Projective dualities for quasiconvex problems Jean-Paul Penot Sorbonne Universités U P M C Université Paris 6 UMR 7598 Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005 PARIS France
[email protected] Abstract We study two dualities that can be applied to quasiconvex problems. They are conjugacies deduced from polarities. They are characterized by the polar sets of sublevel sets. We give some calculus rules for the associated subdi¤erentials and we relate the subdi¤erentials to known subdi¤erentials. We adapt the general duality schemes in terms of Lagrangians or in terms of perturbations to two speci…c problems. First a general mathematical programming problem and then a programming problem with linear constraints. Keywords Conjugacy, Duality, Dual problems, Lagrangian, Optimality conditions, Perturbation, Quasiconvex function, Subdi¤erential
1
Introduction
A duality between two function spaces Y R such that for any family (fi )i2I in Y one has
Y
and Z
Z
R on sets Y and Z is a map f 7! f D from Y to Z
(inf fi )D = sup fiD : i2I
i2I
It allows to de…ne a reverse duality and to transform a problem into a dual one. Such a concept has been used in various contexts, from bare algebraic grounds to highly structured frameworks (see [24], [25], [26], [31], [33], [34], [48], [52] and their references). Numerous applications have been given, from mathematical economics ([8], [9], [16], [14], [22], [34]) to partial di¤erential equations ([1], [5], [42], [43], [41]) and optimization. Many attempts have been proposed to extend the classical convex duality to the case of quasiconvex problems (see [6], [11], [20], [21], [27], [38], [39] and the surveys [8], [22], [31], [35], [36]). In general, in these schemes, the dimension of the dual space Y of Z is greater than the dimension of Z and the dual function is often more intricate than the primal function. However, in some special cases these drawbacks can be eliminated and one gets a rather symmetric approach (see [29], [37], [53]-[56]). It is the purpose of the present paper to study a conjugacy (i.e. a duality f 7! f D such that (f +r)D = f D r for all functions f and all r 2 R) adapted to general quasiconvex functions. In fact, we study two variants that are closely related. They correspond to two natural polarities for convex sets through sublevel sets of functions. For this reason, they are adapted to quasiconvex functions since such functions are characterized by the property that their sublevel sets are convex. These conjugacies are close to the Crouzeix and Diewert conjugacies that were inspired by the passage from utility functions to inverse utility functions in mathematical economics (see [7], [8], [9], [22], [34]). One of these two conjugacies is introduced in [22]; both are special cases of the scheme described in [20]. This scheme is itself a particular instance of the general device proposed by J.-J. Moreau [24] and introduced in quasiconvex analysis in [28], [38], [21], [39]. This general device is now widely recognized as a simple and versatile approach. These two conjugacies are close to the conjugacies studied in [29], [37], [53]-[56], but they address general quasiconvex functions and not just special classes. We point out these links in the next section and we 1
present an interpretation of these conjugacies in terms of the projective space of the dual space. Projective constructions are of fundamental use in algebraic geometry and di¤erential geometry, but their uses are rather limited in optimization (see [2], [3], [30]). Still, such an interpretation explains the singular role of the pair (0; 0) in the dual space Z 0 R we consider. The next section is devoted to a presentation of the polarity under study and its relationships with general concepts. Section 3 deals with subdi¤erentials that are attached to this polarity. We give some elementary calculus rules and we point out some relationships with other subdi¤erentials. In Section 4 we revisit the two classical approaches to duality and we make some adjustments required for the use of polarities. The two last sections are devoted to duality for a general mathematical programming problem and to the case of linearly contrained minimization. In the sequel R denotes the set R [ f 1; +1g of extended real numbers, P (resp. R+ ) the set of positive (resp. nonnegative) real numbers and for a function f : X ! R and r 2 R, ff rg (resp. ff > rg) stands for fx 2 X : f (x) rg (resp. fx 2 X : f (x) > rg); we set ff rg := f f rg and ff < rg := f f > rg. The addition is extended from R to R by setting (+1) + r = +1 for all r 2 R and ( 1) + r = 1 for all r 2 R. Also we set r s := r + ( s) for r; s 2 R.
2
The projective conjugacies
Let us recall a fundamental fact of convex analysis: the Hahn-Banach separation theorem entails that a closed convex subset A of a normed vector space Z (or more generally, of a locally convex topological vector space Z) is the intersection of a family of closed half-spaces fz 0 rg for (z 0 ; r) in some (possibly empty) subset AP of Z 0 R, where Z 0 is the topological dual space of Z. On the other hand, by de…nition, an evenly convex subset Z is the intersection of a family of open half-spaces fz 0 > rg. The family of evenly convex sets is large and enjoy pleasant properties (see [17] f.i.). Such facts make natural the introduction of a polar set. For a subset A of Z we denote by AP (resp. AQ ) the set
(resp.
AP := f(z 0 ; r) 2 Z 0 Q
0
A = f(z ; r) 2 Z
0
hz; z 0 i 8z 2 Ag
R: r
(1)
0
R : r > hz; z i 8z 2 Ag)
where h ; i is the usual coupling of Z with Z 0 . We observe that AP and AQ are convex cones of Z 0 R. Let us place these two instances of polarities in a general framework. It is convenient to introduce polarities with the language of set-valued functions (or multifunctions or relations or correspondences or in short multimaps). Given two sets Y , Z, a multimap R : Z Y identi…ed with its graph G(R) := f(z; y) 2 Z
Y : y 2 R(z)g;
one can de…ne a polarity A 7! AR from the space P(Z) of subsets of Z to the space P(Y ) of subsets of Y by setting: AR := Y nR(A) (2) where R(A) is the image of A under R; i.e. R(A) := fy 2 Y : 9a 2 A; y 2 R(a)g: The polarity A 7! AR satis…es the characteristic property of a polarity P(Z) ! P(Y ); namely: [ \ ( Ai )R = AR i ; i2I
