HAL Id: hal-00243009
https://hal.archives-ouvertes.fr/hal-00243009
Submitted on 6 Feb 2008
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Computing uniform convex approximations for convex envelopes and convex hulls
Rida Laraki, Jean-Bernard Lasserre
To cite this version:
Rida Laraki, Jean-Bernard Lasserre. Computing uniform convex approximations for convex envelopes and convex hulls. Journal of Convex Analysis, Heldermann, 2008, 15 (3), pp.635-654. �hal-00243009�
ECOLE POLYTECHNIQUE
CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE
Computing uniform convex approximations for convex envelopes and convex hulls
Rida Laraki Jean B. Lasserre
Juillet 2005
Cahier n° 2005-021
LABORATOIRE D'ECONOMETRIE
1rue Descartes F-75005 Paris (33) 1 55558215 http://ceco.polytechnique.fr/
mailto:[email protected]
Computing uniform convex approximations for convex envelopes and convex hulls
Rida Laraki
1Jean B. Lasserre
2Juillet 2005 Cahier n° 2005-021
Résumé: Nous établissons une procédure numérique pour approcher uniformément par une suite de fonctions convexes., l'enveloppe convexe d'une fraction rationnelle ayant pour domaine, D, un ensemble de dimension finie supposé compacte et semi- algébrique. Calculer la valeur d'une approximation en un point donné de K=co(D) se résume à résoudre un programme semi-défini. En suite, nous caractérisons K=co(D) comme projection d'un LMI semi-infini, et en plus nous approximons K par une suite décroissante d'ensembles convexes. Tester si un point donné n'est pas dans K se résume à résoudre un nombre fini de programmes semi-définis.
Abstract: We provide a numerical procedure to compute uniform (convex) approximations {f_{r}} of the convex envelope f of a rational fraction f, on a compact semi-algebraic set D. At each point x in K=co(D), computing f_{r}(x) reduces to solving a semidefinite program. We next characterize the convex hull K=co(D) in terms of the projection of a semi-infinite LMI, and provide outer convex approximations {K_{r}}?K. Testing whether x is not in K reduces to solving finitely many semidefinite programs.
Mots clés : Enveloppe convexe, programme semi-défini, dualité, ensemble semi-algébrique
Key Words : Convex envelope, semidefinite program, duality, semi-algbraic set Classification AMS:
1 CNRS, Laboratoire d'Econométrie de l'Ecole Polytechnique, 1 rue Descrates, 75005 Paris, [email protected]
2 CNRS, LAAS, 7 avenue du Colonel Roche, 31077 Toulouse Cédex 4, [email protected]
COMPUTING UNIFORM CONVEX APPROXIMATIONS FOR CONVEX ENVELOPES AND CONVEX HULLS
R. LARAKI AND J.B. LASSERRE
Abstract. We provide a numerical procedure to compute uniform (convex) approximations{fr} of the convex envelopefbof a rational fractionf, on a convex and compact semi-algebraic setD. At each pointx∈K, computing fr(x) reduces to solving a semidefinite program. We next characterize the convex hull K = co(D) in terms of the projection of a semi-infinite LMI, and provide outer convex approximations{Kr} ↓K. Testing whetherx /∈K reduces to solving finitely many semidefinite programs.
1. Introduction
Computing the convex envelope fbof a given function f : Rn → R is a chal- lenging problem, and to the best of our knowledge, there is still no algorithm that approximates fbby convex functions (except for the simpler univariate case). For instance, for a function f on a bounded domain Ω, Brighi and Chipot [4] propose triangulation methods and provide piecewise degree-1 polynomial approximations fh ≥fb, and derive estimates offh−fb(where h measures the size of the mesh).
Another possibility is to view the problem as a particular instance of the general moment problem, and use geometrical approaches as described in e.g. Anastassiou [1] or Kemperman [7]; but, as acknowledged in [1, 7], this approach is only practical for say, the univariate or bivariate cases.
