c 2011 Society for Industrial and Applied Mathematics
SIAM J. OPTIM.
Vol. 21, No. 1, pp. 174–192
NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMA IN
REFLEXIVE SPACES∗
MASSIMO MARINACCI† AND LUIGI MONTRUCCHIO‡
Abstract. We give general necessary and sufficient conditions for the existence of optima of
noncoercive functionals defined on reflexive spaces. Some special cases are then discussed that permit
the establishment of novel sufficient conditions for optimality as well as the recovery of known results.
Key words. minimization of functionals, noncoercive functionals, reflexive spaces, recession
functions, asymptotic directions
AMS subject classifications. 49K, 65K, 90C
DOI. 10.1137/100790264
1. Introduction. Studies of optimization problems led in recent years to the
development of several sufficient conditions for the existence of optima of functionals
with unbounded level sets for which the direct method of the calculus of variations
is not effective. An early important contribution in this literature is from Baiocchi
et al. [7]. Motivated by some variational problems arising in mechanics, they introduced in a general setting recession functions
w
(y) = inf lim inf f (tn yn ) /tn : tn → ∞, yn y
(1)
f∞
n
that describe the behavior at infinity of the objective functions f and that generalize
the classical recession functions of convex analysis (Rockafellar [17] and [18]). Moreover, they formulated sufficient conditions for optima based on two assumptions—the
“compatibility” and “compactness” conditions—that are particularly well suited to
the class of models which they were interested in.
The results of [7] have been subsequently improved by Buttazzo and Tomarelli [9]
and Auslender [4] and [5]. For instance, Auslender in [4] shows that under a suitable
modification of the compatibility condition, the basic sufficient conditions of [7] and
[9] turn out to be necessary as well. However, [4] keeps the compactness condition
required by [7] and [9].
In this paper we introduce necessary and sufficient conditions for the existence of
optima of functionals defined on reflexive spaces that rely on the asymptotic behavior
of minimizing sequences.1 Unlike in [4], they are not formulated through the recession
function, and no compactness conditions are required.
This is accomplished in section 2, where two alternative equivalent conditions are
presented, that is, conditions (A) and (R). Condition (A), though technical, is closely
∗ Received by the editors March 26, 2010; accepted for publication (in revised form) September
8, 2010; published electronically January 11, 2011. The financial support of ERC (advanced grant,
BRSCDP-TEA) is gratefully acknowledged.
http://www.siam.org/journals/siopt/21-1/79026.html
† Department of Decision Sciences, Dondena, and Igier, Università Bocconi, Milan, Italy (massimo.
[email protected]).
‡ Department of Statistics and Applied Mathematics and Collegio Carlo Alberto, Università di
Torino, Italy ([email protected]).
1 Though [7] provides conditions for generic topologies compatible with the pairing, here we
prefer to consider reflexive spaces endowed with the weak topology and sequentially weakly lower
semicontinuous functionals. In any case, most of our results also hold when X is the dual of a
separable Banach space, endowed with the weak∗ topology.
174
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NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMA
175
related to the compatibility condition of [7], whereas condition (R) is simpler and more
natural, though more difficult to check. Our main general existence result, Theorem 1,
shows that (A) and (R) are equivalent necessary and sufficient conditions for optima.
The rest of the paper is devoted to the application of Theorem 1, which provides
a general perspective that allows one to establish novel results, as well as to recover
known results, as detailed in section 4.1. In particular, in section 3 we present sufficient
conditions for optima based on a few notions on the asymptotic behavior of sets and
functions (briefly reported in the appendix). Most of these asymptotic notions are
well known,2 with the exception of the set
w
= {d ∈ V : ∃ {vn }n ⊆ C with vn → ∞ and vn / vn d}
BC
(2)
of the asymptotic weak directions of a set C.
The main result of section 3 is Theorem 4, which provides sufficient conditions for
optima based on the retractivity property of the functional along the weak asymptotic
directions of the minimizing sequences. In particular, Theorem 4 relies on condition
(N) that prevents undesirable asymptotic directions for minimizing sequences.
Theorem 4 is, along with Theorem 1, the main result of the paper. To derive
it we introduce finitely well-positioned sets, which generalize the well-positioned sets
recently introduced by Adly, Ernst, and Thera [1] and [2]. In section 3 we outline a few
important properties that these sets enjoy, and we discuss their relations with other
notions studied in the literature.3 The study of finitely well-positioned sets allows us
to prove Theorem 4, as well as to establish in Theorem 10 some more transparent
sufficient conditions of optimality that show how our sufficient conditions are related
to those of [7], but without their rather strong compactness assumption.
In section 4 we revisit the classical compactness case by establishing some extensions and variants of the original approach of [7]. Finally, in section 5 we apply our
results to some variational problems discussed in the literature. Appendix A contains
some auxiliary material that makes the paper self-contained.
2. General results. Throughout the paper V is a normed vector space
with norm ·. We denote by BV its unit ball {v ∈ V : v ≤ 1} and by SV =
{v ∈ V : v = 1} its unit sphere. Norm convergence of a sequence will be denoted by
xn → x, and xn x indicates weak convergence.
For a set C, χC denotes its indicator function; i.e., χC (x) = 0 if x ∈ C,
/ C. For ease of notation, throughout the paper for funcand χC (x) = +∞ if x ∈
tions f : V → (−∞, ∞] we write arg min f and inf f in place of arg minv∈V f (v)
and inf v∈V f (v), respectively. Even when not mentioned, all functions are proper;
i.e., f (v) < +∞ for some v ∈ V. The notation (f ≤ λ) denotes the sublevel set
{v ∈ V : f (v) ≤ λ}. For brevity, we use sw-lower and sw-closed in place of “sequentially weakly lower” and “sequentially weakly closed,” respectively.
A function f : V → (−∞, ∞] is
(i) sw-lower semicontinuous if all (f ≤ λ) are sw-closed,
(ii) lower semicontinuous if all (f ≤ λ) are norm closed,
(iii) coercive if there is a nonempty sublevel set (f ≤ λ) that is norm bounded.4
We give two equivalent general conditions, (A) and (R), that will turn out to be
necessary and sufficient for existence of optima of functionals f : V → (−∞, ∞]. Both
2 See,
e.g., Rockafellar [18] and Auslender and Teboulle [6].
[14] we carry out a detailed analysis of this class of sets.
4 This notion is weaker than the standard one. Our definition implies that f + χ
(f ≤λ) is coercive
in the usual sense (at least in the reflexive case).
3 In
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176
MASSIMO MARINACCI AND LUIGI MONTRUCCHIO
are conditions on unbounded minimizing sequences, that is, sequences {xn } such that
xn → ∞ and xn ∈ (f ≤ λn ) with λn ↓ inf f .
(A) For any unbounded minimizing sequence {xn }, there is a scalar sequence {αk }
and vector sequences {yk } and {zk } such that, for all k large enough,
(i) yk ≤ xnk , αk ∈ (0, yk ], and zk − yk / yk < 1,
(ii) f (yk − αk zk ) ≤ f (xnk ).
Observe that the sequence {zk } is necessarily bounded with each zk = 0.5
(R) For any unbounded minimizing sequence {xn }, there is a sequence {yk } such
that, for all k large enough, yk < xnk and f (yk ) ≤ f (xnk ).
We use the letter “R” because of the retractive flavor of this condition. We can
now state our general existence result from which most subsequent results will follow.
Theorem 1. Let f : V → (−∞, ∞] be an sw-lower semicontinuous function
defined on a reflexive space V . The following properties are equivalent:
(i) arg min f is nonempty.
(ii) f satisfies (A).
(iii) f satisfies (R).
The proof uses the following elementary geometric fact, observed by [7] and stated
here for later reference.
Lemma 2. Let d ∈ BV and x ∈ V . Then d − x < 1 implies d − tx < 1 for
each t ∈ (0, 1].
Proof. It is enough to observe that
1 ≤ d − tx = (1 − t) d + t (d − x) ≤ (1 − t) + t d − x
implies d − x ≥ 1.
