AW -Convergence and Well-Posedness of Non Convex Functions

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1 Journal of Convex Analysis Volume 10 (2003), No. 2, AW -Convergence Well-Posedness of Non Convex Functions Silvia Villa DIMA, Università di Genova, Via Dodecaneso 35, Genova, Italy Received July 6, 2001 Revised manuscript received July 24, 2002 Let us consider the set of lower semicontinuous functions defined on a Banach space, equipped with the AW -convergence. A function is called Tikhonov well-posed provided it has a unique minimizer to which every minimizing sequence converges. We show that well-posedness of f guarantees strong convergence of approximate minimizers of τ aw -approximating functions (under conditions of equiboundedness of sublevel sets), to the minimizer of f. Moreover we show that a lower semicontinuous function f which satisfies growth conditions at is well-posed iff its lower semicontinuous convex regularization is. Finally we investigate the link between AW -convergence of non convex integrs that of the associated integral functionals. Keywords: AW -convergence, well-posedness, optimization problems 2000 Mathematics Subject Classification: 49K40 1. Introduction In this paper we study non convex minimization problems the AW -convergence. In particular we deal with Tikhonov well-posed functions, i.e. functions with a unique minimum point to which every minimizing sequence converges (see [6] for a complete reference). We are also interested in another concept of well-posedness: the continuous dependence of the minimizer on problem s data. The latter means that approximate minimizers of perturbed functions f n converge to the minimizer of f, that inf f = lim(inf f n ), when we consider an appropriate convergence on the set of lower semicontinuous, proper functions. It turns out that usual notions of convergence (pointwise convergence, uniform convergence) are not well suited in this setting; this explains the attention paid to the AW -convergence, which was deeply studied in the papers of Attouch Wets [1], [2], [3]; in fact this convergence satisfies the desired stability conditions. The two notions of well-posedness are strongly correlated: Theorem 4.1 of [5] establishes the equivalence between Tikhonov well-posedness continuous dependence of the minimizer on problem s data for a proper, lower semicontinuous convex function when the set of such functions is endowed with the AW -convergence. Purpose of this paper is to underst what happens when the functions we consider are proper lower semicontinuous, but not convex. In Section 3 we prove that the previous equivalence is still valid if we require the sublevel Most of this paper is part of the "Tesi di laurea" of the author. ISSN / $ 2.50 c Heldermann Verlag

2 352 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions sets of f n at height near to inf f n to be equibounded. Another way to approach the well-posedness problem of a given function f, is to study the greater lower semicontinuous convex function which minorizes f; this function, denoted by f, is called the lower semicontinuous, convex regularization of f. We establish the equivalence between well-posedness of f of f under the assumption that lim inf x + (f(x)/ x ) (0, + ]. Finally, in Section 4, we investigate the case of non convex integral functionals; more precisely, we prove a dominated convergence theorem for AW -convergence. A similar result is also valid for Mosco convergence in the convex case (see [8]). 2. Definitions preliminaries In order to introduce definitions results, we start with some notations. We always denote by (X, ) a normed linear space by d the distance function generated by the norm. For any subset A of X, d(x, A) := inf x y y A denotes the distance from x to A. In any space we consider, the unit ball is U := {x : x 1} ρu denotes the ball of radius ρ 0. For any set A X ρ 0, we set A ρ := A ρu. Γ 0 (X) denotes the set of all extended real valued, proper, lower semicontinuous convex functions on X. We write v(f) = inf{f(x) : x X}, argminf for the possibly empty set of points {x X : f(x) = v(f)}. For each α R, we denote by lev(f; α) the sublevel set of f at height α, that is, {x X : f(x) α}. For A, B X, the excess of A on B is e(a, B) := sup d(x, B), x A with the convention that e(a, B) = 0 if A = e(a, B) = + if B =. Then, the ρ-hausdorff distance between A B is the following number: haus ρ (A, B) := max{e(a ρ, B), e(b ρ, A).} Definition 2.1. For ρ 0, the ρ-(hausdorff)-distance between two extended real valued functions f g defined on X is haus ρ (f, g) := haus ρ (epif, epig), where the unit ball of X R is the set U := {(x, α) : x 1, α 1}. Definition 2.2. Let f, f n : X [, + ] be lower semicontinuous functions. We say that f n AW - converge to f, we write f = τ aw lim f n, iff: ρ 0 > 0 such that ρ > ρ 0 haus ρ (f, f n ) 0 as n +.

