.44 - Gli abbandoni nell'università: modelli interpretativi.(P.Paolicchi), 1991
Report n.45 - Da un archivio amministrativo a un archivio statistico: una proposta metodologica per i dati
degli studenti universitari.(M.F.Romano), 1991
Report n.46 - Criteri di scelta delle variabili nei modelli MDS: un'applicazione sulla popolazione
studentesca di Pisa. (M.F.Romano), 1991
Report n.47 - Les parcours migratoires en fonction de la nationalité. Le cas de l'Italie.
(O.Barsotti - L.Lecchini), 1991
Report n.48 - Indicatori statistici ed evoluzione demografica, economica e sociale delle province
toscane.(V.Bruno), 1991
Report n.49 - Tangent cones in Optimization. (A.Cambini, L.Martein), 1991
Report n.50 - Optimality conditions in Vector and Scalar optimization: a unified approach.
(A.Cambini, L.Martein), 1991
Report n.51 - Elementi di uno schema di campionamento areale per alcune rilevazioni ufficiali in Italia.
(G.Ghilardi), 1992
Report n.52 - Investimenti e finanziamenti generalizzati. (P.Manca), 1992
Report n.53 - Le rôle des immigrés extra-communautaires dans le marché du travail.
(L.Lecchini, O.Barsotti), 1992
Report n.54 - Alcune condizioni di ottimalità relative ad un insieme stellato.(R.Cambini), 1992
Report n.55 - Uno schema di campionamento areale per le rilevazioni sulle famiglie in Italia.
(G.Ghilardi), 1992
Report n.56 - Studio di una classe di problemi non lineari: un metodo sequenziale.(R.Cambini), 1992
Report n.57 - Una nota sulle possibili estensioni a funzioni vettoriali di significative classi di funzioni
concavo-generalizzate. (R.Cambini), 1992
Report n.58 - Metropolitan Aging Transition and Metropolitan Redistribution of the Elderly in Italy.
(A.Bonaguidi, V.Terra Abrami), 1992
Report n.59 - A comparison of Male and Female Migration Strategies:The Cases of African and Filipino
Migrants to Italy. (O.Barsotti, L.Lecchini), 1992
Report n.60 - Un modello logit per lo studio del fenomeno delle nuove imprese. (G.Ghilardi), 1992
Report n.61 - Generalized Monotonicity. (S.Schaible), 1992
Report n.62 - Dell'elasticità in economia e dell'incertezza statistica. (V.Bruno), 1992
Report n.63 - Alcune classi di funzioni concave generalizzate nell'ottimizzazione vettoriale.
(L.Martein), 1992
Report n.64 - On the Relationships between Bicriteria Problems and Non-linear Programming Problems.
(A.Marchi), 1992
Report n.65 - Considerazioni metodologiche sul concetto di elasticità prefissata. (G.Boletto), 1992
Report n.66 - Soluzioni efficienti e condizioni di ottimalità nell'otti-mizzazione vettoriale.
(L.Martein), 1992
Report n.67 - Le rilevazioni ufficiali ISTAT della popolazione universitaria : problemi e definizioni
alternative. (M.F. Romano), 1992
Report n.68 - La ricerca " Spazio Utilizzato". Obiettivi e primi risultati. (M. Bottai - O. Barsotti), 1993
Report n.69 - Composizione familiare e mobilità delle persone anziane. Una analisi regionale.
(M. Bottai - F. Bartiaux), 1993
Report n.70 - An algorithm for a non-differentiable non-linear fractional programming problem.
(A. Marchi - C.Sodini), 1993
Report n.71 - An finite algorithm for generalized linear multiplicative programming.
(C.Sodini - S.Schaible), 1993
Report n.72 - An Approach to Optimality Conditions in Vector and Scalar Optimization.
(A.Cambini - L. Martein), 1993
Report n.73 - Generalized concavity and optimality conditions in Vector and Scalar Optimization.