Proof of Theorem 1. (R) implies (A). Let {xnk } and {yk } be the sequences that
exist by (R). Then there is a new sequence {yk } and 0 ≤ β k < 1 such that yk = β k yk
with yk = xnk . Notice that yk is uniquely determined if yk = 0, whereas if
yk = 0, one can pick any vector yk with yk = xnk and setting β k = 0. We have
yk
.
yk = β k yk = yk − yk (1 − β k )
yk By assumption,
f
(yk )
yk
= f yk − yk (1 − β k )
≤ f (xnk ) .
yk Setting zk = (1 − β k ) yk / yk and αk = yk , we get f (yk − αk zk ) ≤ f (xnk ), and
then (A) holds because zk − yk / yk = β k < 1.
(A) implies arg min f = ∅. Let λ = inf f and consider a sequence λn ↓ λ. Set
Cn = (f ≤ λn ) for all n ≥ 1. Since f is sw-lower semicontinuous, each Cn is swclosed and nonempty. The norm · is coercive and sw-lower semicontinuous. Hence,
arg minv∈Cn v = ∅ because V is reflexive. Pick a sequence
(3)
xn ∈ arg min v .
v∈Cn
Suppose the monotone sequence {xn } is bounded. Since V is reflexive, there is a
subsequence {xnk } that weakly converges to some x∗ ∈ V . It is easy to check that
x∗ ∈ ∩n Cn . Hence, f (x∗ ) ≤ λn for all n, and so f (x∗ ) = inf f .
5 An important remark on terminology: throughout the paper it will often be claimed the existence
of a sequence {yk } associated with a given sequence {xn }. The important special case yk = xnk
amounts to claiming the existence of a subsequence of {xn } having certain properties.
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NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMA
177
It remains to prove that the sequence {xn } is bounded. Suppose, per contra,
that xn ↑ ∞. By condition (A), there are sequences {yk }, {zk }, and {αk } that satisfy
properties (i) and (ii) in (A). Hence, f (yk − αk zk ) ≤ f (xnk ) for k large enough, and
so, by (3), yk − αk zk ≥ xnk ≥ yk . In turn, this implies
yk
xnk αk
−
z
k
yk yk ≥ yk ≥ 1.
On the other hand, by assumption, zk − yk / yk < 1 and αk / yk ∈ (0, 1]. Hence,
by Lemma 2,
yk
αk
−
z
yk yk k < 1,
a contradiction. We conclude that the sequence {xn }n is bounded.
Finally, to see that arg min f = ∅ implies (R), it suffices to pick yk ≡ x ∈
arg min f .
Theorem 1 provides, via the necessary and sufficient conditions (A) and (R),
a complete characterization of the existence of optima of sw-lower semicontinuous
functions on reflexive spaces without any compactness assumption. This is a major
difference relative to the main general existence results on reflexive spaces available
in the literature (e.g., Baiocchi et al. [7] and Auslender [4]), which instead hold under
suitable compactness conditions. These related results will be discussed in section 4.1.
3. Sufficient conditions for optima. The determination of the sequence {zk }
in condition (A) is, in general, not an easy task. When V is finite dimensional, the
vectors yk / yk belong to the compact unit sphere, and so we can suppose that
yk / yk → d ∈ SV . Moreover, when yk = xnk , the vector d belongs to Sf ; i.e.,
d is a normalized asymptotic direction of f (see Appendix A). Hence, in the finitedimensional case the constant sequence zk = d—or also zk = λd with λ ∈ (0, 2)—turns
out to be a natural choice for the sequence {zk }.
This is no longer the case when V is infinite dimensional. A reasonable substitute in reflexive spaces should be yk / yk d. But, in general, we only have
yk / yk − d ≤ 2. In this section we study this issue and establish conditions that
guarantee the desirable property yk / yk − d < 1. Since the analysis is simpler and
more general on Hilbert spaces, we begin with this important case.
(B) For any unbounded minimizing sequence {xn }, there is an unbounded sequence {yk }, with yk / yk d = 0, and a scalar sequence {αk } such that,
for k large enough,
(i) yk ≤ xnk ,
(ii) f (yk − αk d) ≤ f (xnk ) and αk ∈ (0, (2 − ε) yk ] with ε > 0.
When yk = xnk and αk ≡ α, condition (ii) in (B) is closely related to the compatibility assumption of Baiocchi et al. [7] (see also section 4.1). As in Bertsekas and
Tseng [8] and Ozdaglar and Tseng [16], we can also say that f retracts along d. The
asymptotically linear notion of Auslender [5, Definition 6] is also equivalent.6
Next we show that condition (B) ensures the existence of optimal points for
functions on Hilbert spaces.
Theorem 3. Let f : V → (−∞, ∞] be a sw-lower semicontinuous function defined
on a Hilbert space V . Then arg min f = ∅ under condition (B).
6 Even
the class F in [5] is closely related, though stronger, to the notion of retraction for functions.
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178
MASSIMO MARINACCI AND LUIGI MONTRUCCHIO
Proof. In view of Theorem 1, it is enough to prove that (B) implies (A). Let {xn }
be an unbounded minimizing sequence. Let {yk } and {αk } be sequences that satisfy
2
(B). Set wk = yk / yk , and, given any λ ∈ R, consider wk − λd . We have
2
2
wk − λd = 1 − 2λ wk , d + λ2 d ,
where ·, · indicates the inner product. Therefore, limk wk − λd
=
2
1 − λ (2 − λ) d . Let λ ∈ (0, 2). Since d = 0, it follows that limk wk − λd < 1,
and so wk − λd < 1 for k large enough. Set zk = λd and β k = λ−1 αk . Since
β k λ = αk ∈ (0, (2 −
β k ∈ (0, yk ] by setting λ = 2 − ε. Moreover,
ε) yk ], we have
f (yk − β k zk ) = f yk − λ−1 αk λd = f (yk − αk d) ≤ f (xnk ) for k large enough. We
conclude that (B) implies (A).
The technique used in Theorem 3 is no longer valid when V is not a Hilbert space.
In fact, in this case we have to restrict the analysis to the case where the choice of
sequences is of the kind yk = xnk . The following condition is key.
(N) Given any unbounded sequence {xn }, if xn / xn 0, then lim supn f (xn ) ≥
f (v) for some v ∈ V .
When arg min f = ∅, condition (N) trivially holds by taking v ∈ arg min f . Therefore, condition (N) is necessary for optima. The next result, the main result of this
section, shows what additional properties make it sufficient. The proof will be given
later, after the introduction of some additional notions. As a matter of fact, behind
the next theorem there is the interesting class of finitely well-positioned sets that we
will discuss momentarily.
Theorem 4. Let f : V → (−∞, ∞] be an sw-lower semicontinuous function
defined on a reflexive space V . Then arg min f = ∅ if f satisfies condition (N), and,
for any unbounded minimizing sequence {xn }, there is a subsequence {xnk } and a
scalar sequence {αk } such that
(i) xnk / xnk d,
(ii) f (xnk − αk d) ≤ f (xnk ) with αk ∈ (0, xnk ].
If, in addition, V is Hilbert, then the sequence {αk } can be chosen in
(0, (2 − ε) xnk ] with ε > 0.
3.1. Finitely well-positioned sets. We outline some concepts that are relevant
for our problem of existence of optima. Though the subject may be of independent
interest, it will be here used to complete the proof of Theorem 4. A detailed analysis
is given in [14], which we refer the interested reader to.7
Following [11], we say that a set C of a normed space V allows plastering if a
uniformly positive linear functional exists over C, that is, if there is x∗ ∈ V ∗ such
that x∗ , x ≥ x for all x ∈ C. Though C can be any set, convex cones are the
natural domain for this concept. In fact, it is easily seen that C allows plastering
if and only if its conical hull does. If one considers the Bishop-Phelps cone Kx∗ =
{x : x∗ , x ≥ x}, a set C allows plastering if and only if C ⊆ Kx∗ for some x∗ ∈ V ∗ .