3 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions 353 The following theorem gives a method to compute, or at least estimate, the ρ-hausdorff distance between two functions, see [1]. Theorem 2.3 (Kenmochi conditions). Suppose f, g are proper extended real valued functions defined on a normed real space X, both minorized by α 0 p α 1 for some α 0 0, α 1 R, p 1. Let ρ 0 > 0 be such that (epif) ρ0 (epig) ρ0 are nonempty. 1. Then the following conditions hold: for all ρ > ρ 0 x domf such that x ρ, f(x) ρ, for every ɛ > 0 there exists some x ɛ domg that satisfies x x ɛ haus ρ (f, g) + ɛ, g(x ɛ ) f(x) + haus ρ (f, g) + ɛ as well as a symmetric condition with the roles of f g interchanged. 2. Conversely, assuming that for all ρ > ρ 0 > 0 there exists n(ρ) 0, such that for all x domf with x ρ, f(x) ρ, there exists x domg that satisfies x x n(ρ), g( x) f(x) + n(ρ), the symmetric condition (interchanging the roles of f g), then with ρ 1 := ρ + α 0 ρ p + α 1, haus ρ (f, g) n(ρ 1 ). The next theorem is a sequential characterization of AW -convergence it will be useful in the following, see [10]. Theorem 2.4. Let f n, f : X (, ], n N, be proper lower semicontinuous functions. Assume v(f) >. Then f = τ aw lim f n if only if the following two conditions hold. (i) If x n is a bounded sequence then f n (x n ) is bounded below. Moreover, if f n (x n ) is bounded, then there are sequences y n in X ɛ n 0 such that x n y n 0, ɛ n 0 f(y n ) f n (x n ) + ɛ n, n N. (1) (ii) For any bounded sequence x n such that f(x n ) is bounded above there are sequences y n in X ɛ n 0, such that x n y n 0, ɛ n 0 f n (y n ) f(x n ) + ɛ n, n N. (2) Now, let us recall the definition of Tikhonov well-posedness: Definition 2.5. We say that a function f : X (, + ] is Tikhonov well-posed (shortly, well-posed) if it satisfies the following conditions: 1. there exists a unique global minimum point x 0 for f; 2. if x n is any minimizing sequence, i.e. a sequence in X such that f(x n ) f(x 0 ), then x n x 0. Let f n be a sequence of functions bounded from below. We say that a sequence x n X is asymptotically minimizing iff f n (x n ) inf f n 0. Let us recall some characterizations of well-posedness:

4 354 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions Definition 2.6. A function is called a forcing function iff c : D [0, + ) 0 D [0, + ), c(0) = 0 a n D, c(a n ) 0 a n 0. The proof of the following fact is in [6]. Theorem 2.7. Let f : X (, + ] be proper, bounded from below lower semicontinuous. f is well-posed iff there exists a forcing function c a point x 0 such that f(x) f(x 0 ) + c[d(x, x 0 )] for every x X. Finally we recall a characterization of well-posedness due to Furi Vignoli (see [7]): Theorem 2.8. A proper lower semicontinuous function f : X [, + ] is well-posed if only if inf{diam[lev(f; α)] : α > v(f)} = Well-posedness AW -convergence for non convex functions The main result of this section is a generalization of the following theorem, due to Beer Lucchetti [5], in the case of non convex functions. Theorem 3.1 (Beer-Lucchetti). Let X be a Banach space f Γ 0 (X) bounded from below. Then the following conditions hold: 1. if f is well-posed then the conditions ɛ n > 0, ɛ n 0, f = τ aw lim f n, with f n Γ 0 (X) x n lev(f n ; v(f n ) + ɛ n ) for each n, imply that x