(A.Cambini - L. Martein), 1993
Report n.74 - Alcune nuove classi di funzioni concavo-generalizzate. (R.Cambini), 1993
Report n.75 - On Nonlinear Scalarization in Vector Optimization
(A.Cambini, A.Marchi, L.Martein), 1994
Report n.76 - Analisi delle carriere degli studenti immatricolati dal 1980 al 1982
(M.F. Romano, G. Nencioni), 1994
Report n.77 - Indici statistici della congiuntura.
(G.Ghilardi), 1994
Report n.78 - Condizioni di efficienza locale nella ottimizzazione vettoriale. (R.Cambini), 1994
Report n.79 - Funzioni di utilizzazione dello spazio. (O.Barsotti, M.Bottai), 1994
Report n.80 - Alcuni aspetti dinamici della popolazione dei comuni della Toscana, distinti per ampiezza
demografica e per classi di urbanità e di ruralità. (V.Bruno), 1994
Report n.81 - I numeri indici del potere d’acquisto della moneta. (G.Boletto), 1994
Report n.82 - Some optimality conditions in multiobjective programming.
(A.Cambini, L.Martein, R.Cambini), 1994
Report n.83 - Fractional Programming with Sums of Ratios. (S. Schaible), 1994
Report n.84 - The Minimum-Risk Approach For Continuous Time Linear-Fractional
Programming. (S.Tigan, M.Stancu-Minasian),1994
Report n.85 - On Duality for Multiobjective Mathematical Programming of n-Set
Functions. (V.Preda, I.M.Stancu-Minasian),1994
Report n.86 - Optimality and Duality in Nonlinear Programming Involving Semilocally
Preinvex and Related Functions.(V.Preda, I.M.Stancu-Minasian, A.Batatorescu), 1994
Report n.87 - Una nota storica sulla programmazione lineare: un problema di Kantorovich
rivisto alla luce del problema degli zeri. (E. Melis),1995
Report n.88 - Mobilità territoriale dell'Italia e di tre Regioni tipiche:Lombardia, Toscana,
Sicilia. (V.Bruno),1995
Report n.89 - Bibliografia sulla presenza straniera in Italia. (A.Cortese),1995
Report n.90 - Funzioni Scalari Affini Generalizzate (R.Cambini), 1995
Report n.91 - Modelli epidemiologici: teoria e simulazione. (I) (P.Manfredi - F. Tarini), 1995
Report n.92 - The "OLIVAR" Survey. Methodology and Quality. (M. Bottai - M. Caputo - L. Lecchini),
1995
Report n.93 - Old People and social network. (L. Lecchini - D. Marsiglia - M. Bottai), 1995
Report n.94 - Uno studio empirico sul confronto tra alcuni indici statistici della
congiuntura. (G. Ghilardi), 1995
Report n.95 - Il traffico nei porti italiani negli anni recenti. (V. Bruno) , 1995
Report n.96 - An Analysis of the Falk-Palocsay Algorithm.
(A.Cambini,A.Marchi,L.Martein,S.Schaible) , 1995
Report n 97 - Sulla esistenza di elementi massimali. (A.Cambini, L.Carosi) , 1995
Report n.98 - Generalized Concavity and Generalized Monotonicity Concepts for Vector Valued
(R.Cambini, S.Komlòsi),
Report n.99 - Second Order Optimality Conditions in the Image Space. (R.Cambini) , 1996
Report n.100 - La Stagionalità delle correnti di navigazione marittima. (V.Bruno) , 1996
Report n.101 - A Comparison of Alternative Discrete Approximations of the
Cox- Ingersoll - Ross Model. (E.M.Cleur), 1996
Report n.102 - Sul calcolo del rapporto di concentrazione del Gini. (G. Ghilardi), 1996
Report n.103 - A New Approach to Second Order Optimality Conditions in
Vector Optimization. (A.Cambini, L.Martein, R.Cambini), 1996
Report n.104 - Alcune osservcazioni sull’immunizzazione semideterministica. (F. Gozzi), 1996.
Report n.105 - Innovation and Capital Accumulation in a Vintage Capital Model:
an Infinite Dimensional Control Approach. (E. Barucci, F. Gozzi), 1996.