Concerning convex sets, an important issue raised in Ben-Tal and Nemirovski [3], Parrilo and Sturmfels [12], is to characterize the convex sets that have aLMI (or semidefinite)representation, and calledSDr sets in e.g. [3]. For instance, the epigrah of a univariate convex polynomial is a SDr set; see [3]. So far, and despite some progress in particular cases (see e.g. the recent proof of the Lax conjecture by Lewis et al [11]), little is known. However, Helton and Vinnikov [6] have proved recently thatrigid convexity is a necessary condition for a set to be SDr.
In this paper we consider both convex envelopes and convex hulls for certain classes of functions and sets, namelyrational fractionsand compactsemi-algebraic sets. In both cases we show that one is able to provide relatively simple numerical approximations via semidefinite programming.
Contribution. Our contribution is twofold:
Concerning convex envelopes, we consider the class ofrational fractions f on a compact semi-algebraic setD⊂Rn (and +∞outsideD). We view the problem as a particular instance of the general moment problem, and we provide an algorithm for computingconvex anduniformapproximations of its convex envelope fb. More precisely, withK:= co(D) being the convex hull ofD:
- (a) We provide a sequence of convex functions {fr} that converges tofb,uni- formlyon any compact subset ofKin whichfbis continuous, asrincreases.
1
- (b) At each point x∈Rn, computing fr(x) reduces to solving a semidefinite programQrx.
- (c) For everyx∈intK, the SDP dualQ∗rxis solvable and any optimal solution provides an element of the subgradient∂fr(x) at the pointx∈intK.
Concerning sets, we consider the class of compact semi-algebraic subsetsD⊂Rn, and
- (d) We characterize its convex hull K := co(D) as the projection of a semi- infinite SDr setS∞, i.e., a set defined by finitely many LMIs involving matrices of infinite dimension, and countably many variables. Importantly, the LMI represen- tation ofS∞ is simple and givendirectly in terms of the data defining the original setD.
- (e) We provide outer convex approximations ofK, namely a monotone nonin- creasing sequence of convex sets{Kr}, withKr↓K. Each setKris the projection of a SDr setSr, obtained fromS∞by ”finite truncation”. Then, checking whether x /∈ K reduces to solving finitely many SDPs based on SDr setsSr, until one is unfeasible, which eventually happens for somer. Importantly again, the LMI rep- resentation of Sr is simple and given directly in terms of the data defining the original setD. Other outer approximations are of course possible, like e.g. convex polytopes{Ωr}containingK, but obtaining such polytopes with Ωr↓Kis far from trivial.
2. Notation definitions and preliminary results
In the sequel,R[y](:=R[y1, . . . , yn]) denotes the ring of real-valued polynomials in the variable y = (y1, . . . , yn). Let yi ∈ R[y] be the natural projection on the i-variable that is for everyx∈Rn, yi(x) =xi.For a real-valued symmetric matrix M, the notationM 0 stands forM is positive semidefinite.
LetD⊂Rn be compact, and denote by:
-K, the convex envelope ofD. Hence, by a theorem of Caratheodory,Kis convex and compact; see Rockafellar [14].
- C(D), the Banach space of real-valued continuous functions on D, equipped with the sup-normkfk:= supx∈D|f(x)|,f ∈C(D).
-M(D), its topological dual, i.e., the Banach space of finite signed Borel measures onK, equipped with the norm of total variation.
-M+(D)⊂M(D), the positive cone, i.e., the set of finite Borel measures on D. - ∆(D)⊂M+(D), the set of borel probability measures onD.
- forf inC(D),letfebe its natural extension toRn that is
(1) fe(x) :=
f(x) onD +∞ onRn\D.
Note that feis lowersemicontinuous (l.s.c.), admits a minimum and its domain is non-empty and compact.
- for f in C(D), let fbthe convex envelope of f .e This is the greatest convex function smaller thanfe.