Recently, Adly, Ernst, and Thera [1] and [2] reconsidered this property, under a
different name. In particular, they call a set C well-positioned if there are x0 ∈ V and
x∗ ∈ V ∗ such that x∗ , x − x0 ≥ x − x0 for all x ∈ C. This notion is the property
of allowing plastering for the translated set C − {x0 }. Equivalently, C ⊆ x0 + Kx∗ .
Many nice properties of well-positioned convex sets in reflexive spaces are studied in
[1] and [2].
7 However, to make the paper self-contained, the proofs of Proposition 6 and of Theorem 7 are
given in the appendix.
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NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMA
179
The following weakening of the notion of well-positioned sets will play a key role
in what follows.
Definition 5. A set C of a normed space is said to be finitely well-positioned if
C ⊆ ∪ni=1 Ci , where each Ci is well-positioned.
Finitely well-positioned sets can be viewed as the union of a bounded set and of
finitely many sets that allow plastering, as the following result shows (see [14]).
Proposition 6. An unbounded set C of a normed space is finitely well-positioned
if and only if, for ρ large enough, C ∩ {x ≥ ρ} = ∪ni=1 Ci , where each Ci allows
plastering.
The close link between finitely well-positioned sets and our present theory is
substantiated by the following important characterization of finitely well-positioned
sets (see [14]).
Theorem 7. Let C ⊆ V with V reflexive. The following properties are equivalent:
(i) C is finitely well-positioned.
w
; i.e., there is no unbounded sequence {xn } ⊆ C such that xn / xn (ii) 0 ∈
/ BC
0.
(iii) For any unbounded sequence {xn } ⊆ C, there is a subsequence {xnk } and a
scalar sequence {tk } such that tk → ∞ and xnk /tk d = 0.
That (ii) implies (i) is the nontrivial part of this theorem (the converse actually
holds for any normed space). It follows, for instance, that every set that lives in a
finite-dimensional space is finitely well-positioned.
Point (iii) of Theorem 7 clarifies the relations between finitely well-positioned
sets and other concepts introduced in literature to describe the asymptotic behavior of sets (see [13] and the references therein). From (iii) it follows that the class
of finitely well-positioned sets coincides with the one of recessively weakly compact
sets introduced by [13] when V is reflexive.8 However, the geometric characterization
behind our definition of finitely well-positioned sets will be crucial in our proofs.
The following is another immediate consequence of Theorem 7 (see also [13,
Proposition 2.3(ii)]).
Corollary 8. A set C of a reflexive space V is bounded if and only if it is finitely
well-positioned and Aw
C = {0}.
Proof. Suppose that C is finitely well-positioned and Aw
C = {0}, and suppose, per
w
w
contra, that C is unbounded. As V is reflexive, BC
= ∅. Hence, BC
= {0}, but this a
contradiction by Theorem 7. The converse implication is obvious.
3.2. A more handy sufficient condition. Next we characterize functions that
satisfy the condition (N) previously introduced.
Proposition 9. A function f : V → (−∞, ∞] defined on a reflexive space V
satisfies condition (N) if and only if at least one of the following conditions holds:
(i) arg min f = ∅.
(ii) A nonempty sublevel set (f ≤ λ) is finitely well-positioned.
An implication of this result is that a nontrivial sufficient condition for (N) to
hold is that some nonempty set (f ≤ λ), with λ > inf f , is finitely well-positioned.9
This is the first fact that shows, for our purposes, the importance of this
of sets.
class
Proof. If arg min f = ∅, clearly f satisfies (N). Suppose that f ≤ λ = ∅ is
finitely well-positioned for some λ. We can suppose λ > inf f . Otherwise, we have
8 More precisely, C is recessively weakly compact if it is recessively compact according to the
definition in [13], where V is endowed with the weak topology.
9 For convex functions, the existence of a finitely well-positioned sublevel set (f ≤ λ) implies that
every (f ≤ λ) is finitely well-positioned (even its epigraph is finitely well-positioned; see [14]).
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180
MASSIMO MARINACCI AND LUIGI MONTRUCCHIO
arg min f = ∅, which implies the result. Suppose by contradiction that (N) does
not hold. Then there is an unbounded sequence {xn } suchthat xn / xn 0 and
lim supn f (xn ) ≤ inf f . Since λ > inf f , eventually, xn ∈ f ≤ λ . By Theorem 7,
f ≤ λ would not be finitely well-positioned, a contradiction.
Conversely, suppose that (N) holds for f . Assume, per contra, that neither (i) nor
(ii) holds; i.e., arg min f = ∅ and no sublevel (f ≤ λ) is finitely well-positioned. Let
w
∞
w
λn ↓ inf f . Then 0 ∈ B(f
≤λn ) for all n; i.e., 0 ∈ ∩n=1 B(f ≤λn ) . By Proposition 26(ii),
∞
n=1
w
w
B(f
≤λn ) = B ({(f ≤ λn )}n ) .
Hence, 0 ∈ B w ({(f ≤ λn )}n ); i.e., there is an unbounded sequence {xn } such that
f (xn ) ≤ λn and xn / xn 0. By construction, lim supn f (xn ) ≤ inf f . Since f
satisfies (N), then there is v such that f (v) = inf f . Hence, arg min f = ∅, a contradiction.
Thanks to this result we can formulate more handy sufficient conditions that imply Theorem 4. These conditions make easier the comparison of our results with the
w
is the weak asymptotic funcclassical ones of [7]. Notice that in this theorem f∞
tion (1).
Theorem 10. Let f : V → (−∞, ∞] be an sw-lower semicontinuous function
defined on a reflexive space V . Then arg min f = ∅ if the following conditions hold:
(i) (f ≤ λ) is finitely well-positioned for some λ > inf f .
w
≥ 0.
(ii) f∞
w
(iii) For all d ∈ ker f∞
such that xn / xn d, xn → ∞, and xn ∈ (f ≤ λ),
there is α > 0 such that f (x − αd) ≤ f (x) for all x with x large enough.
Notice that (i) implies condition (N), and (ii) is the classical necessary condition
w
of optimality that implies Aw
f ⊆ ker f∞ (see (11) in Appendix A). Clearly, condition
f (x − αd) ≤ f (x) implies (ii) of Theorem 4.
3.3. Proof of Theorem 4. We close this section by proving the main theorem
(Theorem 4). Its proof relies on Proposition 9 and on the next lemma. Let K =
∪ni=1 Kx∗i , where Kx∗i are Bishop–Phelps cones in V . Through this cone and a scalar
β > 0, we can endow V with the following equivalent norm:
(4)
x1 = x + β
n
|x∗i , x| .
i=1
−1
Lemma 11. If x ∈ K ∩ {x1 ≤ 1}, then x ≤ (1 + β) .
Proof. Let x ∈ K ∩ {x1 ≤ 1}. Then x ∈ Kx∗i for some i. Therefore,
1 ≥ x1 = x + β
n
∗ x , x ≥ x + β x∗ , x ≥ x + β x .
j
i
J=1
−1
Consequently, x ≤ (1 + β) .
Proof of Theorem 4. Suppose first that V is Hilbert. If there exists an unbounded
minimizing sequence {xn } that has a subsequence {xnk } for which xnk / xnk 0,
then (N) implies inf f = lim supk f (xnk ) ≥ f (v) for some v ∈ V , and so arg min f = ∅.
If there are no unbounded minimizing sequences with this property, the result then
follows from Theorem 3.
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NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMA
181
Suppose that V is not necessarily Hilbert. By Proposition 9, either arg min f = ∅
or there is a finitely well-positioned sublevel (f ≤ λ) with λ > inf f . Hence, we can
suppose that (f ≤ λ) is finitely well-positioned. Notice further that we can assume
(f ≤ λ) to be unbounded; otherwise, the theorem trivially holds. By Proposition 6,
n
Kx∗i .
f ≤ λ ∩ {x ≥ ρ} ⊆ K =
i=1
We introduce the equivalent norm (4) associated with the cone K. We will prove that
condition (A) holds under the norm ·1 by taking yk = xnk and with zk as the weak
limit of xnk / xnk 1 . Specifically, consider any minimizing sequence {xn } such that
xn 1 → ∞. Clearly, this is equivalent to xn → ∞. Hence, it is not restrictive
to assume {xn } ∈ (f ≤ λ) ∩ {x ≥ ρ} ⊆ K. By reflexivity, there is a subsequence
{xnk } such that xnk / xnk 1 d. Moreover, xnk / xnk 1 ∈ K. Set uk = xnk / xnk 1 .