i2I
so that AR = \a2A (Y nR(fag)): The subsets Y nR(z) := Y nR(fzg) of Y can be interpreted as remarkable or elementary subsets of Y from which the set AR can be built. In our case these sets are half-spaces. Similarly, 1 the inverse R 1 of R whose graph is f(y; z) 2 Y Z : y 2 R(z)g induces a polarity B 7! B R from P(Y ) to P(Z). For simplicity, by an abuse of notation, we also denote it by B 7! B R . Identifying R with its graph, we observe that y 2 Y nR(A) () (A
fyg) \ R = ? () A \ R 2
1
(y) = ? () A
ZnR
1
(y)
and the family (E(y))y2Y := (ZnR 1 (y))y2Y can be considered as a family of elementary subsets of Z. We observe that the inclusions A ARR := (AR )R hold for all A 2 P(Z) since for B := AR the equivalences z 2 B R () z 2 ZnR 1 (B) () B Y nR(z) () Y nR(A) Y nR(z) () R(z) R(A) show that z 2 B R whenever z 2 A: In the sequel we take Y = Z 0 R = Q where, identifying them with their graphs P := f(z; (z 0 ; r)) 2 Z Q := f(z; (z 0 ; r)) 2 Z
Z0 Z0
R or Y = (Z 0 nf0g)
R, and R = P or
R : r < hz; z 0 ig; R : r hz; z 0 ig:
Then, for every subset A of Z the sets AP and AQ given by AP := Y nP (A) and AQ := Y nQ(A) correspond to the sets de…ned in (1). The fact that these polarities are suited to convexity is illustrated by the following lemma. Lemma 1 A subset A of Z is closed convex (resp. evenly convex) if, and only if there exists some B (Z 0 nf0g) R such that A = B P (resp. A = B Q ). More precisely, one has A = (AP )P (resp. A = (AQ )Q ) if, and only if A is closed convex (resp. A is evenly convex). As shown by Volle [57], to any polarity R one can associate a conjugacy f 7! f R between functions. Conjugacies are best de…ned using a general scheme introduced by Moreau ([24]). Let us recall it and show how it …ts the polarities we de…ned. Given a coupling function c : Z Y ! R taking at least one …nite value between two sets, the c-conjugate of a function f : Z ! R is the function f c : Y ! R de…ned by f c (y) :=
inf (f (z)
z2Z
c(z; y))
y 2 Y:
Given a relation R identi…ed with its graph R Z Y a natural coupling function cR : Z Y ! R can be introduced by setting c := cR := R where R is the indicator function of R whose value at x := (z; y) is 0 if x 2 R and +1 if x 2 = R: Then the conjugate f c of f : Z ! R, also denoted by f R , is given by f R (y) :=
inf z2R
1 (y)
f (z):
(3)
Conversely, given a coupling function c taking its values in f0; 1g we get a polarity by setting R := f(z; y) : c(z; y) = 0g; then for any function f on Z, the conjugate f c is the function f R de…ned by relation (3). For the projective couplings cP and cQ given by cP (z; (z 0 ; r)) := 0 if hz; z 0 i > r; cQ (z; (z 0 ; r)) := 0 if hz; z 0 i r;
cP (z; (z 0 ; r)) := cQ (z; (z 0 ; r)) :=
1 if r hz; z 0 i; 1 if r > hz; z 0 i;
as above, writing f P and f Q instead of f cP and f cQ respectively, we have f P (z 0 ; r) :=
inf f (fz 0 > rg);
f Q (z 0 ; r) :=
inf f (fz 0
rg):
With an abuse of notation, the reverse conjugacies are de…ned for g : Y ! R by g c (z) := so that, for Y := Z 0
R and g : Z 0
g P (z) :=
inf (g(y)
y2Y
c(z; y));
R!R
inffg(z 0 ; r) : hz; z 0 i > rg
g Q (z) :=
inffg(z 0 ; r) : hz; z 0 i
rg.
The fact that (0; 0) is a singular point for our conjugacies is put in light by the following observations. We note that for z 0 6= 0 we have f P (z 0 ; r) = f Q (z 0 ; r) whenever f is radially upper regular in the sense that for 3
any z0 , z1 2 Z, one has f (z0 ) lim inf t!0+ f ((1 t)z0 + tz1 ) and we also observe that this mild condition is satis…ed by convex functions and radially upper semicontinuous functions (i.e. functions whose restrictions to segments are upper semicontinuous). On the other hand, f P (0; r) = f Q (0; r) for all r 2 Rnf0g since f P (0; r) = P
f (0; r) = Thus, for all z 2 Z we have
1 for r 2 R+ ,
inf f (Z) for r 2
f P P (z) = f QQ (z) =
f Q (0; r) =
Q
f (0; r) =
P,
1 for r 2 P
inf f (Z) for r 2 R :
infff P (z 0 ; r) : z 0 6= 0; hz; z 0 i > rg;
infff Q (z 0 ; r) : z 0 6= 0; hz; z 0 i
rg:
Since the duality results we give rely on the relation f P P (0) f (0); f QQ (0) f (0); instead of taking 0 0 0 0 Y := Z R for dual space we can take Y0 := Z0 R with Z0 := Z nf0g and replace the conjugacies f 7! f P and f 7! f Q with the conjugacies de…ned by the relations P0 := f(z; (z 0 ; r)) 2 Z Q0 := f(z; (z 0 ; r)) 2 Z
Z00 Z00
R : r < hz; z 0 ig; R : r hz; z 0 ig:
The space Z 0 R being of dimension greater than Z (when Z is …nite dimensional), it may be of interest to replace it with the projective space Z 0 := ((Z 0 R)nf(0; 0)g) of Z 0 obtained as the quotient of (Z 0 R)nf(0; 0)g for the equivalence relation (z10 ; r1 )~(z20 ; r2 ) if, and only if z20 = tz10 , r2 = tr1 for some t > 0: Here : (Z 0 R)nf(0; 0)g ! Z 0 is the quotient map. From what precedes, we see that we can even replace Z 0 R with the set Z00 = (Z00 R) and set Ap := (AP n(f0g R)), Aq := (AQ n(f0g R)): We note that Y := Z 0 is the union of three disjoint subsets Y+ := (Z 0
P);
Y := (Z 0
( P));
Y0 := (Z00
f0g);
with Z00 := Z 0 nf0g. Correspondingly, AP , AQ , Ap and Aq are the unions of three disjoint subsets that can be related to more classical polarities since Y+ = (Z 0 f1g), Y = (Z 0 f 1g). We can also de…ne the conjugates f p , f q : Z00 ! R of f : Z ! R on Z00 := (Z00 R) by fp
j Z00
R := f P j Z00
fq
R;
j Z00
R = f Q j Z00
R:
Another reason to discard f0g R in the dual space Z 0 R can be found in the approach to duality via a set Yc of linearlike functions giving rise to a set c (Z) of convexlike functions by taking suprema (see [48], [52] for instance). Here the set Yc := fc( ; y) : y := (z 0 ; r) 2 Y g is formed with the opposites of indicator functions of open (or closed) half-spaces and it is natural to exclude the case z 0 = 0 which corresponds to improper half-spaces. In the sequel, even if Z 0 or Z00 is the natural dual space, we rather work with Y := Z 0 R; a more usual space. But we keep in mind the special role of the elements of f0g R. The projective conjugacies ful…ll the general properties of Fenchel-Moreau type conjugacies recalled in the following lemma. Lemma 2 (a) If f1 f2 then f1c f2c ; (b) for any family (fi )i2I one has (inf i2I fi )c = supi2I fic ; (c) for any function f and any r 2 R, (f + r)c = f c r; (d) for any f : Z ! R and any g : Z 0 ! R one has f cc f , g cc g; (e) for any f : Z ! R and any g : Z 0 ! R one has f ccc = f c , g ccc = g c : Note that (e) can be deduced from (a) and (d) since f ccc = (f cc )c f c and f ccc = (f c )cc f c : Property (c) can be extended to the case r 2 R by (f + r)c = ( f c + r): Since the projective conjugacies are deduced from polarities, they enjoy additional properties described in the next statements. For r; s in R we set r _ s := max(r; s), r ^ s := min(r; s), r ? s := r if r < s, r ? s := +1 if r s: 4