Note that the vector spaces M(D) and C(D) are in duality with the duality bracket
hσ, fi:=
Z
K
f dσ, σ∈M(K), f∈C(K)
COMPUTING UNIFORM CONVEX APPROXIMATIONS FOR CONVEX ENVELOPES AND CONVEX HULLS3
Hence, let τ∗ denote the associated weak* topology; this is the coarsest topology onM(D) for which σ→ hσ, fiis continuous for every functionf in C(D).
With f ∈ C(D), and x ∈ K = co(D) fixed, arbitrary, consider the infinite- dimensional linear program (LP):
(2) LPx:
inf
σ∈M+(D) hσ, fi
s.t. hσ, yii =xi, i= 1, . . . , n hσ,1i = 1.
Its optimal value is denoted by inf LPx, and min LPx if the infimum is attained.
Notice that LPxis a particular instance of the generalmoment problem, as described in e.g. Kemperman [7, §2.6]. In particular, the set of x∈Rn such that LPx has a feasible solution, is called themoment space.
Lemma 2.1. Let K = co(D), f ∈ C(D) and febe as in (1). Then the convex envelope fboffeis given by:
(3) fb(x) =
min LPx, x∈K, +∞, x∈Rn\K, so that the domain of fbisK.
Proof. Forx∈ K, let ∆x(D) be the set of probability measures σon D, that are centered at x(that is hσ, yii =xi for i = 1, . . . , n). Let ∆∗x(D) ⊂∆x(D) be the subset of those probability measures that have a finite support. It is well know that
fb(x) = inf
σ∈∆∗x(D)
hσ, fi, ∀x∈K;
see Choquet [5] or Laraki [9]. Next, since ∆∗x(D) is dense in ∆x(D) with respect to the weak* topology, and ∆(D) is metrizable and compact with respect to the same topology (see Choquet [5]), deduce that for everyx∈K
fb(x) = min
σ∈∆x(D)hσ, fi
= min LPx.
Ifx /∈K, there is no probability measure onD, with finite support, and centered in
x; thereforefb(x) = +∞.
Next, letp, q∈R[y], withq >0 onD, and letf ∈C(D) be defined as
(4) y 7→ f(y) = p(y)/q(y), y∈D.
For everyx∈K, consider the LP,
(5) Px:
inf
σ∈M+(D) hσ, pi
s.t. hσ, yiqi =xi, i= 1, . . . , n hσ, qi = 1.
A dual ofPx, is the LP (6) P∗x: sup
γ∈R,λ∈Rn
{γ+hλ, xi: p(y)−q(y)hλ, yi ≥γq(y), ∀y∈D},
where hλ, yi:=Pn
i=1λiyi stands for the standard inner product inRn. The opti- mal value ofP∗x is denoted by supP∗x (and maxP∗x if the supremum is attained).
Equivalently, asq >0 everywhere onD, andf =p/qonD, (7) P∗x: sup
γ∈R,λ∈Rm
{γ+hλ, xi: f(y)− hλ, yi ≥γ, ∀y∈D}.
In view of the definition (4) off, notice that
f(y)− hλ, yi ≥γ, ∀y∈D ⇔ fe(y)− hλ, yi ≥γ, ∀y∈Rn.
Hence P∗x in (7) is just the dual LP∗xof LPx, for every x∈K, and so supP∗x= sup LP∗x, for everyx∈K. In fact we even have the following result:
Theorem 2.2. Let p, q ∈R[x] with q >0 on D, and let f be as in (4). Letx∈ K= co(D)be fixed, arbitrary, and letPxandP∗x be as in (5) and (7), respectively.
ThenPx andLPx are solvable and there is no duality gap, i.e., (8) supP∗x = sup LP∗x = min LPx = minPx = fb(x), x∈K.