n is convergent to the unique minimizer of f; 2. whenever the conditions ɛ n > 0, ɛ n 0, f = τ aw lim f n, with f n Γ 0 (X) x n lev(f n ; v(f n ) + ɛ n ) for each n imply that x n is convergent, then f is well-posed. We start with the following lemma, which obtains the continuity of the value function v. Lemma 3.2. Let ɛ n > 0 be such that ɛ n 0, let f, f n : X (, + ] be lower semicontinuous proper. Assume v(f) > suppose there exists α > v(f) such that lev(f; α) is bounded. Moreover assume there is m > 0 such that lev(f n ; v(f n ) + ɛ n ) mu for each n sufficiently large f = τ aw lim f n. Then v(f n ) v(f). Proof. By definition of v(f), there exists ɛ (0, 1] such that v(f) + ɛ α therefore there is ρ such that lev(f; v(f) + ɛ) ρu. Let x lev(f; v(f) + ɛ). Then the pair (x, f(x)) epif ru, with r max{ρ, v(f) + 1}. From Theorem 2.4 we get the existence of y n X, δ n 0, such that x y n 0, δ n 0 f n (y n ) f(x)+δ n. Hence we have: v(f n ) f n (y n ) f(x) + δ n v(f) + ɛ + δ n. (3) Passing to the upper limit we have: lim sup v(f n ) v(f). (4)

5 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions 355 Now consider x n lev(f n ; v(f n ) + ɛ n ). By hypothesis x n is bounded, so, from Theorem 2.4 we get that f n (x n ) is bounded below; moreover, from f n (x n ) v(f n ) + ɛ n (3) we obtain that f n (x n ) is also bounded above therefore, by Theorem 2.4, we have that there exist y n X, δ n 0, such that y n x n 0, δ n 0 f(y n ) f n (x n ) + δ n. Then we have: v(f) f(y n ) f n (x n ) + δ n v(f n ) + ɛ n + δ n. Passing to the lower limit we get: Hence, recalling (4), we have v(f n ) v(f). v(f) lim inf v(f n ). Remark 3.3. Notice that we haven t used the hypothesis of equiboundedness of sublevel sets to prove upper semicontinuity of the value function v, which therefore is valid more generally for proper lower semicontinuous functions only. We shall use the following assumption repeatedly on a sequence ɛ n on a sequence f n : X (, + ]: Assumption ɛ n > 0 ɛ n 0; 2. f n are proper, lower semicontinuous there exists m > 0 such that lev(f n ; v(f n )+ ɛ n ) mu for each n sufficiently large. Now we are able to prove the main result of this section: Theorem 3.5. Let X be a Banach space f : X (, + ] be bounded from below, proper lower semicontinuous. 1. Suppose that for every sequences ɛ n, f n satisfying Assumption 3.4 with f := τ aw lim f n, every asymptotically minimizing sequence corresponding to f n is convergent. Then f is well-posed. 2. Suppose f is well-posed. Then for every sequences ɛ n f n satisfying Assumption 3.4 f := τ aw lim f n, whenever x n is asymptotically minimizing, then x n is convergent to the unique minimizer of f. Proof. 1. Let us define f n = f for each n N. Consider ɛ n > 0, ɛ n 0 two sequences x n y n belonging to lev(f; v(f) + ɛ n ) for each n N. Suppose lim y n = y lim x n = x. Now consider the sequence: { x n if n is even, z n := if n is odd. y n The sequence z n is minimizing, thus convergent thus x = y. All the sequences which satisfy the hypothesis are therefore convergent to the same point x 0. Now, let x n be a minimizing sequence x nk := y k any subsequence. By the definition of minimizing sequence there exists a subsequence y kh := z h such that z h lev(f; v(f) + ɛ h ), ɛ h 0, which is convergent to x 0. Since the subsequence was arbitrary, x n is itself