Lemma 11 implies
uk − d1 = uk − d + β
n
|x∗i , uk − d| ≤ uk + d + β
i=1
−1
≤ 2 (1 + β)
+β
n
n
|x∗i , uk − d|
i=1
|x∗i , uk − d| .
i=1
n
As i=1 |x∗i , uk − d| → 0 as k → ∞, it follows that uk − d1 < 1 for k large enough,
provided β > 1. Notice further that 0 < αk ≤ xnk ≤ xnk 1 if αk ∈ (0, xnk ]. This
proves that condition (A) holds, and so, by Theorem 1, the proof is completed.
4. Asymptotic compactness conditions. In this section we briefly study
some compactness conditions that extend those used by Baiocchi et al. [7] (see also
[9] and [4]). The next condition is the version of condition (B) in which norm convergence replaces the weak one. This is a major difference that gives the condition a
compactness flavor.
(C) For any unbounded minimizing sequence {xn }, there is an unbounded sequence {yk }, with yk / yk → d, and a scalar sequence {αk } such that, for k
large enough,
(i) yk ≤ xnk ,
(ii) f (yk − αk d) ≤ f (xnk ) and αk ∈ (0, (2 − ε) yk ] with ε > 0.
Proposition 12. Let f : V → (−∞, ∞] be an sw-lower semicontinuous functional defined on a reflexive space V . Under (C), arg min f = ∅.
Proof. In view of Theorem 1, it is enough to prove that (C) implies (A). Set
zk = λd with λ ∈ (0, 2). Observe that yk / yk − λd ≤ yk / yk − d + |1 − λ|.
Therefore, yk / yk − λd < 1 for k large enough and λ ∈ (0, 2). It is easy to see
that this is equivalent to the condition αk ∈ (0, (2 − ε) yk ] for some ε > 0.
Next consider the following compactness condition.
(K) For any unbounded minimizing sequence {xn }, there is a sequence {yk } such
that yk / yk norm converges, and, for all k large enough, f (yk ) ≤ f (xnk )
and yk ≤ xnk .
Under this compactness condition we can prove the following result, which relies on conditions (5) and (6).10 The proof shows that these conditions imply our
10 These
conditions use some asymptotic concepts for functions collected in the appendix.
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182
MASSIMO MARINACCI AND LUIGI MONTRUCCHIO
basic condition (A), and so also this result follows from our main existence result,
Theorem 1.
Proposition 13. Let f be an sw-lower semicontinuous function defined on a
reflexive space V . If f satisfies (K), then arg min f = ∅, provided
Sf ⊆ −cone (Pf ) ,
(5)
or, equivalently,
Af = cone (Pf ) = −cone (Pf ) .
(6)
Moreover,
(7)
Aarg min f = Af = cone (Parg min f ) = −cone (Parg min f ) ,
and Af = {0} if and only if arg min f is bounded.
Proof. We prove that the conditions of this theorem imply (A). Let {xn } be an
unbounded minimizing sequence. Set f (xn ) = λn . By (K), there is a sequence {yk }
such that yk ∈ (f ≤ λnk ), xnk ≥ yk , and yk / yk → d ∈ SV . By Proposition 28,
d ∈ Sf . By (5), d ∈ cone (−Pf ); i.e., −αd ∈ Pf for some α > 0. This implies yk − αd ∈
(f ≤ λnk ); i.e., f (yk − αd) ≤ f (xnk ). As yk / yk − d → 0, yk / yk − d < 1
holds for k large enough. Hence, (A) is satisfied.
Let us prove that (5) and (6) are equivalent. From Sf ⊆ −cone (Pf ), it follows
that cone (Sf ) = Af ⊆ −cone (Pf ). Hence,
(8)
cone (Pf ) ⊆ Af ⊆ −cone (Pf ) .
Clearly, cone (Pf ) ⊆ −cone (Pf ) implies −cone (Pf ) ⊆ cone (Pf ). Hence, cone (Pf ) =
−cone (Pf ). By (8), we get (6). As to the converse, it is enough to observe that, by
(6), Sf ⊆ Af = −cone (Pf ).
By Propositions 28 and 29, arg min f = ∅ implies Aarg min f ⊆ Af and Parg min f =
Pf . Hence, Aarg min f ⊆ Af = cone (Parg min f ) = −cone (Parg min f ). On the other
hand, Aarg min f ⊇ cone (Parg min f ) and so (7) holds.
Clearly, Af = {0} implies Aarg min f = {0}. Let us prove that arg min f is bounded.
Suppose, per contra, that arg min f is unbounded. Then there is a sequence xn ∈
arg min f such that xn → ∞. As f (xn ) ↓ inf f , by (K), there is a sequence {yk } ⊆
arg min f such that yk / yk → d ∈ SV . Hence, d ∈ Aarg min f , which contradicts
Aarg min f = {0}. Conversely, arg min f bounded implies Aarg min f = {0}. By (7), we
then have Aarg min f = Af = {0}. This completes the proof.
4.1. Related works. Let us compare our results with those of Baiocchi et al.
[7]. Translated into our setting, the sequential version of their main existence result,
[7, Theorem 3.9], shows the existence of optima for sw-lower semicontinuous f on a
reflexive space V under the following assumptions:
(B1 ) Compactness: let {zn } be a sequence such that zn d. If tn ↑ ∞ and
f (tn zn ) ≤ λ for all n and some λ, then zn → d.
(B2 ) Compatibility: (i) fw∞ ≥ 0; (ii) if d ∈ ker fw∞ , then there is α > 0 such that
f (x − αd) ≤ f (x) for all x ∈ V .
We show how their result can be seen as a particular case both of Theorem 10 and
of Proposition 12. In particular, point (i) of Proposition 14 implies that [7, Theorem
3.9] is a consequence of Theorem 10, and point (iii) implies that it also follows from
Proposition 12.
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NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMA
183
Proposition 14. LetV be reflexive.
Then
(i) (B1 ) implies that f ≤ λ is finitely well-positioned;
(ii) (B1 ) implies (K);
(iii) (B1 ) and (B2 ) imply
(C).
Proof. (i) Let xn ∈ f ≤ λ , xn → ∞, and xn / xn d. Set zn = xn / xn and tn = xn . We have tn zn ∈ f ≤ λ and zn d. By (B1 ), xn / xn → d. Hence,
d = 1 and d = 0. By Proposition 7, f ≤ λ is finitely well-positioned.
(ii) Suppose
(B1 ), and let {xn } be an unbounded minimizing sequence. Clearly,
{xn } ⊆ f ≤ λ for n large enough. Since V is reflexive, there is a subsequence of
{xnk }k such that xnk / xnk weakly converges to some d ∈ V . Set tk = xnk and
zk = xnk / xnk . By (B1 ), zk = xnk / xnk norm converges to d. Condition (K) is
thus satisfied by setting yk = xnk for all k.
(iii) Suppose (B2 ), and let {xn } be an unbounded minimizing sequence. Set λn =
f (xn ) so that xn ∈ (f ≤ λn ) and λn ↓ inf f . By the reflexivity of V , there is a
w
subsequence xnk / xnk d ∈ Aw ((f ≤ λn )) ∈ ker f∞
. By (B2 ), f (xnk − αd) ≤
f (xnk ) for nk . Hence, point (ii) of condition (C) is satisfied by setting yk = xnk and
αk = α. The rest of condition (C) follows from (B1 ) , as easily seen by proceeding as
in (i).
We show that our compactness condition (K) is weaker than (B1 ).
