Conference ADGO 2013 October 16 , 2013

Brøndsted-Rockafellar property of subdifferentials of prox-bounded functions

Marc Lassonde Universit´edes Antilles et de la Guyane

Playa Blanca, Tongoy, Chile SUBDIFFERENTIAL OF CONVEX FUNCTIONS

∗ Everywhere X is a Banach space. A set-valued operator T : X ⇒ X , or graph T ⊂ X × X∗, is monotone provided

hy∗ − x∗, y − xi ≥ 0, ∀(x, x∗), (y, y∗) ∈ T, and maximal monotone provided it is monotone and not properly contained in another monotone operator.

∗ The subdifferential ∂f : X ⇒ X of a convex f : X → ]−∞, +∞] is n o ∂f(x) := x∗ ∈ X∗ : hx∗, y − xi + f(x) ≤ f(y), ∀y ∈ X , ∗ and the duality operator J : X ⇒ X is n o J(x) := x∗ ∈ X∗ : hx∗, xi = kxk2 = kx∗k2 . It is easily verified that J(x) = ∂j(x) where j(x) = (1/2)kxk2.

Theorem (Rockafellar, 1970) Let X be a Banach space. The sub- differential ∂f of a proper convex lower semicontinuous function f : X → ]−∞, +∞] is maximal monotone.

2 PROOF WHEN X = H IS HILBERT (taken from Brezis, 1973)

∗ By Hahn-Banach, f ≥ ` + α for some ` ∈ X and α ∈ R, and j + ` is coercive (j(x) + `(x) = (1/2)kxk2 + `(x) → +∞ as kxk → +∞), so

f + j is coercive. Hence f + j attains its minimum at somex ¯ ∈ H, so 0 ∈ ∂(f + j)(¯x).

Since ∂j = ∇j = I (identity on H), we readily get 0 ∈ (∂f + I)(¯x), so 0 ∈ R(∂f + I). We conclude that

X∗ = R(∂f + I). This is easily seen to imply that ∂f is maximal monotone.

(This is the elementary part in Minty’s characterization of maximal monotonicity (1962).)

3 PROOF IN THE GENERAL BANACH CASE: STEP 1

Claim: 0 ∈ R(∂f + J). (1)

∗ First, f ≥ `+α for some ` ∈ X and α ∈ R, and j +` bounded below, so f + j is bounded below.

Next, let ε > 0 arbitrary and let yε ∈ dom f such that 2 (f + j)(yε) ≤ (f + j)(y) + ε , ∀y ∈ X. By Brøndsted-Rockafellar approximation theorem (1965), ∗ ∗ ∗ ∗ ∃xε ∈ X with kxεk ≤ ε and zε ∈ X such that xε ∈ ∂(f + j)(zε). ∗ By Rockafellar’s sum rule (1966), xε ∈ ∂f(zε) + J(zε).

∗ ∗ Conclusion: ∃xε ∈ R(∂f + J) with kxεk ≤ ε, proving the claim.

4 PROOF: STEP 2

Let (x, x∗) ∈ X × X∗ such that hy∗ − x∗, y − xi ≥ 0, ∀(y, y∗) ∈ ∂f. (2)

Applying (1) to f(x + .) − x∗, we get x∗ ∈ R(∂f(x + .) + J). ∗ ∗ ∗ ∗ Thus, there are (xn) ⊂ X with xn → x and (hn) ⊂ X such that ∗ ∗ ∗ xn ∈ ∂f(x + hn) + J(hn). Let (yn) ⊂ X such that ∗ ∗ ∗ yn ∈ ∂f(x + hn) and xn − yn ∈ J(hn).

By definition of J, we have ∗ ∗ ∗ ∗ 2 2 hxn − yn, hni = kxn − ynk = khnk . (3) ∗ ∗ ∗ From (2) and yn ∈ ∂f(x + hn), we get hx − yn, x + hn − xi ≤ 0, so 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ khnk = hxn−x , hni+hx −yn, x+hn−xi ≤ hxn−x , hni ≤ kxn−x kkhnk.