Report n.106 - A survey of bicriteria fractional problems.
(A.Cambini, L.Martein, I.M.Stancu-Minasian), 1996
Report n.107 - Viscosità dei salari, offerta di lavoro endogena e ciclo
(L. Fanti, P. Manfredi), 1996
Report n.108 - Crescita con ciclo, ritardi nei piani di investimento ed effettivi popolazione.
(L.Fanti, P. Manfredi), 1996
Report n.109 – Ciclo di vita di nuovi prodotti: modellistica non lineare (P. Manfredi),1996
Report n.110 - Un modello “classico” di ciclo con crescita ed offerta di lavoro
endogena. (L.Fanti, P. Manfredi), 1996
Report n.111 - On the Connectedness of the Efficient Frontier: Sets Without Local Maxima.
(A. Marchi), 1996
Report n.112 - Generalized Concavity for Bicriteria Functions (R. Cambini), 1996
Report n.113 - Variazioni dinamiche (1971-1981-1991) dei fenomeni demografici dei comuni (urbani
e rurali) della Lombardia,in relazione ad alcune caratteristiche di mobilità territoriale
(VBruno), 1996
Report n.114 - Infectious diseases: epidemiology, mathematical models, and immunization policies
(P. Manfredi, F. Tarini, J.R. Williams, A. Carducci, B. Casini), 1997
Report n.115 - One dimensional SDE models, low order numerical methods and simulation
based estimation: a comparison of alternative estimators
(E.M. Cleur, P. Manfredi), 1997
Report n.116 - Point stability versus orbital stability (or instability): remarks on policy
implications in classical growth cycle model (L. Fanti, P; Manfredi), 1997
Report n.117 - Transition into adulthood, marriage, and timing of life in a stable
framework (P. Manfredi, F. Billari) 1997
population
Report n.118 - Una nota sul concetto di estremo superiore di insiemi ordinati da coni convessi
(L. Carosi), 1997
Report n.119 - Reti sociali degli anziani: selezione e qualità delle relazioni
(L.Lecchini e D.Marsiglia), 1997
Report n.120 - Gestation lags and efficiency wage mechanisms in a Goodwin
type growth model (P. Manfredi e L. Fanti), 1997
Report n.121 - La metodologia statistica multilevel come possibile strumento per
lo studio delle interazioni tra il comportamento procreativo individuale
e il contesto (G. Rivellini), 1997
Report n.122 - Una nota sugli insiemi C-limitati e L-limitati
(L. Carosi ), 1997
Report n.123 - Sull’estremo superiore di una funzione lineare fratta ristretta ad un insieme
chiuso e illimitato (L. Carosi ), 1997
Report n.124 - A demographic framework for the evaluation of the impact of imported
infectious diseases (P. Manfredi), 1997
Report n.125 - Calo della fecondità ed immigrazioni: scenari e considerazioni sul caso
italiano (A. Valentini), 1997
Report n.126 - Second order optimality conditions (A.Cambini - L. Martein),1997
Report n.127 - Populations with below replacement fertility: theoretical considerations and
scenarioes from the italian laboratory (P. Manfredi and A. Valentini), 1998
Report n.128 - Programmazione frazionaria e problemi bicriteria (A. Cambini - L. Martein - E. Moretti),
1998
Report n.129 - Incentive compatibility constraints and dynamic programming in continuous time
(E. Barucci - F. Gozzi A. Swiech), 1998
Report n 130 - Impatto delle immigrazioni sulla popolazione italiana: confronto tra scenari alternativi
(A. Valentini), 1 9 9 9
Report n.131 -Recent developement of migrations from Poland to Europe with a special emphasis on
Italy (K. Iglicka) Le migrazioni est-ovest : Le unioni miste in Italia - (O.Barsotti –
L.Lecchini),1999
Report. n.132 - Proiezioni demografiche multiregionali a due sessi, con immigrazioni internazionale e
vincoli di consistenza (A. Valentini), 1999.