Example 15. Given the Hilbert space l2 , consider the function f : l2 → R given
by f (x) = −e−x . If 0 > λ > −1, the set (f ≤ λ) is nonempty and bounded, and in
turn, this implies that (K) is trivially satisfied. Consider the sequence {en }n of the
standard orthonormal basis in l2 . Clearly, en 0, though the sequence en is not
strongly convergent. On the other hand, if tn ↑ ∞, we have supn f (tn en ) = 0. This
shows that this function does not satisfy the compactness condition (B1 ).
5. More examples.
5.1. Nested sequences of sets. Using the results derived in the paper, we can
state sufficient conditions for the nonemptiness of nested sequences of sets. Following
Bertsekas and Tseng [8, Definition 1], we introduce the following definitions.
Definition 16. Given a nested sequence of sets {Cn } and d ∈ B w ({Cn }), we
say that {Cn } retracts along d if, for any unbounded sequence xk ∈ Cnk such that
xk / xk d, there exists a subsequence {xkr } and a bounded sequence αr > 0 such
that xkr − αr d ∈ Cnkr for all r.
A nested sequence {Cn } is called retractive if it retracts along all d ∈ B w ({Cn }).11
Notice that when Cn = C for all n, analogous definitions hold for a fixed set C
w
and where B w ({Cn }) ≡ BC
. The next result is an infinite-dimensional extension of
[8, Proposition 1].
Proposition 17. A retractive nested sequence of nonempty sw-closed sets of
a reflexive space has nonempty intersection provided at least one of the following
conditions holds:
/ Cm for all n ≥ n0 and some m.
(i) xn → ∞ and xn / xn 0 implies xn ∈
(ii) Some Cm is finitely well-positioned.
∞
Proof. Let α = n=1 αn be a summable series
∞with αn > 0 for each n. Consider the sw-lower semicontinuous function f = − n=1 αn 1Cn . Clearly, inf f = −α,
and the inf is attained if and only if ∩n Cn = ∅. We can apply Theorem 4 to
the function f . Points (i) and (ii) are clearly equivalent to condition (N). Consider
11 We can also define retractivity along the directions d of Aw ({C }) in place of those in
n
B w ({Cn }). This possibility, for brevity not considered here, is pursued in [14].
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184
MASSIMO MARINACCI AND LUIGI MONTRUCCHIO
unbounded minimizing sequences {xk }. We can suppose that there exist such sequences; otherwise, the result is trivial. Moreover, it is not restrictive to suppose that
they all satisfy f (xk ) > −α = inf f for all k. Otherwise, f (xk0 ) = −α for some k0 ,
and thus, xk0 ∈ ∩n Cn and we are done. Consequently, with an unbounded minimizing sequence {xk }, we can associate a subsequence {Cnk } such that xk ∈ Cnk and
/ Cnk+1 . Passing to a subsequence if necessary, we can suppose that xk / xk d.
xk ∈
Such a subsequence is still denoted by {xk }. By the hypothesis of retractivity for the
sequence {Cn }, we have xkr − αr d ∈ Cnkr for a subsequence {xkr } and a bounded
sequence αr > 0. Hence, f (xkr − αr d) ≤ f (xkr ) and αr ∈ (0, xkr ], since αr are
bounded. Theorem 4 concludes the proof.
We give an application of this result to the closedness of linear images, a problem
well studied in the literature.12
Proposition 18. Let T : V → W be a continuous linear mapping with V reflexive. If C ⊆ V is an sw-closed finitely well-positioned set, then T (C) is sw-closed,
w
∩ ker T .
provided C retracts along all directions d ∈ BC
Proof. Suppose first that W is separable and yn ∈ T (C) with yn y. We must
show that y ∈ T (C). The sequence is bounded; i.e., yn ∈ ρBW . As W is separable, it
is weakly metrizable over the bounded set ρBW . Denote by δ such a metric. Consider
the sequence of sets in W given by
Wn = {y ∈ W : δ (y, y) ≤ δ (yn , y) and y ∈ ρBW } .
As δ (yn , y) → 0, passing to a subsequence, we can suppose δ (yn , y) ↓ 0. Hence, the
sequence Cn = C ∩ T −1 (Wn ) in V decreases. Clearly, the sets Wn are weakly closed,
and thus, the Cn are
sw-closed and nonempty by construction. If their intersection is
∞
nonempty and x ∈ n=1 Cn , then T (x) = y.
w
w
Let us prove that B w ({Cn }) ⊆ BC
∩ ker T . Let d ∈ B w ({Cn }); that is, d ∈ BC
n
w
for all n. Clearly, d ∈ BC . Let {xm } be a sequence such that {xm } ⊆ Cn , xm → ∞,
and xm / xm d. Consequently, T xm / xm T d. On the other hand, T (xm ) ∈
ρBW , so it is bounded. We have T (d) = 0 and d ∈ ker T , and thus, the inclusion
w
∩ ker T is proved.
B w ({Cn }) ⊆ BC
w
If BC ∩ ker T = ∅, then B w ({Cn }) = ∅. That means that the sets {Cn } are
bounded because Cn are finitely well-positioned. Clearly, in this case their intersection
is not empty since V is reflexive. We can thus assume {Cn } to be unbounded, and so,
w
∅ = B w ({Cn }) ⊆ BC
∩ ker T .
To invoke Proposition 17, we must show that the sequence {Cn } is retractive.
Take any d ∈ B w ({Cn }), and let {xn } be a sequence such that xn ∈ Cn , xn → ∞
w
and xn / xn d. Clearly, d ∈ BC
∩ ker T . By hypothesis, C retracts along the
w
directions in BC ∩ ker T . Hence, xnk − αk d ∈ C for a subsequence {xnk } and for a
sequence {αk }. On the other hand, T (xnk − αk d) = T (xnk ) ∈ T (Cnk ). It follows
that xnk − αk d ∈ Cnk . We have proved that Cn are retractive, and thus, ∩∞
n=1 Cn = ∅,
provided W is separable.
Suppose now that W is any space and yn y with yn ∈ T (C). It suffices
to consider the separable linear space W1 = span {yn }n and the linear mapping
T : T −1 (W1 ) → W1 . It is easy to see that all the conditions of the result hold for the
set C ∩ T −1 (W1 ), and so, the result follows from the first part of the proof.
12 In view of the equivalence of finitely well-positioned sets with recessively weakly compact sets,
the next result is partly related to [13, Corollary 3.6].
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NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMA
185
Proposition 18 has the following interesting subcases:
w
(i) BC
∩ ker T = ∅ if and only if Aw
C ∩ ker T = {0}. An example is when T is
injective.
(ii) Aw
C ∩ ker T ⊆ LC .
(iii) C is retractive (e.g., C is an asymptotic convex polyhedron).
w
5.2. Quasi-coercive functions. We call a function f quasi-coercive if f∞
≥0
w
w
and ker f∞ = {0}. By (11), A(f ≤λ) = {0} for all nonempty level sets (f ≤ λ). If the
dimension of V is finite, these functions are clearly coercive. In infinite-dimensional
spaces, quasi coercivity does not even ensure that a function be bounded from below.
Next, we give standard examples of quasi-coercive functionals.
Example 19. Consider a continuous and strictly positive bilinear form a : V ×V →
R defined on a Banach space V ; that is, a (u, v) satisfies
(i) |a (u, v)| ≤ k u v,
(ii) a (u, v) ≥ 0 with a (u, u) = 0 if and only if u = 0.
Fix u∗ ∈ V ∗ . The convex functional f (u) = 2−1 a (u, u) − u∗ , u is quasi-coercive.
w
w
Actually, f∞
(0) = 0 and f∞
(d) = +∞ if d = 0. In general, f may fail to be coercive.13
For instance, the bilinear form a : 2 × 2 → R given by
a (x, y) =
∞
1
xi yi
i2
i=1
yields noncoercive functionals over 2 . Another example is the bilinear map (u, v) →
1
uvdt, defined indifferently over C (0, 1), L∞ (0, 1) , or W 1,p (0, 1) with p ≥ 1.
0
Proposition 20. Let f be sw-lower semicontinuous and quasi-coercive on a reflexive space V . Then arg min f = ∅ if and only if f satisfies (N). In particular, it is
coercive if and only if its sublevel sets are finitely well-positioned.