Lemma 3 ([23]) For R = P or R = Q the conjugacy f 7! f R satis…es the following properties: (a) for any function f : Z ! R and any nondecreasing upper semicontinuous function h : R!R one has (h f )R = h ( f R ); (b) for any function f : Z ! R and any r 2 R one has (f _ r)R = f R ^ ( r); (c) for any function f : Z ! R and any r 2 R one has (f ? r)R = (( f R ) ? r); (d) if X is a topological vector space, if g : X ! Z is a continuous linear map, and if f : Z ! R is an arbitrary function, then (f g)R (g T (z 0 ); r) = f R (z 0 ; r) for all (z 0 ; r) 2 Z 0 R such that (g T (z 0 ); r) 6= (0; 0). Properties (b), (c) can be deduced from assertion (a) by considering the functions h : t 7! t _ r and h : t 7! t ? r: A more speci…c property has to be pointed out. Proposition 4 For any functions f : Z ! R and g : Z 0 R !R the functions f P and g P are quasiconvex and lower semicontinuous in the sense that for all s 2 R their sublevel sets ff P sg, fg P sg are closed, Q Q convex and the functions f and g are evenly quasiconvex in the sense that their sublevel sets ff Q sg, fg Q sg are evenly convex for all s 2 R. More precisely ff P
sg = ff
s ff
t0 g
sg = fg
sg for all s 2 R. Now,
tgR = \t>s ff < tgRR
ff < t00 g we note that
tgRR :
Let St := ff tg. If StRR = St for all t 2 R we get ff RR sg = ff sg and f RR = f: Conversely, if f = f RR then Ss := ff sg = ff RR sg is the intersection of the sets StRR for t > s hence satis…es SsRR = Ss : For all z 0 2 Z 0 the functions fzP0 := f P (z 0 ; ) and fzQ0 := f Q (z 0 ; ) are non increasing and for all r 2 R one has f P (z 0 ; r) = sup f Q (z 0 ; s) f Q (z 0 ; r) s>r
since for any nonempty subset T of R and any t 2 T , r 2 R one has t > r if, and only if there exists s > r such that t s: Let us point out a link with other conjugacies that are suited to quasiconvex analysis since their de…nitions involve sublevel sets. These “musical conjugacies” are de…ned by f [ (z 0 ; q) := supfhz; z 0 i : z 2 ff < qgg;
f ] (z 0 ; q) := supfhz; z 0 i : z 2 ff
qgg:
They play an important role in the study of Hamilton-Jacobi equations ([1], [5], [41], [43], [42], [58]). The most interesting relationship is the following one: for any function f and any (z 0 ; q; r) 2 Z 0 R R one has f [ (z 0 ; q)
r () 5
f P (z 0 ; r)
q
since ff < qg fz 0 rg is equivalent to fz 0 > rg ff qg: With the terminology issued from [40] that P means that r 7! fz0 (r) is the hypo-epi-inverse of fz[0 : From this relation one can deduce some implications with f ] and f Q similar to the ones in [34, Prop. 24]. Let us give some examples providing some familiarity with these polarities. Example 1. For a nonempty subset A of Z and R := P the polar AP of A can be interpreted as the epigraph of the usual support function hA := supa2A ha; i of A: It is a closed convex cone, whereas the polar AQ of A is an evenly convex cone without such a simple interpretation. However, AQ contains the strict epigraph of hA and is contained in the epigraph of hA . Example 2. For a nonempty subset A of Z and any coupling c we have c A (y)
=
inf (
z2Z
A (z)
c(z; y)) =
inf
z2A
c(z; y) = sup c(z; y); z2A
the c-support function z 7! supz2A c(z; y) of A: Example 3. More precisely, if c = cR for some polarity R, we have cA (y) = R(A) (y). In particular P 0 0 0 0 R : r < hA (z 0 )g is the strict hypograph hypos hA (z ; r); where hypos hA := f(z ; r) 2 Z A (z ; r) = 0 of the support function hA of A: In fact, we have P A (z ; r) = 0 if, and only if there exists some z 2 A 0 0 such that (z ; r) 2 P (z) i.e. r < hz; z i if, and only if r < hA (z 0 ): On the other hand, by (2), we have Q 0 0 (Z 0 R)nAQ (z ; r): A (z ; r) = Example 4. If A is the open half-space A = P 1 (w0 ; q) or the closed half-space A = Q 1 (w0 ; q) for some (w0 ; q) 2 (Z 0 nf0g) R, then the Farkas-Minkowski Lemma shows that AP = R+ (f w0 g ( q + R+ )); the cone generated byf( w0 ; r) : r qg [ f(0; 0)g: Similarly, for A = P 1 (w0 ; q) one has AQ = P(f w0 g ( q + R+ )). Example 5. If f : Z ! R is the linear form w0 2 Z 0 nf0g, then f P (z 0 ; r) = f Q (z 0 ; r) = r=t if z 0 = tw0 for some t 2 P; f P (0; r) = 1 for r 0, f P (0; r) = +1 for r < 0, f P (z 0 ; r) = +1 else. One has f Q (z 0 ; r) = f P (z 0 ; r) but for (z 0 ; r) = (0; 0) since f Q (0; 0) = +1, f P (0; 0) = 1. Using the preceding example, we check that ff P sg = ff < sgP : Example 6. If f : R ! R is the function z 7! z 3 ; then f P (z 0 ; r) = f Q (z 0 ; r) = (r=z 0 )3 if z 0 > 0, f P (z 0 ; r) = f Q (z 0 ; r) = +1 if z 0 = 0; r < 0, f P (z 0 ; r) = f Q (z 0 ; r) = 1 if z 0 = 0, r > 0, f P (z 0 ; r) = f Q (z 0 ; r) = +1 if z 0 < 0 but f P (0; 0) = 1; f Q (0; 0) = +1. The computation for this example can be deduced from the preceding one and the fact that for f := h g; with g : Z ! R, h : R ! R nondecreasing and upper semicontinuous, and any polarity R one has f R (y) = h( g R (y)) for all y 2 Y: Example 7. If R is any polarity and if f := A for a subset A of Z; then f R = Y nR(ZnA) : Example 8. If R is any polarity and if for a subset A of Z; f := vA is the valley function of A given by vA (z) := 1 for z 2 A; vA (z) := +1 for z 2 ZnA; then f R = vY nR(A) : Example 9. If R is any polarity, if f : Z!R and if s 2 R, the s-truncated function fs is given by fs (z) = f (z) if f (z) s; fs (z) = s if f (z) > s: Then one has (fs )R = f R _ ( s): Note that when f is quasiconvex, fs is still quasiconvex but if f is convex, fs is no more convex in general. Thus the relation (fs )RR (0) = fs (0) may hold in cases (fs ) (0) 6= fs (0); and the duality results of Sections 4 and 5 apply in cases convex duality cannot be applied.