Proof. Observe thatPxis equivalent to LPx. Indeed, withσ an arbitrary feasible solution ofPx, the measuredµ:=qdσis feasible in LPx, with same value. Similarly, withµan arbitrary feasible solution of LPx, the measuredσ:=q−1dµ, well defined onDbecauseq >0 onD, is feasible inPx, and with same value. Finally, it is well known thatfbis the Legendre-Fenchel biconjugate1offe, and sofb(x) = supP∗x, for allx∈K. Indeed, letf∗:Rn →Rbe the Legendre-Fenchel conjugate offe, i.e.,
λ7→ f∗(λ) := sup
y∈Rn
:{hλ, yi −fe(y)}.
In view of the definition offe,
f∗(λ) = sup
y∈D
:{hλ, yi −f(y)}, and therefore,
supP∗x = sup
λ
{hλ, xi+ inf
y∈D
{f(y)− hλ, yi}}
= sup
λ
{hλ, xi −sup
y∈D
{hλ, yi −f(y)}}
= sup
λ
{hλ, xi −f∗(λ)} = (f∗)∗(x) = fb(x).
In fact, we have the following.
Corollary 2.3. Let x∈Kbe fixed, arbitrary, and let P∗xbe as in (7).
(a) P∗x is solvable iff ∂fb(x) 6= ∅2. In that case, any optimal solution (λ∗, γ∗) satisfies:
(9) λ∗∈∂fb(x), and γ∗ = −f∗(λ∗).
(b) If f is a rational fraction onD as in (4), then∂fb(x)6=∅ for every x inK so that, in this caseP∗x is solvable and (a) holds for everyxinK.
1See Section 2 in Benoist and Hiriart-Urruty [2]
2∂fb(x)6=∅at least for everyxin the relative interior ofK(see Rokafellar 1970, Theorem 23.4)
COMPUTING UNIFORM CONVEX APPROXIMATIONS FOR CONVEX ENVELOPES AND CONVEX HULLS5
Proof. Suppose that for some x ∈ K, P∗x is solvable (that is, the supremum is achieved, say atλ∗(x) andγ∗(x)). Then, for everyy∈K,
fb(x) = hλ∗(x), xi+γ∗(x) fb(y) = sup
λ,γ
{hλ, yi+γ:f(z)− hλ, zi ≥γ, ∀z∈D}, y ∈K
≥ hλ∗(x), yi+γ∗(x),
and in view of (3), the latter inequality also holds for everyy in Rn; therefore, fb(y)−fb(x)≥ hλ∗(x), y−xi, ∀y∈Rn.
Hence, λ∗(x) ∈ ∂fb(x). Finally, from the standard Legendre-Fenchel equality, we deduce thatγ∗(x) =−f∗(λ∗(x)) wheref∗ is the Legendre-Fenchel conjugate offe. Conversely, ifλ∗(x)∈∂fb(x) then, by the Legendre-Fenchel equality we have,
fb(x) =hλ∗(x), xi −f∗(λ∗(x)).
Therefore, we have:
(10) supP∗x = f(x) =b hλ∗(x), xi −f∗(λ∗(x)).
Next, from
f∗(λ∗(x)) = sup
y∈D
hλ∗(x), yi −f(y), we have
−f∗(λ∗(x)) ≤ f(y)− hλ∗(x), yi, ∀y∈D,
which shows that the pair (λ∗(x),−f∗(λ∗(x))) is a feasible solution ofP∗x, and in view of (10), an optimal solution.
Now, if f is a rational fraction on D, then it is differentiable and Lipschitz on Dso that, from Theorem 3.6 in Benoist and Hiriart-Urruty [2],∂fb(x) is uniformly bounded as x varies on the relative interior of K. Since fbis l.s.c. (see below), we deduce that ∂fb(x)6=∅ for every xin K. Actually, let x∈K and let xn be a sequence in the relative interior ofKthat converges toxand letλn∈∂fb(xn) such that λn →λ(which is possible (passing to a subsequence if needed) since ∂fb(xn) is uniformly bounded). Hence, for everyy inRn,
fb(y)≥fb(xn) +hλn, y−xni, so that,
fb(y) ≥ lim inf
n→∞ fb(xn) +hλ, y−xi
≥ fb(x) +hλ, y−xi
consequently,λ∈∂fb(x), the desired result.