Proof. Suppose f quasi-coercive. Hence, Aw
(f ≤λ) = {0}. By Proposition 9, (N)
implies that either arg max f = ∅ or some level (f ≤ λ) is finitely well-positioned. In
the latter case, (f ≤ λ) is bounded by Proposition 6. The rest of the proof is obvious.
In view of Example 19, f (u) = 2−1 a (u, u) − u∗ , u + χK satisfies the condition
of Proposition 20 under weaker conditions. For instance, it suffices that a ≥ 0 and
(9)
{u : a (u, u) = 0} {u : u∗ , u ≥ 0} Aw
K = {0} .
w
(d) = − u∗ , d if
In fact, setting g (u) = 2−1 a (u, u) − u∗ , u, we easily obtain g∞
w
w
w
, (9) follows.
a (d, d) = 0, and g∞ (d) = ∞ otherwise. From (g + χK )∞ ≤ g∞ + χAw
K
Proposition 20 generalizes Corollary 3.10 of Baiocchi et al. [7], with its Lions–
Stampacchia-type condition (see [12]) based on a compactness condition stronger
than condition (N). Indeed, under compactness conditions, a weaker notion of quasi
w
.
coercivity can be invoked via the asymptotic function f∞ in place of f∞
Proposition 21. Let f be sw-lower semicontinuous and quasi-coercive on a reflexive space V . Suppose f∞ ≥ 0 with f∞ (d) = 0 if and only if d = 0. Under (K),
arg min f = ∅ and every minimizing sequence is bounded.
Proof. By (11), Af = {0}. Hence, the relation (6) is trivially verified. Proposition
13 provides the first result. Moreover, we can deduce that arg min f is bounded. We
can also easily get the stronger property that all minimizing sequences are necessarily
bounded.
13 Though
f is strictly convex, it is not necessarily uniformly convex.
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186
MASSIMO MARINACCI AND LUIGI MONTRUCCHIO
5.3. Weak coercivity. Following Auslender [4], f is called weakly coercive if
w
w
w
f∞
≥ 0 and ker f∞
⊆ Tf (i.e., f∞
(d) = 0 implies that f is constant on each line with
direction d).
Clearly, quasi-coercive functions are weakly coercive. If f is convex and lower
semicontinuous on a finite-dimensional space, weak coercivity is equivalent to 0 ∈
rint (dom f ∗ ) (see [18, Corollary 13.3.4(b)] and [4]).
w
Proposition 22. If f is weakly coercive, then ker f∞ = ker f∞
= Lf .
w
Proof. We have Tf ⊆ Lf ⊆ ker f∞ ⊆ ker f∞ ⊆ Tf , which implies the result.
This result implies that for convex lower semicontinuous functions weak coercivity
is equivalent to the condition that ker f∞ be a subspace of V (see also [7, Theorem
3.12]).
Proposition 23. Let f be sw-lower semicontinuous and weakly coercive on a
reflexive space V . Then arg min f = ∅ if f satisfies condition (N) or condition (K).
Proof. Suppose (N) holds. Let {xn } be an unbounded minimizing sequence. By the
w
reflexivity of V , there is a subsequence {xnk } such that xnk / xnk d ∈ ker f∞
=
Lf . Hence, f (x + λd) = f (x) for all x and λ ∈ R. Theorem 4 or Theorem 10 provides
the desired result.
Suppose (K) holds. By Proposition 22, Af = Lf . Hence,
Af ⊆ ker f∞ = Lf ⊆ −cone (Pf ) ⊆ Af ,
Af ⊆ ker f∞ = Lf ⊆ cone (Pf ) ⊆ Af ,
and (6) is true. The result follows from Proposition 13.
Example 24. This is the infinite-dimensional version of an example from Auslender
[4]. Consider a bounded linear operator J : V → W between normed spaces. Let
g : W → [−∞, ∞) be quasi-coercive and sw-lower semicontinuous. Define the function
f (x) = g (Jx + v) on V , where v is a fixed element of W such that v ∈ dom g. It is
w
= ker J.
easy to check (see [4]) that f is weakly coercive. Specifically, ker f∞
∗
Example 25. Let f (x) = J (x)−x , x+χC , where J is lower semicontinuous and
convex and C is closed and convex. In this case f is weakly coercive if ker (J∞ − x∗ ) ∩
AC and J∞ (u) ≥ x∗ , u for each u ∈ AC . Such conditions appear in [7, Theorem
3.14] under a Fichera-type compactness condition [10]. Proposition 23 thus generalizes
Baiocchi et al.’s result [7].
Appendix A. Asymptotic behavior.
A.1. Sets. Throughout this section C denotes a subset of a normed space V.
(i) The recession cone RC of C is defined by
RC = {w ∈ V : v + tw ∈ C for all v ∈ C and all t ≥ 0} ,
with the convention R∅ = V .
(ii) The asymptotic cone AC and the weak asymptotic cone Aw
C of C are, respectively, defined by
AC = {z ∈ V : ∃tn → ∞ and {vn }n ⊆ C such that vn /tn → z} ,
Aw
C = {z ∈ V : ∃tn → ∞ and {vn }n ⊆ C such that vn /tn z}
with the convention A∅ = Aw
∅ = V.
(iii) The integer recession cone PC of C is defined by PC = {z ∈ V : C + z ⊆ C}
with the convention P∅ = V .
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NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMA
187
It is easy to check that
AC \ {0} = {0 = z ∈ V : ∃ {vn }n ⊆ C with vn → ∞ and vn / vn → z/ z} .
This observation leads us to consider the following set:
SC = {d ∈ SV : ∃ {vn }n ⊆ C with vn → ∞ such that vn / vn → d} ,
which is the set of normalized asymptotic directions. If C = V , it reduces to the
sphere SV .
The basic properties of these sets can be found in [18, Theorem 8.4].
w
of SC , given by (2)—i.e.,
In section 3 the weak version BC
w
= {d ∈ V : ∃ {vn }n ⊆ C with vn → ∞ and vn / vn d}
BC
w
may not be normalized with,
—plays a central role. Clearly, the vectors d ∈ BC
w
w
possibly, d = 0. We have BC = ∅ when C is bounded. Moreover, BC
⊆ BV and
w
w
w
w
w
= ∅, and
SC ⊆ BC ⊆ AC . It is easy to check that AC = cone (BC ), provided BC
w
w
BC
= ∅ implies Aw
=
{0}.
Whenever
V
is
reflexive
and
C
unbounded,
BC
is a
C
nonempty weakly compact set (see [14]).
The lineality space LC of C is defined by
LC = {w ∈ V : v + tw ∈ C for all v ∈ C and all t ∈ R}
with the convention L∅ = V . It is easy to check that LC is a vector subspace of V
with LC = RC ∩ (−RC ) = RC ∩ R−C . In a similar vein, define the integer lineality
space
TC = {z ∈ V : C ± z ⊆ C} = {z ∈ V : C + z = C} .
The set TC is an integer vector subspace of V with TC = PC ∩ (−PC ) = PC ∩ P−C .
In fact, P−C = {z ∈ V : C − z ⊆ C}. Moreover, C ± z ⊆ C implies that C + z ⊆
C ⊆ C + z; i.e., C + z = C. We therefore have the alternative definition TC =
{z ∈ V : C + z = C}. It holds that LC ⊆ TC with equality if C is closed and convex.
Following Bertsekas and Tseng [8], we can also define asymptotic cones for nested
sequences {Cn }. For instance,
A ({Cn }) = {z ∈ V : ∃tn → ∞ and vn ∈ Cn such that vn /tn → z}
with similar definitions for Aw ({Cn }) and B w ({Cn }). Notice that an equivalent definition is
A ({Cn }) = {z ∈ V : ∃tk → ∞ and vk ∈ Cnk such that vk /tk → z} .
We list some properties.
Proposition 26. (i) A ({Cn }) = ∩n ACn .
w
.
(ii) B w ({Cn }) = ∩n BC
n
w
(iii) A ({Cn }) = cone [B w ({Cn })].
w
w
(iv) Aw ({Cn }) ⊆ ∩n Aw
Cn and A ({Cn }) = ∩n ACn , provided V is reflexive.