3
Projective Subdi¤erentials
We follow a well-known general scheme associating a subdi¤erential to a conjugacy c : Z Y ! R (see [4], [22]). For a function f : Z ! R and z 2 f 1 (R), the c-subdi¤erential of f at z is the set @ c f (z) of those y 2 Y such that c(z; y) is …nite and 8z 2 Z
f (z)
c(z; y)
f (z)
c(z; y):
When c is the coupling function associated with a polarity R; y 2 @ c f (z) if and only if c(z; y) = 0 and z is a minimizer of f c( ; y) if, and only if c(z; y) = 0 and f c (y) = f (z) c(z; y) if, and only if c(z; y) = 0 and f c (y) + f (z) = 0.
6
1
De…nition 6 For a function f : Z ! R and z 2 f
(R) the projective subdi¤ erential of f at z is the set
@ p f (z) := fy 2 Z 0 : f (z) + f p (y) = cp (z; y) = 0g = (@ P f (z)); with @ P f (z) := f(z 0 ; r) 2 Z 0
R : hz; z 0 i > r; f (z) = inf f (fz 0 > rg)g:
The evenly projective subdi¤ erential of f at z is the set @ q f (z) := fy 2 Z 0 : f (z) + f q (y) = cq (z; y) = 0g = (@ Q f (z)); with @ Q f (z) := f(z 0 ; r) 2 Z 0
R : hz; z 0 i
r; f (z) = inf f (fz 0
rg)g:
The sets @ P f (z) and @ Q f (z) are convex cones, the set @ Q f (z) is a closed convex cone and one has @ P f (z) cl(@ Q f (z)): in fact, for all (z 0 ; r) 2 @ P f (z) and all (rn ) ! r with rn 2]r; hz; z 0 i] one has (z 0 ; rn ) 2 @ Q f (z). If f is radially upper regular one has @ P f (z) @ Q f (z). Moreover @ P ( f )(z) = @ P f (z) and @ Q ( f )(z) = @ Q f (z) for all > 0. The de…nition of @ P f (z) being rather restrictive, it may be of interest to introduce the variant @ O f (z) := f(z 0 ; r) 2 Z 0 so that @ P f (z) [ @ Q f (z)
R : hz; z 0 i
r; f (z)
inf f (fz 0 > rg)g;
@ O f (z): If f is radially upper regular, one has @ O f (z)n(f0g
@ Q f (z)
R)
since for (z 0 ; r) 2 @ O f (z)n(f0g R) and z 2 fz 0 rg we can …nd w 2 Z such that hw; z 0 i > 0; hence 0 hz + tw; z i > r for all t > 0; so that f (z) lim inf t!0+ f (z + tw) f (z). The set @ Q f (z) is closely related to the sets @r f (z) := fz 0 2 Z 0 : hz; z 0 i 1; f (z) = inf f (fz 0 1g)g @ f (z) := fz 0 2 Z 0 : hz; z 0 i 1; f (z) = inf f (f z 0
1g)g
considered in [37]. In fact one has
Q
@ Q f (z) \ (Z 0
@ f (z) \ (Z
0
P) = P(@r f (z)
( P)) =
P(@ f (z)
f1g)
f1g):
One also has a link (see [22, Cor. 6.6]) with the Greenberg-Pierskalla’s subdi¤ erential @ f (z) given by @ f (z) := fz 0 2 Z 0 : hz
z; z 0 i < 0 8z 2 ff < f (z)gg
in view of the relation z 0 2 @ f (z) () (z 0 ; hz; z 0 i) 2 @ Q f (z):
Since the Fenchel-Moreau subdi¤erential @ F M is contained in the Greenberg-Pierskalla’s subdi¤erential @ , one also gets z 0 2 @ F M f (z) =) (z 0 ; hz; z 0 i) 2 @ Q f (z): A similar implication holds with the Plastria’s subdi¤ erential ([47]) de…ned by @ < f (z) := fz 0 2 Z 0 : f (z)
f (z)
hz
z; z 0 i 8z 2 ff < f (z)gg:
In fact, if z 0 2 @ < f (z); for r := hz; z 0 i one has hz; z 0 i < r for all z satisfying f (z) < f (z) or equivalently f Q (z 0 ; r) f (z) or f Q (z 0 ; r) = f (z): Another similar implication holds with @ Q f (z) replaced with @ O f (z) < and @ f (z) replaced with the infra-di¤ erential @ f (z) of f at z de…ned by Gutiérrez [13] as @ f (z) := fz 0 2 Z 0 : f (z)
f (z) 7
hz
z; z 0 i 8z 2 ff
f (z)gg:
Finally, let us make a comparison with the directional subdi¤ erential (or Dini-Hadamard subdi¤erential or contingent subdi¤erential) @D f (z) given by z 0 2 @D f (z) , (8w 2 Z
1 (f (z + tv) (t;v)!(0+ ;w) t lim inf
f (z))
hw; z 0 i):
Given a quasiconvex function f , z 2 domf , z 0 2 @D f (z), for all z 2 ff f (z)g; setting r := hz; z 0 i, we have 0 hz z; z i lim inf t!0+ (1=t)(f (z + t(z z)) f (z)) 0; hence, f (w) > f (z) for all w 2 fz 0 > rg; so that z 0 2 @D f (z) =) (z 0 ; hz; z 0 i) 2 @ O f (z): If f is quasiconvex and radially upper semicontinuous, or even radially upper regular, since @ O f (z)n(f0g R) @ Q f (z), one has z 0 2 @D f (z)nf0g =) (z 0 ; hz; z 0 i) 2 @ Q f (z): As in convex analysis, these subdi¤erentials can characterize minimizers. Proposition 7 For a function f : Z ! R and z 2 f 1 (R) the following assertions are equivalent: (a) f (z) = min f (Z); (b) for all z 0 2 Z 0 and all r < hz; z 0 i (resp. r hz; z 0 i) one has (z 0 ; r) 2 @ P f (z) (resp. (z 0 ; r) 2 @ Q f (z)); (c) for all r < 0 (resp. r 0) one has (0; r) 2 @ P f (z) (resp. (0; r) 2 @ Q f (z)); (d) there exists some r < 0 (resp. r 0) such that (0; r) 2 @ P f (z) (resp. (0; r) 2 @ Q f (z)). Proof. (a))(b) When f (z) = min f (Z), given (z 0 ; r) 2 Z 0 R such that r < hz; z 0 i (resp. r hz; z 0 i) we have f (z) inf f (fz 0 > rg) (resp. f (z) inf f (fz 0 rg)). (b))(c) and (c))(d) are obvious. (d))(a) Since f0 > rg = Z when r < 0; if f (z) = inf f (f0 > rg) one has f (z) = inf f (Z); the case r 0 and f (z) = inf f (fz 0 rg) with z 0 = 0 is similar. Optimality conditions for constrained optimization problems in terms of the subdi¤erentials @ P and @ Q will be given later on. Let us give some calculus rules. The proofs are easy consequences of the de…nitions. Proposition 8 (a) For any functions f; g : Z ! R …nite at z with f (z) = g(z), for R = P or Q one has co(@ R f (z); @ R g(z)) @ R (f _ g)(z): (b) For any function f : Z ! R …nite at z and s := f (z) one has @ R (f ^ s)(z) = @ R f (z): Proposition 9 (a) For any function f : Z ! R, any nondecreasing function h : R!R, and any z 2 domf \ dom(h f ), for R = P or Q one has (z 0 ; r) 2 @ R f (z) =) (z 0 ; r) 2 @ R (h f )(z): (b) If X is a topological vector space, if g : X ! Z is a continuous linear map, and if f : Z ! R is an arbitrary function, then for any z 2 domf and any x 2 g 1 (z); for R = P or Q one has (z 0 ; r) 2 @ R f (z) =) (g T (z 0 ); r) 2 @ R (f g)(x): If g is surjective this implication is an equivalence. The following existence result is another analogy with the case of the Fenchel-Moreau subdi¤erential for convex continuous functions. Proposition 10 Let f : Z ! R be a quasiconvex function …nite at z 2 Z: If ff < f (z)g is open, in particular if f is upper semicontinuous, then @ Q f (z) is nonempty. Proof. When f (z) = inf f (Z) Proposition 7 yields the conclusion. Suppose f (z) > inf f (Z): Since ff < f (z)g is convex, open and nonempty and z 2 = ff < f (z)g the separation theorem yields some z 0 2 Z 0 nf0g and 0 some r 2 R such that hz; z i r and 8z 2 ff < f (z)g Then, for all z 2 fz 0
rg we have f (z)
hz; z 0 i < r:
f (z) or (z 0 ; r) 2 @ Q f (z) since z 2 fz 0 8
rg:
4
Perturbations and Lagrangians
Let us consider the (apparently) unconstrained minimization problem (P )
Minimize f (x)
subject to x 2 X
where X is a set and f : X ! R takes at least one …nite value. A sub-perturbation (resp. a perturbation) of f is a map M : X Z ! R such that M ( ; 0Z ) f (resp. M ( ; 0Z ) = f ), where Z, the parameter space, is a normed vector space (or a locally convex topological vector space) and 0Z is a base point of Z (the nominal parameter). We associate to M the value function (or performance function) m : Z ! R given by m(z) := inf M (x; z): x2X
Given a coupling c : Z
Y ! R, the inequality mcc (0Z )
(DM )
m(0Z ) leads to the introduction of the dual problem (mc (y)
Maximize dM (y) :=
c(0Z ; y))
y2Y
or of the adjoint problem 0 (PM )
Minimize
dM (y) = mc (y)
c(0Z ; y)
y2Y
so that the following weak duality inequality holds: sup(DM )
inf(P ):
If equality holds we say that there is no duality gap. In the sequel we assume that c takes its values in R[f 1g and we set Yc := fy 2 Y : c(0Z ; y) > 1g: Then we have dM (y) = c(0Z ; y)
mc (y)
if y 2 Yc ;
dM (y) =
1 if y 2 Y nYc :
When c is the coupling function associated with a polarity R (with a nonempty graph) we have YR := Yc := fy 2 Y : c(0Z ; y) = 0g and dM (y) = mc (y) if y 2 YR , dM (y) = 1 if y 2 Y nYR : Let us identify the set of solutions to the dual problem. Proposition 11 Suppose the value mcc (0Z ) of the dual problem is …nite and c = cR for a polarity R. Then the set SD of solutions to the dual problem (DM ) coincides with @ c mcc (0Z ) := fy 2 Y : 8z 2 Z mcc (z)
c(z; y)
mcc (0Z ); c(0Z ; y) = 0g:
Proof. By our convention about the operations in R, since mcc (0Z ) is …nite, if y 2 SD we must have c(0Z ; y) < +1 or equivalently c(0Z ; y) = 0: By de…nition of @ c mcc (0Z ) we have y 2 @ c mcc (0Z ) if, and only if c(0Z ; y) = 0 and mcc (0Z ) = mccc (y) or inf y2Y mc (y) c(0Z ; y) = mc (y) and c(0Z ; y) = 0, or supy2Y dM (y) = dM (y) and y 2 Yc , or y 2 SD : Another way of obtaining duality results uses the concept of Lagrangian for (P ). Recall that a function L : X Y ! R is called a Lagrangian (resp. a sub-Lagrangian) for (P ) if one has f ( ) = supy2Y L( ; y) (resp. f ( ) supy2Y L( ; y)). Then the Lagrangian dual problem is (DL )
maximize dL (y) := inf L(x; y) x2X
y 2 Y:
When the function L( ; y) is easier to handle than f the computation of dL ( ) may be simple enough and lead to useful global estimates derived from the obvious inequality sup inf L(x; y)
inf sup L(x; y)
y2Y x2X
x2X y2Y
9
meaning that sup(DL ) inf(P ): Moreover, if y is a multiplier in the sense that dL (y) = inf(P ), i.e. y is a solution to (DL ) and there is no duality gap, the set S of solutions to (P ) is the set fx 2 Sy : L(x; y) = f (x)g where Sy is the set of solutions to (Qy )
Minimize L(x; y)
x 2 X;
an unconstrained (or simply constrained) problem (see [35, Prop. 1.4]). Lagrangians can be devised directly or obtained from perturbations. Here we improve [35, Prop. 1.4] in order to drop its assumption (S) that c(0Z ; ) = 0, or even its assumption (F) that c(0Z ; ) is …nitely valued since in the case the conjugacy is associated to a polarity R we may have c(0Z ; y) = 1. Thus, the Lagrangian we introduce is slightly di¤erent from the one in [35, Prop. 1.4]. Proposition 12 Let M : X Z ! R be a sub-perturbation for (P ) and let c : Z function. Then L : X Y ! R given by (Mxc (y)
L(x; y) :=
Y ! R be a coupling
c(0Z ; y));
with Mx := M (x; ) is a sub-Lagrangian for (P ): Moreover the objective dL of (DL ) satis…es dL dM and if c takes its values in R[f 1g one has dL = dM and the values of (DL ) and (DM ) are equal. The function L is a Lagrangian for (P ) if M is a perturbation of (P ) and if for all x 2 X, Mx is c-convex in the sense that Mxcc = Mx : Proof. The …rst assertion stems from the relations Mxcc sup L(x; y) = y2Y