In other words, any optimal solution ofP∗xprovides an element of the subgradient of fbat the point x. Corollary 2.3 should be viewed as a refinement for convex envelopes of rational fractions, of Theorem 2.20 in Kemperman [7, p. 28] for the general moment problem, where strong duality results are obtained for the interior of the moment space (here intK) only.
3. Preservation of continuity
Later we will construct a sequence fr that approximates fbuniformly at each compact set on whichfbis continuous. Hence it is natural to first define conditions on the data, to ensure thatfbis continuous. The question was solved in Laraki [9]
whenDis convex (K= co(D)).Here we extend this to our general framework.
Definition 3.1. The compact set D of Rnis Splitting-Continuous if and only if x7→∆x(D) is continuous when∆(D)is equipped with the weak* topology.
Examples : this is exactly the Splitting-Continuous condition defined in Laraki [9] when D is convex. Note that ifD is a polytope or is strictly-convex then it is Splitting-Continuous (see [9][Theorem 1.16]. In fact, if D is convex then there are equivalence between Splitting-continuous and the more natural condition faces- closed. The latter concept means (whenDis convex) that any Hausdorff converging sequence of faces ofDis also a face ofD(see Laraki 2004, Theorem 1.16).
Lemma 3.2. Let f be continuous on the compact D of Rn. Let fbbe as in (3).
Then,fbis always l.s.c. with respect toK= co(D)and is continuous on any compact K that is strictly included in the relative interior of K. Moreover, D is Splitting- Continuous if and only iffbis continuous on K, for everyf which is the restriction onD of some continuous function onK.
Proof. From Theorem 10.1 in Rockafellar [14], sincefbis convex, it is always con- tinuous in the relative interior ofK. Also, since the correspondencex7→∆x(D) is always uppersemicontinuous (u.s.c.), and sincef is continuous, fbis always l.s.c. ; see e.g. Theorem 6 in Laraki and Sudderth [8].
Again, from Theorem 6 in Laraki and Sudderth [8],x7→∆x(D) is continuous if and only iffbis continuous onK, for everyf which is the restriction onDof some
continuous function onK.
4. Uniform convex approximations of fbby SDP-relaxations In this section, we assume that f is defined as in (4) for some polynomials p, q ∈ R[x], with q > 0 on K, where D ⊂ Rn is the convex and compact semi- algebraic set defined as
(11) K:={x∈Rn:| gj(x)≥0, j= 1, . . . , m},
for some polynomials{gj} ⊂ R[x]. Depending on its parity, let 2rj−1 or 2rj be the total degree ofgj, for all j= 1, . . . , m. Similarly, let 2rp,2rq or 2rp−1,2rq−1 be the total degree ofpandqrespectively.
We next provide a sequence{fr}rof functions such that for all r -fr is convex;
- the domain offr isKr⊃K= co(D);
-fr≤fband for everyx∈K,fr(x)↑f(x) asb r→ ∞. In fact, we even have
r→∞lim :kfb−frk
K → 0,
that is,fr converges tofb,uniformly on any compactK⊂Kin whichfbis continu- ous. Consequently, ifDis Splitting-Continuous and ifq >0 on K, then we obtain uniform convergence onK. Also, ifKis strictly included in the relative interior of Kthen we also obtain uniform convergence.
To do this we first introduce some additional notation.
COMPUTING UNIFORM CONVEX APPROXIMATIONS FOR CONVEX ENVELOPES AND CONVEX HULLS7
4.1. Notation and definitions. Lety ={yα}α∈Nn be a sequence indexed in the canonical basis{zα}ofR[z], and letLy:R[z]→Rbe the linear functional defined by
h(:= X
α∈Nn
hαzα) 7→Ly(h) := X
α∈Nn
hαyα.