If the elements of {Cn } are unbounded and V is reflexive, point (ii) implies that
B w ({Cn }) is nonempty and weakly compact.
Proof. We omit the simple proofs of (i) and (iii). As to (ii), clearly, B w ({Cn }) ⊆
w
∩n BC
. Suppose first that V ∗ is separable. Then BV is metrizable, and we denote
n
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188
MASSIMO MARINACCI AND LUIGI MONTRUCCHIO
w
by δ the metric on BV . Let d ∈ ∩n BC
and εn ↓ 0. For all n, there is vn ∈ Cn
n
such that δ (vn / vn , d) < εn and vn ≥ n. It follows that d ∈ B w ({Cn }). Now
w
, there are sequences {xnm } such that
let V be any normed space. If d ∈ ∩n BC
n
n
n
n
xm / xm d as m → ∞, xm ∈ Cn for all m, and xnm → ∞ as m → ∞. Consider
the separable subspace W = span {xnm }n,m . The dual space W ∗ is separable. By
using the previous result, for a sequence εn ↓ 0, for all n, there is some m(n) such
that δ(xnm(n) /xnm(n) , d) < εn and xnm(n) ≥ n. Hence, xnm(n) /xnm(n) → d in the
σ (W, W ∗ ) topology. But it agrees with the σ (V, V ∗ ) topology relativized on W . Hence,
xnm(n) /xnm(n) d, and thus, d ∈ B w ({Cn }).
w
Then (iv) is easily obtained from (iii) and the fact that BC
are weakly compact
n
sets with nonempty intersection.
Though easy to prove, the next properties are very useful.
Proposition 27. Let {Cα }α be any collection of closed sets with ∩α Cα = C = ∅.
Then AC ⊆ ∩α ACα and PC ⊇ ∩α PCα . In particular, ∩α PCα = PC = AC = ∩α ACα if
PCα = ACα for all α.
A.2. Functions. We can extend to functions f : V → [−∞, ∞) the asymptotic
notions earlier introduced for sets. Specifically, given f : V → (−∞, ∞], we consider
(i) the recession cone Rf , given by Rf = ∩λ>inf f R(f ≤λ) ;
w
w
(ii) the asymptotic cones Af , Aw
f , and Bf , given by Af = ∩λ>inf f A(f ≤λ) , Af =
w
w
w
∩λ>inf f A(f ≤λ) , and Bf = ∩λ>inf f B(f ≤λ) , respectively;
(iii) the integer recession cones Pf , given by Pf = ∩λ>inf f P(f ≤λ) ;
(iv) the lineality space Lf , given by Lf = ∩λ>inf f L(f ≤λ) ;
(v) the integer lineality space Tf , given by Tf = ∩λ>inf f T(f ≤λ) ;
(vi) the asymptotic direction Sf , given by Sf = ∩λ>inf f S(f ≤λ) .
Here the restriction λ > inf f ensures that all sets (f ≤ λ) are nonempty. Moreover, unless either inf f = −∞ or (f ≤ inf f ) = ∅, the two chains {(f ≤ λ)}λ>inf f and
{(f ≤ λ)}λ∈R differ.
We have Rf ⊆ Pf ⊆ Af ⊆ Aw
f and Lf ⊆ Tf with equality if f is lower semicontinuous and quasi-convex. Moreover, Sf = Af ∩ SV and Rf = Pf = Af = {0} if all
sublevel sets of f are bounded.
Next we give some useful properties. Recall that if {Cα }α∈R is a chain, with
Cβ ⊆ Cα if β ≥ α, a countable subchain {Cαn }n with each Cαn nonempty is said to
be cofinal if, for each Cα , some n exists such that Cαn ⊆ Cα .
Proposition 28. We have Af = A ((f ≤ λn )n ) and Bfw = B w ((f ≤ λn )n ),
where (f ≤ λn )n is any cofinal subchain of {(f ≤ λ)}λ>inf f . Moreover, Af ⊇ Aarg min f
if arg min f = ∅, and Aw ((f ≤ λn )n ) ⊆ Aw
f with equality if V is reflexive.
Proof. Clearly ∩n A(f ≤λn ) = ∩λ>inf f A(f ≤λ) since (f ≤ λn ) is a cofinal subchain.
By Proposition 26(i), A (f ≤ λn ) = ∩n A(f ≤λn ) = ∩λ>inf f A(f ≤λ) = Af . If arg min f =
∅, then Aarg min f = ∩λ∈R A(f ≤λ) ⊆ ∩λ>inf f A(f ≤λ) = Af . The proof of the last claim
w
w
is similar since Aw ((f ≤ λn )n ) ⊆ ∩n Aw
(f ≤λn ) = ∩λ>inf f A(f ≤λ) = Af .
A stronger result is true for integer recession cones. Similar relations are satisfied
for Rf , Lf , and Tf .
Proposition 29. It holds that
P(f ≤λ) =
P(f ≤λ) .
(10)
Pf =
λ>inf f
λ∈R
In particular, Pf = Parg min f provided arg min f = ∅.
Proof. Clearly, ∩λ>inf f P(f ≤λ) ⊇ ∩λ≥inf f P(f ≤λ) = ∩λ∈R P(f ≤λ) . Let z ∈
∩λ>inf f P(f ≤λ) ; i.e., z ∈ P(f ≤λ) for all λ > inf f . This means that (f ≤ λ)+z ⊆ (f ≤ λ)
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NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMA
189
for all λ > inf f . Hence, ∩λ>inf f (f ≤ λ) + z ⊆ ∩λ>inf f (f ≤ λ). On the other
hand, ∩λ>inf f (f ≥ λ) = (f ≤ inf f ). Thus, z ∈ P(f ≤inf f ) , and so, ∩λ>inf f P(f ≤λ) ⊆
∩λ≥inf f P(f ≤λ) .
A.3. Asymptotic functions. Though conceptually important, the asymptotic
objects associated with functions discussed in the previous subsection are often hard to
compute. The asymptotic functions extend the basic Rockafellar’s asymptotic function
introduced in the convex setting (see [18] and [17]). The advantage is that they are
simpler to handle, though less informative, aside from the convex case. Definitions
below are due to [7] (here we only treat the sequential case).
Given a proper f : V → [−∞, ∞), define the asymptotic function f∞ by
f∞ (y) = inf lim inf f (tn yn ) /tn : tn → ∞, yn → y
n
w
and the weak asymptotic function f∞
by (1). We have the following known properties:
w
(i) f∞ ≤ f∞ , and
(11)
A(f ≤λ) ⊆ (f∞ ≤ 0) ,
Aw
f
⊆
w
(f∞
w
Aw
(f ≤λ) ⊆ (f∞ ≤ 0) ,
Af ⊆ (f∞ ≤ 0) ,
and
≤ 0) .
w
(ii) inf f > −∞ implies f∞
≥ 0.
w
(iii) If f is convex and lower semicontinuous, then f∞ = f∞
and (f∞ ≤ 0) =
A(f ≤λ) for all λ > inf f .
w
By (ii), f∞
≥ 0 is a necessary condition for the existence of optima. In view of (i)
w
w
and (ii), the cones ker f∞ and ker f∞
will be important when f∞ ≥ 0 and f∞
≥ 0,
respectively.
A.4. Proofs of section 3.1.
Proof of Proposition 6. First, let C be well-positioned; i.e., x∗ , x − x0 ≥ x − x0 for all x ∈ C. Then
x
x0 ∗
∗
∗
∗ x0
+ x ,
−
x , x = x , x − x0 + x , x0 ≥ x .
x x x
As x → ∞, x∗ , x0 / x → 0 and x/ x − x0 / x → 1. It follows that x∗ , x ≥
(1 − η) x for x ≥ ρ large enough. Hence, C ∩ {x ≥ ρ} allows plastering. If
now C is finitely well-positioned, then C = ∪ni=1 Ci , where each Ci is well-positioned.