inf
y2Y
(Mxc (y)
Mx for all x 2 X and
c(0Z ; y)) = Mxcc (0Z )
Mx (0Z )
f (x):
These relations are equalities when M is a perturbation of (P ) and Mxcc (0Z ) = Mx (0Z ): Since for all x 2 X we have m(z) Mx (z), hence Mxc (y) mc (y), the objective dL of (DL ) satis…es (mc (y)
dL (y) := inf L(x; y) x2X
c(0Z ; y)) =: dM (y)
For y 2 Y nYc we have L(x; y) = 1 for all x 2 X hence dL (y) = mc := (inf x2X Mx )c = supx2X Mxc , we have dM (y) = c(0Z ; y)
sup Mxc (y) =
x2X
sup (Mxc (y)
1 and dL (y) = dM (y): For y 2 Yc since
c(0Z ; y)) = inf
x2X
x2X
(Mxc (y)
c(0Z ; y)) = dL (y);
so that dM = dL on Y and sup(DL ) = sup(DM ):
5
Application to general mathematical programming problems
A noticeable case in which a perturbation of (P ) can be introduced concerns the case (P ) is in fact a constrained problem in which the feasible set is the value at some point 0Z of a parameter space Z of a multimap F :Z X: (C) Minimize f (x) x 2 F (0Z );
where f : X ! R is some objective function, so that (C) coincides with (P ) if one replaces f in (P ) with fC := f + F (0Z ) where F (0Z ) is the indicator function of the feasible set F (0Z ): In such a case, it is natural to consider the perturbed problems (Cz )
Minimize f (x)
Identifying F with its graph and setting G := F (C) by setting M (x; z) := f (x) + Mcc (x; z) := f (x) +
F (z) (x) = cc G(x) (z):
1
f (x) +
x 2 F (z):
we can de…ne a perturbation and a sub-perturbation of F (z; x)
10
= f (x) +
G (x; z)
= f (x) +
G(x) (z);
c Clearly M (x; 0Z ) = fC (x) and Mcc (x; 0Z ) fC (x) since cc G(x) (0Z ): Since (Mcc (x; )) (y) = G(x) (0Z ) ccc f (x) = cG(x) (y) f (x) = (M (x; ))c (y), the associated sub-Lagrangians coincide and are given by G(x) (y) c G(x) (y)
L(x; y) := f (x) + c(0Z ; y)
if y 2 Yc ,
L(x; y) =
1 if y 2 Y nYc :
c Since the objective function of (C) is fC ( ) = f ( ) + G( ) (0Z ) and since supfc(0Z ; y) G(x) (y) : y 2 Yc g = cc G(x) (0Z ) , we know from Proposition 12 that L is a sub-Lagrangian of (C) and a Lagrangian G(x) (0Z ) when cc (0 ) = G(x) (0Z ): When c takes its values in f 1; 0g, using Example 2, we have for L the simpli…ed Z G(x) forms c G(x) (y)
L(x; y) := f (x) = f (x)
if y 2 Yc ,
sup c(z; y) if y 2 Yc ,
L(x; y) =
1 if y 2 Y nYc
L(x; y) =
1 if y 2 Y nYc :
z2G(x)
If moreover c := cR is the coupling associated to a polarity R, by Example 3 we get the Lagrangian LR given by LR (x; y) = f (x) + R(G(x)) (y) if y 2 YR , LR (x; y) = 1 if y 2 Y nYR :
In the sequel we suppose Z is a locally convex topological vector space with dual Z 0 , 0Z is the origin of Z, and c = cR for some polarity R: In particular, for R = P we have YP := YcP = Z 0 ( P), and for R = Q we have YQ := YcQ = Z 0 R . By Example 3, for x 2 X, y := (z 0 ; r) 2 Z 0 R we get LP (x; y) = f (x) + LQ (x; y) = f (x) +
hypos hG(x) (z Q(G(x)) (y)
0
; r) if (z 0 ; r) 2 Z 0 0
if (z ; r) 2 Z
0
( P),
LP (x; y) =
R ;
LQ (x; y) =
1 if (z 0 ; r) 2 Z 0 0
1 if (z ; r) 2 Z
R+ ; 0
P:
In the usual case, for a map g : X ! Z and a nonempty subset C of Z we have G(x) := g(x)
C
and we can rewrite (C) as the mathematical programming problem (C)
Minimize f (x)
subject to g(x) 2 C:
Then, since hG(x) (z 0 ) = hg(x); z 0 i + hC ( z 0 ), the Lagrangian LP is given by LP (x; y) = Z 0 R+ and, for y := (z 0 ; r) 2 Z 0 ( P) LP (x; y) = f (x) if hg(x); z 0 i > r
hC ( z 0 ),
1 for y := (z 0 ; r) 2
LP (x; y) = +1 otherwise.
Thus the dual objective function dP associated with the sub-Lagrangian LP is given by dP (z 0 ; r) = infff (x) : x 2 X; hg(x); z 0 i > r On the other hand, setting w := g(x) y := (z 0 ; r) 2 Z 0 R LQ (x; y) = f (x) if hg(x); z 0 i L0Q
hC ( z 0 )g if r < 0;
z, we have LQ (x; y) = r + hw; z 0 i for some w 2 C,
dP (z 0 ; r) =
1 if r
1 for y := (z 0 ; r) 2 Z 0
0: P and, for
LQ (x; y) = +1 otherwise.
Let us make a change of variables, setting s := r hC ( z 0 ); and let us introduce a simpli…ed sub-Lagrangian such that L0Q (x; (z 0 ; s)) LQ (x; (z 0 ; r)) by setting L0Q (x; s) = 1 for s > hC ( z 0 ) and, for s hC ( z 0 ) L0Q (x; (z 0 ; s)) = f (x) if hg(x); z 0 i
Since the inequality s given by
L0Q (x; (z 0 ; s)) = +1 otherwise.
s,
hC ( z 0 ) is equivalent to (z 0 ; s) 2
d0Q (z 0 ; s) = infff (x) : x 2 X; hg(x); z 0 i
C P ; the dual function d0Q associated with L0Q is
sg for (z 0 ; s) 2 11
CP ;
d0Q (z 0 ; s) =
1 otherwise:
1 for y := (z 0 ; s) 2 =
Let us introduce a similar simpli…cation of LP ; setting L0P (x; y) = y := (z 0 ; s) 2 C Q , L0P (x; (z 0 ; s)) = f (x) if hg(x); z 0 i > s,
C Q and, for
L0P (x; (z 0 ; s)) = +1 otherwise.
This time we have L0P (x; (z 0 ; s)) LP (x; (z 0 ; r)) for s := r hC ( z 0 ) since (z 0 ; s) 2 C Q whenever (z 0 ; s) satis…es s > hC ( z 0 ): However L0P is still a sub-Lagrangian: given x 2 B := g 1 (C) and (z 0 ; s) 2 C Q we have hg(x); z 0 i < s; hence L0P (x; (z 0 ; s)) = f (x): The dual function associated with L0P is given by the function d0P de…ned by d0P (z 0 ; s) := infff (x) : x 2 X; hg(x); z 0 i > sg for (z 0 ; s) 2
d0P (z 0 ; s) =
C Q,
1 otherwise.