Report. n 133 - Backward- Forward stochastic differential utility: Existence, consumption and
equilibrium analysis (F.Antonelli - E. Barucci - M.E. Mancino), 1999
Report. n.134 - Asset pricing with endogenous Aspirations (E. Barucci - M.E. Mancino), 1999
Report n. 135 - Estimating a class of diffusion models: An evaluation of the effects of sampled discrete
observations. ( E. M. Cleur), 1999.
Report n. 136 - Labour supply, time delays, and demoeconomic oscillations in a Solow-type growth
model (L. Fanti - P. Manfredi), 1999.
Report n.137 - Some Results on Partial Differential Equations and Asian Options ( E. Barucci S.Polidori-V.Vespri), 1999
Report n.138 - Hedging European Conitngent Claims in a Markovian Incomplete Market (E. Barucci
M. Elvira Mancino), 1999
Report n.139 - L’applicazione del modello multiregionale-multistato alla popolazione in Italia mediante
l’utilizzo del Lipro: procedura di adattamento dei dati e particolarità tecniche del
programma. (A. Valentini), 1999.
Report n.140 - Optimality Conditions and Duality in Fractional Programming- Involving Semilocally
Preinvex and Related Functions ( I.M.Stuncu-Minasian), 1999
Report n.141- Proiezioni demografiche con algoritmi di consistenza per la popolazione in Italia nel
periodo 1997 2142: presentazione dei risultati e confronto con metodologie di stima
alternative (A. Valentini), 1999
Report n. 142 - Competitive equilibria with money and restricted participation (L. Carosi), 1999
Report n. 143 - Monetary policy and Pareto improvability in a financial economy with restricted
Participation (L. Carosi ), 1999
Report n. 144 -Misurare il benessere e lo sviluppo dai paradossi del Pil a misure di benessere economico
sostenibile, con uno sguardo allo sviluppo umano (B. Cheli), 1999
Report n.145 - The old people’s perception of well-being: the role of material and non material resources
(B.Cheli, L. Lecchini, L. Masserini), 1999
Report n. 146 - Maximum likelihood estimation of one-dimensional stochastic differential equation
models from discrete data: some computational results ( Eugene M. Cleur), 1999
Report n. 147- Utilizzi empirici di modelli multistato continui con durate multiple (Alessandro Valentini
Francesco Billari Piero Manfredi), 1999
Report n.148- Transition into adulthoold: its macro-demographic consequences in a multistatew stable
population framework ( F. Billari , P. Manfredi, A. Valentini , A. Bonaguidi), 1999
Report n.149- Becoming Adult and its Macro-Demographic Impact: Multistate Stable Population Theory
And an Application to Italy (F. Billari, P. Manfredi, A. Valentini), 1999
Report n.150 - Le previsioni demografiche in presenza di immigrazioni: confronto tra modelli alternativi
e loro utilizzo empirico ai fini della valutazione dell'equilibrio nel sistema pensionistico
(A.Valentini), 1999
Report n.151 - Diffusion processes for asset prices under bounded rationality (E. Barucci. R. Monte)
Report n.152 - Reti neurali e analisi delle serie storiche: un modello per la previsione del BTP future
(E.Barucci, P.Cianchi, L.Landi, A.Lombardi), 1999
Report n.153 - On the supremum in fractional programming (A. Cambini, L. Carosi, L. Martein),
1999
Report n.154 - First and second order characterizations of a class of pseudoconcave vector functions.