Therefore, for ρ large enough,
C ∩ {x ≥ ρ} =
n
[Ci ∩ {x ≥ ρ}] ,
i=1
where each Ci ∩ {x ≥ ρ} allows plastering.
As to the converse, suppose first that C ∩ (ρB)c allows plastering, where ρB is the
c
open ball of radius ρ. That is, C ∩ (ρB) ⊆ Kx∗ . We can assume x∗ > 1 so that Kx∗
has a nonempty interior. Fix d ∈ int Kx∗ . Clearly, d is a recession direction of Kx∗ .
Hence, Kx∗ +λd ⊆ Kx∗ and that implies Kx∗ ⊆ Kx∗ −λd. Hence, C ∩(ρB)c ⊆ Kx∗ −λd
for all λ ≥ 0. On the other hand, ρB + λd ⊆ Kx∗ for λ large enough.14 It follows that
c
ρB ⊆ Kx∗ − λd, and so, C ∩ ρB ⊆ Kx∗ − λd. As C ∩ (ρB) ⊆ Kx∗ − λd, we conclude
that C ⊆ Kx∗ − λd if λ is large enough. Namely, C is well-positioned.
14 This
is actually equivalent to (ρ/λ) B + d ⊆ Kx∗ , which is true if d ∈ int Kx∗ .
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190
MASSIMO MARINACCI AND LUIGI MONTRUCCHIO
c
To complete the proof, suppose that C ∩ (ρB) ⊆ ∪ni=1 Kx∗i for some ρ. Clearly,
C ⊆ [Kx∗1 ∪ ρBV ] ∪ (∪ni>1 Kx∗i ). From the earlier part of the proof, we get C ⊆
(∪ni>1 Kx∗i ) ∪ (Kx∗1 − λd), and so, C is finitely well-positioned.
To prove Theorem 7, we need the following lemma.
Lemma 30. Let C be a bounded set of a normed space V with 0 ∈
/ C. Then
w
0∈
/ C if and only if C = ∪ni=1 Ci , where each Ci allows plastering. If, in addition, V
is reflexive or with separable dual, we can replace weak closure with sw-closure.
w
Proof. Suppose that C = ∪ni=1 Ci , where each Ci allows plastering. Let d ∈ C .
w
w
w
w
As C = (∪ni=1 Ci ) = ∪ni=1 Ci , it follows that d ∈ Ci for some i. Let xα d be
∗
a net with {xα } ⊆ Ci . Since Ci allows plastering, u , xα ≥ xα for some u∗ ∈ V ∗ .
As 0 ∈
/ C, xα ≥ η > 0. Taking the limit, we get u∗ , d ≥ η > 0. Hence, d = 0, and
w
so, 0 ∈
/C .
w
Conversely, suppose that 0 ∈
/ C . There will be a weak neighborhood of zero that
n
does not meet C. In other words, there is ε > 0 and a finite sequence {x∗i }i=1 of
∗
∗
/ C. Equivalently, x ∈
/ C if
elements of V such that |x, xi | < ε for each i implies x ∈
x, ±x∗i < ε for each i. Consider the finite set D = {±x∗i : i = 1, . . . , n}. The above
property can then be equivalently described as follows: for each x ∈ C, there is u∗ ∈ D
such that u∗ , x ≥ ε. Define the possibly empty sets Cu∗ = C ∩ {x : u∗ , x ≥ ε} for
each u∗ ∈ D. The above arguments imply C = ∪u∗ ∈D Cu∗ . It remains to check that
every Cu∗ = ∅ allows plastering. In fact, u∗ , x ≥ ε for all x ∈ Cu∗ . Since C is
bounded, x ≤ N for x ∈ Cu∗ . Hence, u∗ , x ≥ ε = (ε/N ) N ≥ (ε/N ) x, which
shows that Cu∗ allows plastering.
seq w
w
= C under our hypotheses. Since
The proof is completed by noticing that C
∗
if V is separable, the bounded sets of V are weakly metrizable (see, e.g., [3, Theorem
seq w
w
3.35]), and thus, C
= C . If V is reflexive, C is relatively weakly compact if
w
is bounded. By Day’s lemma (see, e.g., [15, Lemma 2.8.5]), if d ∈ C , there is a
sequence in C that converges weakly to d. Also, in this case the desired property thus
holds.
w
Proof of Theorem 7. The result holds for a bounded set C since BC
= ∅. We will
thus suppose that C is unbounded.
(i) implies (ii): suppose C = ∪ni=1 Ci , where each Ci is well-positioned and let
xn → ∞ and xn / xn d. Without loss of generality, we can suppose {xn } ⊆ Ci0
for some i0 . Moreover, by Proposition 6, we can suppose that Ci0 allows plastering.
Therefore, x∗ , xn ≥ xn ; i.e., x∗ , xn / xn ≥ 1. This implies x∗ , d ≥ 1 and so
d = 0.
(ii) implies (i): suppose first that the dual V ∗ is separable and that (ii) holds.
Fix a radius ρ > 0, and define the set ∅ = Sρ = {x/ x : x ∈ C and x ≥ ρ} ⊆
seq w
SV . We claim that 0 ∈
/ Sρ
for ρ large enough. Suppose not. Then there is a
seq w
sequence ρn ↑ ∞ such that 0 ∈ S ρn for all n. Taking n = 1, there is a sequence
{un } ⊆ Sρ1 for which un 0. On the other hand, un = x1n /x1n with x1n ≥ ρ1 .
By hypothesis, the sequence x1n is necessarily bounded (otherwise, x1n /x1n 0,
thus contradicting (i)). Therefore, there is some ρn2 such that x1n < ρn2 for all n.
Iterating the same argument for the set Sρn2 , we obtain a new sequence {x2n } having
the properties x2n ≥ ρn2 , x2n /x2n 0, and x2n < ρn3 (and so on). Consequently,
we get countably many sequences {xkn } for which xkn /xkn 0 as n → ∞, and
ρnk ≤ xkn < ρnk+1 . Since V ∗ is separable, the unit ball of V is weakly metrizable.
Denote by δ such a metric. For all k, there is an element xknk /xknk of the sequence
{xkn /xkn }n such that δ(xknk /xknk , 0) < 1/k. Therefore, by construction, for the
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NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMA
191
sequence {xknk /xknk }k , it holds xknk /xknk 0 as k → ∞ and xknk → ∞. This
seq w
contradicts (i). Hence, 0 ∈
/ Sρ
for ρ sufficiently large.
By Lemma 30, Sρ = ∪ni=1 Ci , where each Ci allows plastering. On the other hand,
we have {x ∈ C : x ≥ ρ} ⊆ cone Sρ = cone ∪ni=1 Ci = ∪ni=1 cone Ci . Clearly, each
cone Ci allows plastering. Thus, C ⊆ (C ∩ ρB) ∪ ∪ni=1 cone Ci , and so, C is finitely
well-positioned.
Suppose now that V is reflexive. The proof proceeds in a similar way until the
construction of the sequences {xkn }. Now consider the separable and reflexive subspace
W = span{xkn }n,k of V . Its dual W ∗ is separable, and the σ (W, W ∗ ) convergence of
sequences in W is equivalent to their σ (V, V ∗ ) convergence (see, e.g., [15, Proposition
2.5.22]). By using the existing metric on BV ∩ W , we can then extract an unbounded
sequence {xknk }k such that xknk 0. This leads to a contradiction, and the proof
proceeds as in the previous case.
We prove that (iii) is equivalent to (ii), provided V is reflexive. Assume (iii), and
let {xn } be an unbounded sequence. Consider the sequence xn / xn . As V is reflexive,
there is a convergent subsequence xnk / xnk d, and d does not vanish by (ii).
Now assume (iii), and let {xn } ⊆ C with xn → ∞ and xn / xn d. By (iii),
there is a subsequence {xnk } and a sequence {tk } for which tk → ∞ and xnk /tk d1 = 0. It is easy to see that there is a subsequence {xnkr } for which xnkr /xnkr λd1 = 0. Hence, xn /xn d = 0 and (ii) holds.
Acknowledgment. We thank two anonymous referees for some very helpful
comments and suggestions.
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