6 356 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions convergent. On the other h, from the definition of minimizing sequence of lower semicontinuity of f, we get: lim inf (v(f) + ɛ h ) lim inf f(z h ) f(x 0 ) v(f), so x 0 argminf. 2. Let x be the minimum point of f. By well-posedness Lemma 3.2, v(f n ) f(x). Pick x n lev(f n ; v(f n ) + ɛ n ) fix ɛ > 0: we will find N ɛ such that for each n > N ɛ, x n x < ɛ. Since f is well-posed, from the Furi-Vignoli characterization, Theorem 2.8, we get the existence of δ > 0 such that diam[lev(f; v(f) + δ)] < ɛ/2. (5) Moreover there exists N N such that, when n > N we have: - ɛ n < δ/3, - v(f n ) < v(f) + δ/3. Since f n (x n ) is bounded, by Theorem 2.4, there exist y n in X, γ n 0, such that y n x n 0, γ n 0 f(y n ) f n (x n ) + γ n. Choosing N ɛ > N such that for each n > N ɛ we have x n y n < ɛ/2 γ n < δ/3, we get: Therefore, by (5) we have: f(y n ) f n (x n ) + γ n v(f n ) + γ n + ɛ n v(f) + δ. x n x x n y n + y n x < ɛ. Thus the sequence x n goes to x the proof is complete. The hypothesis of equiboundedness of sublevel sets cannot be omitted, as the following example shows: Example 3.6. Let f, f n : R R be such that f(x) = x : x if x n, f n (x) = x + 2n if n < x 2n, x + 2n if x > 2n. Clearly f is well-posed 0 is the unique minimizer, f = τ aw lim f n since f n converges uniformly to f on bounded sets (for a proof of this fact, see e.g. [4]), but taking x n = 2n we have f n (x n ) = min f n = 0, however x n 0. Notice that the condition on the equiboundedness of sublevel sets is satisfied when the approximating functions are convex: this follows from Lemma 3.1 Theorem 3.6 of [5]. The same condition is also satisfied when the functions f n are equicoercive: Proposition 3.7. Let f, f n : X (, + ] with lim sup v(f n ) < +. Moreover suppose there exists ϕ : X (, + ] such that: f n (x) ϕ(x) for each n lim ϕ(x) = + x + let f = τ aw lim f n, ɛ n > 0, ɛ n 0. Then there exists m > 0 such that lev(f n ; v(f n ) + ɛ n ) mu if n is sufficiently large.

7 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions 357 Proof. By Remark 3.3 we get v(f n ) < v(f) + 1 if n is sufficiently large; moreover we can assume without loss of generality ɛ n 1 for each n. Taking x n lev(f n ; v(f n ) + ɛ n ), with n sufficiently large, we have: ϕ(x n ) f n (x n ) v(f n ) + 1 v(f) + 2. (6) Since ϕ(x) + if x +, there exists R > 0 such that ϕ(x) > v(f) + 2 if x > R. Thus, from (6), we deduce x n R, i. e. equiboundedness of sublevel sets. Let us change point of view, find the relations between well-posedness of a given function f well-posedness of its convex envelope f. Definition 3.8. Let f : X (, + ] be proper, bounded from below lower semicontinuous. We define the convex, lower semicontinuous regularization of f to be the function f such that epi f = cl co(epif). The definition is consistent, in the sense that the convex hull of an epigraph is still an epigraph. Moreover f is the largest convex lower semicontinuous function minorizing f. Remark 3.9. Let f : X (, + ] be as in the definition above suppose that min f = 0. This means that epif X [0, + ] which is a closed convex subset of X (, + ] so, by the definition of f, we have epi f X [0, + ]. The following theorem establishes the relation