Let Pk ⊂ R[z] be the space of polynomials of total degree less than k, and let r0 := max[rp, rq + 1, r1, . . . , rm]. Then forr ≥r0, consider the optimization problem:
(12) Qrx:
infyLy(p)
s.t. Ly(ziq) = xi, i= 1, . . . , n, Ly(h2)≥0, ∀h∈ Pr,
Ly(h2gj)≥0, ∀h∈ Pr−rj,:j = 1, . . . , m, Ly(q) = 1.
Problem Qrx is a convex optimization problem, in fact, a so-called semidefinite programming problem, called a SDP-relaxation ofPx. For more details on semi- definite programming and its applications, the reader is referred to Vandenberghe and Boyd [16].
Indeed, giveny={yα}, letMr(y) be themoment matrix associated withy, that is, the rows and columns ofMr(y) are indexed in the canonical basis ofPr, and the entry (α, β) is defined by Mr(y)(α, β) =Ly(zα+β) =yα+β, for all α, β ∈N, with
|α|,|β| ≤r. Then
Ly(h2)≥0, ∀h∈ Pr ⇔ Mr(y)0.
Similarly, writing
z 7→ gj(z) := X
γ∈Nn
(gj)γzγ, j= 1, . . . , m,
thelocalizing matrixMr(gjy) associated with y and gj ∈R[z], is the matrix also indexed in the canonical basis ofPr, and whose entry (α, β) is defined by
Mr(gjy)(α, β) = Ly(gj(z)zα+β) = X
γ∈Nn
yα+β+γ(gj)γ, for allα, β∈N, with|α|,|β| ≤r. Then, for everyj= 1, . . . , m,
Ly(gjh2)≥0, ∀h∈ Pr ⇔ Mr(gjy)0.
Observe that ify has arepresenting measure µy, i.e. if yα =
Z
xαdµy, ∀α∈Nn,
then, withh∈ Pr, and also denoting byh={hα} ∈Rs(r)its vector of coefficients in the canonical basis,
hh, Mr(gjy)hi = Z
h2gjdµy.
Therefore, ifµy has its support contained in the level set{x∈Rn: gj(x)≥0}, we must haveMr(gjy)0.
One also denotes by M∞(y) and M∞(gjy) the (obvious) respective ”infinite”
versions ofMr(y) andMr(gjy), i.e., moment and localizing matrices with countably
many rows and columns indexed in the canonical basis{zα}, and involvingall the variablesy (as opposed to finitely many inMr(y) andMr(gjy)).
For more details on moment and localizing matrices, the reader is referred to Lasserre [10].
4.2. SDP-relaxations. Hence, using the above notation, the optimization prob- lemQrx defined in (12) is just the SDP
(13) Qrx:
infyLy(p)
s.t. Ly(ziq) = xi, i= 1, . . . , n Mr(y) 0,
Mr−rj(gjy) 0, j= 1, . . . , m, Ly(q) = 1,
with optimal value denoted infQrx, and minQrx if the infimum is attained.
If we write Mr(y) = P
αBαyα, and Mr−rj(gjy) = P
αCαjyα, for appropriate symmetric matrices{Bα, Cαj}, then the dual ofQrx is the SDP
(14)
Q∗rx:
sup
λ,γ,X,Zj
γ+hλ, xi s.t. hBα, Xi+
m
X
j=1
hCαj, Zji+γqα+
n
X
i=1
λi(ziq)α =pα, |α| ≤2r X, Zj 0.
In fact,Q∗rx is the same as (lettingg0≡1)
(15) Q∗rx:
sup
γ∈R,λ∈Rn,uj∈R[z]
γ+hλ, xi s.t. p−γq− hλ, ziq =
m
X
j=0
ujgj
uj s.o.s., degujgj≤2r, j= 0, . . . , m (where s.o.s. stands forsum of squares).