Thus, the dual problems turn out to be 0
(DP0 ) Maximize dP (z 0 ; s) = inf f (fz 0 g > sg) : 0
0 (DQ ) Maximize dQ (z 0 ; s) = inf f (fz 0 g
0
z2Z
C Q;
(z 0 ; s) 2
sg) :
Introducing the function fg : Z ! R given by fg (z) := infff (x) : x 2 g fg (z) := +1 if g 1 (z) = ?) and noting that for (z 0 ; s) 2 C P dQ (z 0 ; s) = inf finfff (x) : g(x) = zg : hz; z 0 i
(z 0 ; s) 2 1
CP :
(z)g (with the usual convention that
sg = inf ffg (z) : hz; z 0 i z2Z
sg =
(fg )Q (z 0 ; s);
0
we see that solving the dual problem (DQ ) is reduced to the computation of fg and to solving Maximize
(fg )Q (z 0 ; s) : (z 0 ; s) 2
epi hC :
A similar observation holds for the dual problem (DP0 ): When g is linear and continuous a more direct route is possible. We take it in the next section.
6
Duality for linearly constrained problems
When X is a normed space or a locally convex topological vector space and g : X ! Z is linear and continuous, using the transpose g T of g; so that g T (z 0 ) := z 0 g; the Lagrangian L0Q can be written for (z 0 ; s) 2 C P L0Q (x; (z 0 ; s)) = f (x) if x 2 fg T (z 0 ) Again L0Q (x; (z 0 ; s)) = (z 0 ; s) 2 = C P and
1 for (z 0 ; s) 2 = d0Q (z 0 ; s) =
sg; L0Q (x; (z 0 ; r)) = +1 if x 2 = fg T (z 0 )
sg
C P : Thus, the dual function d0Q is given by d0Q (z 0 ; s) = f Q (g T (z 0 ); s)
(z 0 ; s) 2
CP ;
where, by analogy with the conjugacy de…ned above, for f : X ! R, (x0 ; s) 2 X 0 f Q (x0 ; s) :=
inf f (fx0
1 for
R we set
sg):
In the sequel we consider essentially the problem 0 (DQ ) Maximize
Its feasible set is just
CP =
f Q (g T (z 0 ); s) : (z 0 ; s) 2
CP :
epi hC : We shall also consider the problem (DP0 ) and a variants of it.
Proposition 13 (Weak duality) One always has infff (x) : x 2 g
infff (x) : x 2 g
1 1
(C)g (C)g
supf f Q (g T (z 0 ); s) : (z 0 ; s) 2 P
T
0
0
supf f (g (z ); s) : (z ; s) 2 12
C P g; Q
C g:
(4) (5)
Proof. The result follows from the fact that L0Q and L0P are sub-Lagrangians of (C): A direct proof is as follows. Given x 2 g 1 (C) and y := (z 0 ; s) 2 C P we have hg(x); z 0 i hC ( z 0 ) s hence 0 Q T 0 1 f (x) inf f (fz g sg) =: f (g (z ); s): Taking the in…mum on x 2 g (C) and the supremum on (z 0 ; s) 2 C P we get relation (4). Since for x 2 g 1 (C) and y := (z 0 ; s) 2 C Q we have hg(x); z 0 i < s hence f (x) inf f (fz 0 g > sg) =: f P (g T (z 0 ); s), the second inequality follows similarly. Remark. Since C Q C P relation (4) entails the relation inf f (g
1
(C))
supf f Q (g T (z 0 ); s) : (z 0 ; s) 2
C Qg
that is less exacting. On the other hand, the inequality inf f (g 1 (C)) supf f P (g T (z 0 ); s) : (z 0 ; s) 2 C P g may not hold as the following example shows. Example. Let X = Z := R, C := R+ , and let f; g be the identity map IR identi…ed with 1 2 X 0 . Then CP = R R+ and, by Example 5, f P (0; 0) = +1 whereas inf f (C) = 0: Remark. Since ( C P ) \ (f0g P) = ? and since f Q (0; s) = inf f (X) for s 2 R the really informative part of inequality (4) is the relation infff (x) : x 2 g
1
(C)g
supf f Q (g T (z 0 ); s) : (z 0 ; s) 2
C P ; (g T (z 0 ); s) 6= (0; 0)g:
Such a fact incites to introduce another dual problem 0
(DP )
Maximize
f P (g T (z 0 ); s) :
with f P (x0 ; s) := inf f (fx0 > sg) for (x0 ; s) 2 X 0 of interest when f is regular enough.
(z 0 ; s) 2
C P ; (g T (z 0 ); s) 6= (0; 0);
R. Although it is not associated with a Lagrangian it is
Corollary 14 Suppose f is upper semicontinuous (or just radially upper regular). Then inf f (g
1
(C))
0
sup(DP ):
(6)
Proof. For all (z 0 ; s) 2 C P with (g T (z 0 ); s) 6= (0; 0) we have fg T (z 0 ) sg = clfg T (z 0 ) > sg hence T 0 T 0 Q T 0 P T 0 inf f (fg (z ) sg) = inf f (fg (z ) > sg) or f (g (z ); s) = f (g (z ); s) and the inequality follows from relation (4). 0 If C = Z one has B = X and (0Z ; 0) is a solution to the dual problem (DQ ) with f Q (0Z ; 0) = inf f (B ); this case can be discarded since then problem (C) is an unconstrained problem. Another case that can be discarded is expounded in assertion (a) of the next statement, one of the assumptions of this case being very demanding. Theorem 15 Let C be convex, nonempty, and let B := g 1 (C); with g linear and continuous. Suppose either (a) C is evenly convex, C 6= Z and $ := inf f (B) = inf f (X) or (b) inf f (X) < inf f (B), g(X) = Z, g is open from X onto Z, and f is upper semicontinuous and quasiconvex. 0 0 Then there is no duality gap, the dual problems (DQ ) and (DP ) have a solution (z 0 ; s) with z 0 6= 0. Moreover, 0 0 (z ; s) is a multiplier for the sub-Lagrangian LQ : When X and Z are Banach spaces, the assumption that g is open from X onto Z is satis…ed if g(X) = Z in view of the Banach open mapping theorem. Then, the assumption that B is convex nonempty is ful…lled whenever C is convex and nonempty. Proof. If (a) holds, taking z 2 ZnC we can …nd some z 0 2 Z 0 nf0g; s 2 R such that C fz 0 > sg and 0 0 0 Q P Q T T hz; z i s; C being evenly convex. Then (z ; s) 2 C C and f (g (z ); s) = infff (fg (z 0 ) sg) inf f (X ) = $; so that equality holds by weak duality. Now let us suppose (b) holds. Since B and ff < $g are convex, nonempty and disjoint and since ff < $g is open, f being upper semicontinuous, the Eidelheit separation theorem yields some (x0 ; s) 2 (X 0 R)nf(0; 0)g such that for all x 2 B and all w 2 ff < $g hx; x0 i
s > hw; x0 i: 13