between well-posedness of a given function f well-posedness of its convex, lower semicontinuous regularization: Theorem Let f : X (, + ] be proper lower semicontinuous. Suppose min f = f(0) = 0. The following properties are equivalent: (i) (ii) f(x) f is well-posed lim inf x + x f is well-posed. (0, + ]; Proof. Suppose condition (i) is satisfied. Then there exists M > 0 such that lim inf x + 4M. By Remark 3.9 we have that min f inf f. On the other h f(x) f(x) f(x) x for every x X, so inf f = f(0) = f(0). We observe that, since f is well-posed, by Theorem 2.7, there exists a forcing function c such that f(x) c( x ). Without loss of generality we can assume that lim inf t + c(t)/t 2M. In fact, if lim inf t + c(t)/t = 0, we can replace c by the function c 1 defined in this way: c 1 (t) = { c(t) if t r 2Mt if t > r, with r such that f(x) 2M x when x > r. It s enough to prove that c is forcing. In fact we have: f(x) c( x ),

8 358 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions which means f is well-posed in the case that c is forcing. First observe that c(x) 0 for every x X c(0) = c(0) = 0 from the previous argument. By contradiction, let us suppose that there exist t n [0, + ) such that c(t n ) 0, but t n 0. Since t n 0, we can assume (maybe considering a subsequence) that there exists a > 0 such that t n 2a for each n. Since a forcing function is well-posed, thanks to the Furi-Vignoli characterization of well-posedness, Theorem 2.8, we can find δ > 0 such that lev(c; δ) [0, a). Moreover, since lim inf c(t)/t = 2M, there exists R > 0 such that c(t) Mt for all t > R. We define g : [0, + ) [0, + ) to be the following function: 0 if t a g(t) = δ if a < t K Mt + δ MK if t > K. Choose K such that K max{ δ + a, R}. By construction, we get g(t) c(t) for every M t. In this way g is the following function: 0 if t a δ g(t) = K a t δa if a < t < K K a Mt + δ MK if t K. g is a convex lower semicontinuous function which minorizes c, so we have c g, min c = g(0) = 0. The sequence t n is therefore minimizing also for g; but t n 2a g(t n ) 0, a contradiction. We then have that c is forcing f is well-posed. Now suppose f is well-posed. We get min f = min f = f(0) = f(0). Let x n be a minimizing sequence for f: then x n is a minimizing sequence also for f therefore x n 0. This means that f is well-posed. For the sake of contradiction, assume lim inf x + (f(x)/ x ) = 0. Then it s possible to find a sequence y k in X such that y k > k f(y k ) < 1 k y k. Recalling the definition of epigraph (y k, 1 k y k ) epif. Since epi f = cl co(epif), the segments joining (0, 0) to (y k, 1 k y k ) are entirely contained in epi f. Consider the sequence z k := t k y k, with t k = 1 Then t k y k = t k y k = 1, so z k 0. Moreover (t k y k, (t k /k) y k ) epif t kk y k = 1 0; that is z k k is minimizing for f, but z k 0, which is a contradiction, because f is well-posed. 4. A dominated convergence theorem It is useful to obtain criteria which guarantee that a sequence f n of lower semicontinuous functions converges in the AW -sense to a certain function f. This is relevant e. g. in order to apply Theorem 3.5. In this section we study some class of integral functionals, we prove that under suitable hypotheses convergence of the sequence of the integrs is a sufficient condition y k.