We next make the following assumption on the polynomials {gj} ⊂R[z] in the definition of the setDin (11).
Assumption 4.1. There is a polynomialu∈R[z], positif onD, which can written
(16) u = u0+
m
X
j=1
ujgj,
for a family of s.o.s. polynomials {uj}mj=0 ⊂ R[z], and the level set {z ∈ Rn| : u(z)≥0} is compact.
Assumption 4.1 is not very restrictive. For instance, it is satisfied if:
- all thegj’s are linear (and so,Dis a polytope), or if
- the level set{z∈Rn|gj(z)≥0} is compact, for somej∈ {1, . . . , m}.
Moreover, if one knows that the compact set D is contained in a ball {z ∈ Rn|:kzk ≤M}, for someM ∈R, then it suffices to add the redundant quadratic constraintM2−kzk2≥0 in the definition (11) ofD, and Assumption 4.1 holds true.
COMPUTING UNIFORM CONVEX APPROXIMATIONS FOR CONVEX ENVELOPES AND CONVEX HULLS9
Under Assumption 4.1, every polynomialv∈R[x], strictly positive onD, can be written as
v = v0+
m
X
j=1
vjgj,
for some family of s.o.s. polynomials{vj}mj=0⊂R[x]. This is Putinar’s Positivstel- lensatz, an particular version of Schm¨udgen’s Positivstellensatz (see Putinar [13]).
Then we have the following result:
Theorem 4.2. Let D be as in (11), and let Assumption 4.1 hold. Let f be as in (4) with p, q ∈ R[z], and with q > 0 on D. Let fbbe as in (3), and with x ∈ K
= co(D)fixed, consider the SDP-relaxations{Qrx} defined in (12) (or equivalently in (13)). Then:
(a) For every x∈Rn,
(17) infQrx ↑ fb(x), as r→ ∞.
(b) The function fr:Rn→R∪ {+∞} defined by (18) x 7→ fr(x) := infQrx, x∈Rn, is convex, and asr→ ∞,fr(x)↑fb(x) pointwise, for allx∈Rn.
(c) IfKhas a nonempty interiorintK, then
(19) supQ∗rx = maxQ∗rx= infQrx = fr(x), x∈intK, and for every optimal solution (λ∗r, γ∗r)of Q∗rx,
fr(y)−fr(x) ≥ hλ∗r, y−xi, ∀y∈Rn, that is,λ∗r∈∂fr(x).
For a proof see§6.1. As a consequence, we also get:
Corollary 4.3. Let Dbe as in (11), and let Assumption 4.1 hold. Letf andfbbe as in (4) and (3) respectively, and letfr:K→R, be as in Theorem 4.2. Then fr is always l.s.c. Moreover, for every compact K⊂ Kin whichfbis continuous,
(20) lim
r→∞: sup
x∈K
:|fb(x)−fr(x)| = 0,
that is, the monotone nondecreasing sequence {fr} converges to fb, uniformly on every compact in whichfbis continuous. Consequently, if in additionDis Splitting- continuous and if q >0 onK, then the convergence is uniform onK.
Proof. The lowersemicontinuity of fr may be obtained using Laraki & Sudderth [8]. This is due to the facts that:
- the objectif function ofQrxdoes not depend onx, and
- the feasible set ofQrxas a function ofx, is u.s.c., in the sense of Kuratowski.
By Theorem 4.2, we already have thatfr↑fbonK.
(1) The convergencefr↑fbonK, (2) the fact thatfr is l.s.c., (3) that the limit fbis continuous, and finally (4) thatK is compact, imply that the convergence is uniformonK. Actually, by (1) and (2) the functionfb(x)−fr(x) is always positive and u.s.c. onK, and therefore, by (4), it admits a global maximizer xr∈K. Let α= lim supr→∞fb(xr)−fr(xr). Without loss of generality and using (4), suppose