Since x0 is bounded below on B; it is bounded below on Kerg: given w 2 Kerg and b 2 B; we have b + w 2 B; hence hw; x0 i s hb; x0 i: Thus x0 is null on this subspace. Then, since g is open from X onto Z there exists a continuous linear form z 0 on Z such that x0 = z 0 g: Since C = g(B) we see that (z 0 ; s) 2 C P and for any w 2 fx0 sg we have w 2 = ff < $g, i.e. f (w) $ for all w 2 fx0 sg, hence f Q (x0 ; s) $ and 0 0 0 P P Q T f (x ; s) $. By weak duality we get f (x ; s) = f (g (z ); s) = $ and (z 0 ; s) is a solution to the dual problems. Remark. If for some x in the solution set S of (C) f is semi-strictly quasiconvex at x in the sense that for all x0 2 ff < f (x)g and all t 2 [0; 1[ one has f ((1 t)x0 + tx) < f (x); then one can take s = hx; g T (z 0 )i since for any sequence (tn ) ! 1 with tn < 1 for all n we have xn := (1 tn )x0 + tn x 2 ff < f (x)g and (xn ) ! x; hence s limn hxn ; x0 i = hx; x0 i s: Remark. If f (0) < $ we see that s > 0 and setting fr (z 0 ) :=
inf f (fz 0
1g)
C r := fz 0 2 Z 0 : C
fz 0
z0 2 Z 0;
1gg;
for z00 := z 0 =s; we have z00 2 C r , fr (z00 ) := f Q (z00 ; 1) = f Q (z 0 ; s), so that z00 is a solution to the dual problem (D r ) maximize
fr (z 0 )
z0 2 C r
considered in [37, Corollary 3.3] under the assumption that f is radiant and upper semicontinuous. Remark. If C is absorbant, the inequality hz; z 0 i s for all z 2 C implies that htz; z 0 i s for all 0 z 2 Z and all t > 0 small enough. Since z 6= 0; we get s < 0 and z00 := z 0 =s 2 C := fz 0 2 Z 0 : C fz 0 1gg so that z00 is a solution to the dual problem (D ) maximize
f (z 0 )
z0 2 C
considered in [37, Corollary 3.5], with f (z 0 ) := f Q ( z00 ; 1). In order to give a variant, let us recall that a subset S of a convex subset B of X is extremal in B if for any x0 , x1 2 B such that (1 t)x0 + tx1 2 S for some t 2]0; 1[ one has either x0 2 S or x1 2 S; equivalently S is extremal in B if, and only if BnS is convex. Theorem 16 Let C be convex, nonempty, and let B := g 1 (C); with g linear, continuous and surjective. Suppose X is …nite dimensional, f is quasiconvex, upper semicontinuous, and inf f (X) < inf f (B). If the set S of solutions to (C) is extremal (in particular if it is empty), and the interior of S in B is 0 0 empty, then there is no duality gap for the dual problem (DP ) and (DP ) has a solution (z 0 ; s) with z 0 6= 0, s = hx; g T (z 0 )i for all x 2 S: Proof. We have ff $g \ (BnS) = ?: Since BnS and ff $g are convex and nonempty, the separation theorem in the …nite dimensional space X yields some (x0 ; s) 2 (X 0 nf0g) R such that for all x 2 BnS and all w 2 ff $g hx; x0 i s hw; x0 i: Since the interior of S in B is empty, for all x 2 S we can …nd a sequence (xn ) ! x with xn 2 BnS for all n 2 N, so that hx; x0 i = limn hxn ; x0 i s for all x 2 B. Again, since x0 is bounded below on Kerg, it is null on this subspace. Then, since g is open from X onto Z there exists some z 0 2 Z 0 such that x0 = z 0 g: Since C = g(B) we get hz; z 0 i s for all z 2 C, or (z 0 ; s) 2 C P . Moreover, since S is contained in ff $g we have hx; x0 i = s for all x 2 S. For all w 2 fx0 > sg we have w 2 = ff $g, or, equivalently, f (w) > $ for all w 2 fx0 > sg hence f P (x0 ; s) $: By the weak duality inequality (6) we get f P (g T (z 0 ); s) = $ so that 0 (z 0 ; s) is a solution to the dual problem (DP ) and there is no duality gap. We can deduce optimality conditions from the previous duality results. We denote by N (C; z) the normal cone to C at z 2 C in the sense of convex analysis: N (C; z) := fz 0 2 Z 0 : 8z 2 Z hz 14
z; z 0 i
0g:
Proposition 17 In order that x 2 B := g 1 (C) be a solution to (C) it su¢ ces that there exists some z 0 2 N (C; g(x)) and s 2 R such that (g T (z 0 ); s) 2 @ Q f (x): If assumption (b) of Theorem 15 is satis…ed then these conditions are necessary in order that x 2 g 1 (C) be a solution to (C). Proof. Suppose there exists some z 0 2 N (C; g(x)) such that (g T (z 0 ); s) 2 @ Q f (x) for some s 2 R. Then, for x := g T (z 0 ) we have cQ (x; (x0 ; s)) = 0 or hx; x0 i s: Then, for all z 2 C we have hz; z 0 i hz; z 0 i = hx; x0 i s, hence for all x 2 B we have hx; x0 i = hg(x); z 0 i s: Since (g T (z 0 ); s) 2 @ Q f (x) we have inf f (fx0 sg) = Q 0 f (x ; s) = f (x): In particular, for all x 2 B we have f (x) f (x) : x is a solution to (C): Now suppose the second set of assumptions of Theorem 15 is satis…ed and x 2 B is a solution to (C). 0 Then the conclusion of this theorem yields a solution (z 0 ; s) to the dual problem (DQ ): we have (z 0 ; s) 2 C P and f Q (g T (z 0 ); s) = inf f (C) = f (x): Thus, s hC ( z 0 ) or s hz; z 0 i for all z 2 C and in particular s hg(x); z 0 i = hx; g T (z 0 )i so that cQ (x; g T (z 0 )) = 0; and since f Q (g T (z 0 ); s) = f (x) we get (g T (z 0 ); s) 2 @ Q f (x): 0
Taking Z = X; g = IX the identity map, we get the following consequence. Corollary 18 Let f : X ! R be upper semicontinuous and quasiconvex and let C X be convex, nonempty, with inf f (X) < inf f (C) 2 R. Then x 2 C is a minimizer of f on C if, and only if there exists some x0 2 N (C; x) and s 2 R such that (x0 ; s) 2 @ Q f (x): Acknowledgement. The author is grateful to an anonymous referee for a minute reading and for criticisms that allowed to prevent the reader to stumble on some obscurities.
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