( R.Cambini and L. Martein), 1999
Report n. 155 - Embedding population dynamics in macro-economic models. The case of the
goodwin's growth cycle ( P. Manfredi and L. Fanti), 1999
Report n. 156 - Migrazioni dei preti dalla Polonia in Italia (Laura Lecchini e Odo Barsotti), 1999
Report n. 157 - Analisi dei prezzi, in Italia, dal 1975 in poi (Vincenzo Bruno), 1999
Report n.158 - Analisi del commercio al minuto in Italia (Vincenzo Bruno), 1999
Report. n.159 -.Aspetti ciclici della liquidità bancaria, dal 1971 in poi (Vincenzo Bruno), 1999
Report n. 160 - A separation theorem in alternative theorems and vector optimization (Anna Marchi),
1999
Report n. 161- Labour suppley, population dynamicics, and persistent oscillations in a Goodwin-type
growth cycle model (P.Manfredi and L.Fanti), 2000
Report n. 162- Neo-classical labour market dynamics and chaos (and the Phillips curve revisited)
(P. Manfredi and L.Fanti) 2000
Report n.163- Detection of Hopf bifurcations in continuous-time macro-economic models, withan
application to reducible delay-systems ( P. Manfredi and Luciano Fanti)2000
Report n.164 – The dynamics of pareto efficient allocations with stochastic differential utility (F.
Antonelli E.Barucci), 2000
Report n. 165 - Computing maximum likelihood estimates of a class of One-Dimensional stochastic
differential equatin models from discrete Data* (Eugene M. Cleur), 2000
Report n. 166 - Estimating the drift parameter in diffusion processes more efficiently at discrete times: a
role of indirect estimation (Eugene M. Cleur), 2000
Report n. 167 – “Forecasting the forecasts of others” e la politica di inflation targeting (E.Barucci –
V.Valori), 2000
Report n. 168 – First and second order optimality conditions in vector optimization (A.Cambini –L.
Martein), 2000
Report.n. 169 – Theorems of the Alternative by way of separation Theorems (A. Marchi ) 2000
Report. n. 170 – Asset Pricing and Diversification with Partially Exchangeable Random Variables
(E.Barucci
M. Elvira Mancino), 2000
Report.n. 171 – Lon Term Effects of the Efficiency Wage Hypothesis in Goodwin-Type Economies (P.
Manfredi and L. Fanti), 2000
Report. n.172 - Long term effects of the efficiency wage hypothesis in Goodwin-type economies: a reply
(P. Manfredi and L. Fanti), 2000
Report. n.173 – Innovazione Finanziaria e domanda di moneta in un modello dinamico IS-LM con
accumulazione (L. Fanti), 2000
Report n. 174 – Social Heterogeneities in Classical New product Diffusion Models. I: “External” and
“Internal” Models. ( P.Manfredi, A.Bonaccorsi, A. Secchi), 2000
Report n. 175 – Modelli per formazione di coppie e modelli di Dinamica familiare. (P. Manfredi,
E.Salinelli), 2000
Report n. 176 – Long term interference between Demography and Epidemiogy: the case of tuberculosis.
(P. Manfredi, E. Salinelli, A. Melegaro, A. Secchi), 2000
Report n. 177 – Toward the development of an age structure teory for family dynamics I: general frame
(P. Manfredi – E. Salinelli), 2000
Report n. 178 – Population heterogeneities, nonlinear oscillations and chaos in some goodwin-type demoeconomic models paper to be presented at the: second workshop on “nonlinear
demography” max planck institute for demographic research Rostock, Germany , May
31-June 2, 2000 - ( P.Manfredi and Luciano Fanti), 2000
Report n. 179 – Volatility estimation via fourier analysis ( E. Barucci, M.E.Mancino, R. Renò), 2000
Report n. 180 – Minimum Principle Type Optimality Conditions (Riccardo Cambini ), 2000
Report n. 181 – Asset Prices under Bounded Rationality and Noise Trading (Emilio Barucci,
Massimiliano Giuli, Roberto Monte), 2000
Report n. 182 – Order Preserving Transformations and application (A. Cambini, D.T.Luc and L.
Martein), 2000
Report n. 183 – Variazioni dinamiche (1971-1981-1991) dei fenomeni demografici dei comuni urbani e
rurali della Sicilia, in relazione ad alcune caratteristiche di mobilità territoriale (V.Bruno),
2000
Report n. 184 – Asset Pricing with a Backward-Forward Stochastic Differential Utility (F. Antonelli,
E.Barucci, M.E.Mancino), 2000
Report n. 185 - Coercivity Concepts and Recession Function in Constrained Problems (R. Cambini, L.
Carosi), 2000