9 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions 359 to assure convergence of the sequence of the integrals (both in the AW -sense). Let (T, A, µ) a measure space let µ be a positive finite measure on A which is complete. Definition 4.1. An integr is a function f : T R k (, + ]. If for each t T, the set epif(t, ) is closed nonempty, if the multifunction t epif(t, ) is measurable, we say that f is a normal integr. Let f be a normal integr let L be a class of measurable functions from T to R k ; then for each u L, the function t f(t, u(t)) is measurable; if it is summable or it is majorized by a summable function, a natural value can be assigned to the integral F (u) = f(t, u(t)) dµ. (7) T Otherwise, we set F (u) = +. In this way, F is a well-defined extended real-valued functional on the space L; we say that F is the integral functional associated with the normal integr f. See [9]. Let h be a function on R k bounded from below lower semicontinuous; then for each z R k H z (x) = h(x) + x z has at least one global minimum point on R k. Let us define Prox[z, h] = argminh z. Given f a normal integr defined on T R k bounded from below a measurable function u : T R k, we can consider the multifunction P given by: P (t) = Prox[u(t), f(t, )]. Since the function (t, x) f(t, x) + x u(t) is itself a normal integr, we get that P is a measurable multifunction from Theorem 2K of [9]. Moreover, by Theorem 1C of [9], there exists a measurable selection of P, which we will denote by t prox[u(t), f(t, )]. Lemma 4.2. Let f, f n : T R m [0, + ) be normal integrs with ρ 0 > 0. Fix G T, G measurable write for all ρ > ρ 0, Consider x : T R m such that, for some ρ, sup haus ρ (f n (t, ), f(t, )) := h n (ρ). t G {x(t) : t G} ρu {f(t, x(t)) : t G} ρu let G n T be measurable x n : T R m be a sequence such that: {x n (t) : t G n, n N} ρu {f n (t, x n (t)) : t G n, n N} ρu. Then, for every ɛ > 0, t G for each n N, we can find x n,ɛ (t) R m such that: x(t) x n,ɛ (t) h n (ρ) + ɛ

10 360 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions f n (t, x n,ɛ (t)) f(t, x(t)) + h n (ρ) + ɛ for every ɛ > 0, t G n, n N we can find y n,ɛ (t) such that: x n (t) y n,ɛ (t) h n (ρ) + ɛ f(t, y n,ɛ (t)) f n (t, x n (t)) + h n (ρ) + ɛ. Proof. By Theorem 2.3, for every ɛ > 0, t G for each n N, we can find x n,ɛ (t) R m such that: In particular we obtain: x(t) x n,ɛ (t) haus ρ (f(t, ), f n (t, )) + ɛ f n (t, x n.ɛ (t)) f(t, x(t)) + haus ρ (f(t, ), f n (t, )) + ɛ. x(t) x n,ɛ (t) h n (ρ) + ɛ f n (t, x n,ɛ (t)) f(t, x(t)) + h n (ρ) + ɛ. Finally, by Theorem 2.3, for every ɛ > 0 t G n there exist y n,ɛ (t) such that: x n (t) y n,ɛ (t) h n (ρ) + ɛ f(t, y n,ɛ (t)) f n (t, x n (t)) + h n (ρ) + ɛ. Given f, f n : T [0, + ) normal integrs we can consider the associated integral functionals F, F n defined by (7) on L p = L p (T ; R m ), with p (1, + ). In the sequel we denote by p the conjugate exponent of p. Using Lemma 4.2 we can prove the following Theorem: Theorem 4.3. Let f, f n : T R m [0, + ) be normal integrs. Suppose there exist ϕ : T [0, + ) in L p ψ : T [0, + ) in L 1 such that f n (t, x) ϕ(t) x + ψ(t) f(t, x) ϕ(t) x + ψ(t) for each n N, t T, x R m. Moreover assume f(t, ) = τ aw lim f n (t, ) for almost every t. Then: F = τ aw lim F n. Proof. We shall prove:

11 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions 361 (i) For all r > 0 ɛ > 0 there exists η ɛ N such that for all u L p with u p r F (u) r, for each n > η ɛ we can find w n L p which satisfy: u w n p ɛ F n (w n ) F (u) + ɛ. (ii) For all r > 0 ɛ > 0 there exists ν ɛ N such that for all u n L p with u n r F n (u n ) r, for each n > ν ɛ we can find y n L p which satisfy: u n y n p ɛ F (y n ) F n (u n ) + ɛ. Since F, F n are proper, lower semicontinuous bounded from below by the same constant, we can apply part 2 of Theorem 2.3, therefore the previous conditions will imply that for every r > 0 ɛ > 0, there exists N ɛ := η ɛ + ν ɛ such that haus r (F, F n ) < ɛ if n > N ɛ, that is exactly what we have to prove. (i) Fix r > 0 an let u L p (T ; R m ) such that u p r also F (u) r. (8) Fix ɛ > 0 let K > 0 to be chosen later. Define: S K := {t T : u(t) K} T K := {t T : f(t, u(t)) K}. We have ( u p ) p = T u(t) p dµ = u(t) p dµ + T \S K u(t) p dµ r p. S K Hence: S K rp K, p (9) where denotes measure. In the same way, from (8): f(t, u(t)) dµ = T f(t, u(t)) dµ + T \T K f(t, u(t))dµ r T K therefore: T K r K. By assumptions ψ L 1 so, by the absolute continuity of the integrals, for each δ > 0 there exists m δ > 0 such that, given a measurable set G for which G m δ, we have ψ(t) dµ δ. G On the other h ϕ p L 1 ; thus for all δ > 0 there exists c δ > 0 such that, given a measurable set G for which G c δ, we get (t) dµ δ. G ϕp Now, choose K to have r p K + r { p K < min m(ɛ/3), c } (ɛ/3r) p. (10) 2 2 By hypothesis haus ρ (f(t, ), f n (t, )) 0 for every ρ almost every t; by the Severini- Egoroff Theorem we have that if λ > 0 there exists a measurable set E λ T such that T \ E λ λ haus ρ (f(t, ), f n (t, )) approaches 0 uniformly on E λ. In particular, choosing λ = min{ m (ɛ/3), c (ɛ/3r) p } we can find some measurable E 2 2 ɛ such that: T \ E ɛ min { m(ɛ/3) 2, c (ɛ/3r) p 2 } (11)

12 362 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions haus ρ (f(t, ), f n (t, )) goes to 0 uniformly on E ɛ. Let t E ɛ \ (S K T K ), then: {(u(t), f(t, u(t)))} KU [0, K]. From Lemma 4.2, when t E ɛ \ (S K T K ), it s possible to find u n,ɛ (t) such that u(t) u n,ɛ (t) < h n (K) + 1 { } ɛ 2 min 2 T, ɛ 1/p 6 T where h n (K) = sup t G haus K (f n (t, ), f(t, )) with G := E ɛ \ (S K T K ) f n (t, u n,ɛ (t)) < f(t, u(t)) + h n (K) + 1 { } ɛ 2 min 2 T, ɛ. (13) 1/p 6 T On the other h, from uniform convergence there exists η ɛ such that, given n > η ɛ, we get: h n (K) 1 { } ɛ 2 min 2 T, ɛ. (14) 1/p 6 T Now, using (14), rewriting (12) (13), we obtain: { } ɛ u(t) u n,ɛ (t) < min 2 T, ɛ 1/p 6 T { } ɛ f n (t, u n,ɛ (t)) < f(t, u(t)) + min 2 T, ɛ. (16) 1/p 6 T Consider the functions: v n (t) := prox[u(t), max{0, f n (t, ) f(t, u(t))}]. Now fix n > η ɛ. For any t E ɛ \ (S K T K ), by (15) (16) we have: (12) (15) 0 u(t) v n (t) + max{0, f n (t, v n (t)) f(t, u(t))} u(t) u n,ɛ (t) + max{0, f n (t, u n,ɛ (t)) f(t, u(t))} { } ɛ < min 3 T, ɛ T 1/p Define C K,ɛ := (T \ E ɛ ) S K T K { v n (t) if t E ɛ \ (S K T K ), w n (t) := u(t) if t C K,ɛ. (17) By construction, the functions w n are measurable we have w n u < ɛ T 1/p which implies w n u p < ɛ. Moreover, by (9), (10) (11), we get: C K,ɛ = S K T K (T \ E ɛ ) rp K + r p K + T \ E ɛ min{m (ɛ/3), c }. (ɛ/3r) p

13 Hence: S. Villa / AW -Convergence Well-Posedness of Non Convex Functions 363 ϕ(t) p dµ C K,ɛ ( ) p ɛ 3r C K,ɛ ψ(t) dµ ɛ 3. Then, using Hölder inequality recalling (8) (17), it follows that: F n (w n ) = f n (t, w n (t)) dµ T = f n (t, v n (t)) dµ + f n (t, u(t)) dµ T \C K,ɛ C K,ɛ (f(t, u(t)) + ɛ T \C K,ɛ 3 T ) dµ + (ϕ(t) u(t) + ψ(t)) dµ C K,ɛ f(t, u(t)) dµ + ɛ T 3 C + ϕ(t) u(t) dµ + ψ(t) dµ K,ɛ C K,ɛ F (u) + ɛ ( ) 1/p 3 + ϕ(t) p dµ u p + ɛ C K,ɛ 3 F (u) + ɛ. Therefore the condition (i) is satisfied. (ii) Let u n L p (T ; R m ) such that u n p r F n (u n ) r. Considering SK n := {t T : u n (t) K}, for each n it follows that: SK n rp /K p. Similarly, defining TK n = {t T : f n(t, u n (t)) K}, for each n it happens that TK n r. K Choose K such that { m(ɛ/3) r p K + r p K min, c (ɛ/3r) p 2 2 for each n N. Let E ɛ as in (i). Following the same proof as in (i), since haus ρ (f n (t, ), f(t, )) 0 uniformly with respect to t on E ɛ for each ρ > 0, as a consequence of Lemma 4.2 there exists ν ɛ such that, given n > ν ɛ, for all t E ɛ \ (SK n T K n) we can find z n,ɛ(t) such that { ɛ z n,ɛ (t) u n (t) < min } 2 T 1/p, ɛ 6 T } (18) { } ɛ f(t, z n,ɛ (t)) f n (t, u n (t)) + min 2 T, ɛ. (19) 1/p 6 T Define v n (t) := prox[u n (t), max{0, f(t, ) f n (t, u n (t))}]. Now fix n > ν ɛ. For any t E ɛ \ (SK n T K n ), by (18) (19), we have: Let us construct the functions 0 u n (t) v n (t) + max{0, f(t, v n (t)) f n (t, u n (t))} u n (t) z n,ɛ (t) + max{0, f(t, z n,ɛ (t)) f n (t, u n (t))} { } ɛ < min 3 T, ɛ. (20) T 1/p y n (t) := { v n (t) if t E ɛ \ (S n K T n K ) u n (t) if t (T \ E ɛ ) S n K T n K.

14 364 S. Villa / AW -Convergence Well-Posedness of Non Convex Functions For every n > ν ɛ we have: F (y n ) = = u n y n p < ɛ f(t, y n (t)) dµ T f(t, v n (t)) dµ + E ɛ \(SK n T K n ) (f n (t, u n (t)) + 2 ɛ T 6 T ) dµ + +ψ(t)) dµ F n (u n ) + ɛ. Taking N ɛ = η ɛ + ν ɛ, the proof is complete. (T \E ɛ ) S n K T n K (T \E ɛ ) S n K T n K f(t, u n (t)) dµ (ϕ(t) u n (t) + Acknowledgements. useful suggestions. The author wishes to thank T. Zolezzi for continuous advice several References [1] H. Attouch, R. Wets: Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 3 (1991) [2] H. Attouch, R. Wets: Quantitative stability of variational systems: II. A framework for nonlinear conditioning, SIAM J. Optim. 3 (1993) [3] H. Attouch, R. Wets: Quantitative stability of variational systems: III. ɛ-approximate solutions, Math. Programming 61 (1993) [4] G. Beer: Topologies on Closed Closed Convex Sets, Kluwer, Dordrecht (1993). [5] G. Beer, R. Lucchetti: Convex optimization the epi-distance topology, Trans. Amer. Math. Soc. 327 (1991) [6] A. L. Dontchev, T. Zolezzi: Well-Posed Optimization Problems, Lecture Notes in Mathematics 1543, Springer-Verlag, Berlin (1993). [7] M. Furi, A. Vignoli: About well-posed pptimization problems for functionals in metric spaces, J. Optim. Theory Appl. 5 (1970) [8] J. L. Joly, F. De Thelin: Convergence of convex integrals in L p spaces, J. Math. Anal. Appl. 54 (1976) [9] R. T. Rockafellar: Integral Functionals, Normal Integrs Measurable Selections, Lecture Notes in Mathematics 543, Springer-Verlag (1976). [10] P. Shunmugaraj, D. V. Pai: On stability of approximate solutions of minimization problems, Numer. Funct. Anal. Optim. 12 (1991)