∗ ∗ ∗ ∗ Hence, hn → 0, so, by (3), kxn−ynk → 0, therefore yn → x . Since ∂f ∗ ∗ has closed graph and yn ∈ ∂f(x + hn), we conclude that x ∈ ∂f(x). 5 OTHER PROOFS OF MAXIMALITY OF ∂f FOR CONVEX f

1/ f everywhere finite and continuous: • Minty (1964), Phelps (1989), using mean value theorem and link between subderivative and subdifferential

2/ f lsc, X Hilbert: • Moreau (1965), via prox functions and duality theory, • Brezis (1973), showing directly that ∂f + I is onto

3/ f lsc, X Banach: all proofs use a variational principle and an- other tool • Rockafellar (1970): continuity of (f + j)∗ in X∗ and link between (∂f)−1 and ∂f ∗ in X∗∗ × X∗, • Taylor (1973) and Borwein (1982): subderivative mean value in- equality and link between subderivative and subdifferential, • Zagrodny (1988?), Simons (1991), Luc (1993), etc: subdifferen- tial mean value inequality, • Thibault (1999): limiting convex subdifferential calculus, • Marques Alves-Svaiter (2008), Simons (2009): conjugate func- tions and Fenchel duality formula or subdifferential sum rule. 6 BEYOND THE CONVEX CASE: MAIN TOOLS

Let be given a ’subdifferential’ ∂ that associates a subset ∂f(x) ⊂ X∗ to each x ∈ X and each function f on X so that ∂f(x) coincides with the convex subdifferential when f is convex.

The two main tools in the convex situation were: • Brøndsted-Rockafellar’s approximation theorem (1965) • Rockafellar’s subdifferential sum rule (1966).

They will be respectively replaced by:

Ekeland Variational Principle (1974). For any lsc function f on X, x¯ ∈ dom f and ε > 0 such that f(¯x) ≤ inf f(X)+ε, and for any λ > 0, there is xλ ∈ X s.t. kxλ − x¯k ≤ λ, f(xλ) ≤ f(¯x), and x 7→ f(x) + (ε/λ)kx − xλk has a minimum at xλ.

Subdifferential Separation Principle. For any lsc functions f, ϕ on X with ϕ convex Lipschitz nearx ¯ ∈ dom f ∩ dom ϕ, f + ϕ has a local minimum atx ¯ =⇒ 0 ∈ ∂f(¯x) + ∂ϕ(¯x).

7 SUBDIFFERENTIALS SATISFYING THE SEPARATION PRINCIPLE

The main examples of pairs (X, ∂) for which the Subdifferential Separation Principle holds are:

• the Clarke subdifferential ∂C in arbitrary Banach spaces, • the limiting Fr´echetsubdifferential ∂bF in Asplund spaces, • the limiting Hadamard subdifferential ∂bH in separable spaces, • the limiting proximal subdifferential ∂bP in Hilbert spaces.

For more details, see, e.g., Jules-Lassonde (2013, 2013b).

8 COMBINING THE TOOLS

Set dom f ∗ = {x∗ ∈ X∗ : inf(f − x∗)(X) > −∞}.

Proposition Let X Banach, f : X → ]−∞, +∞] proper lsc, ϕ : X → R convex loc. Lispchitz. Then, dom (f + ϕ)∗ ⊂ cl (R(∂f + ∂ϕ)).

Proof. Let x∗ ∈ dom (f + ϕ)∗ and let ε > 0. There isx ¯ ∈ X s.t. (f + ϕ − x∗)(¯x) ≤ inf(f + ϕ − x∗)(X) + ε2, so, by Ekeland’s variational principle, there is xε ∈ X such that ∗ x 7→ f(x)+ϕ(x)+h−x , xi+εkx−xεk attains its minimum at xε. Now, applying the Separation Principle with the convex locally Lipschitz ∗ ∗ ψ : x 7→ ϕ(x) + h−x , xi + εkx − xεk we obtain xε ∈ ∂f(xε) such that ∗ ∗ ∗ −xε ∈ ∂ψ(xε) = ∂ϕ(xε) − x + εBX∗. So, there is yε ∈ ∂ϕ(xε) such ∗ ∗ ∗ ∗ that kx − yε − xεk ≤ ε. Thus, for every ε > 0 the ball B(x , ε) ∗ ∗ contains xε + yε ∈ ∂f(xε) + ∂ϕ(xε) ⊂ R(∂f + ∂ϕ). This means that x∗ ∈ cl (R(∂f + ∂ϕ)).

The case ϕ = 0 and f = δC with C nonempty closed convex set says that the set ∗ R(∂δC) of functionals in X that attain their supremum on C is dense in the set ∗ dom δC of all those functionals which are bounded above on C (Bishop-Phelps). 9 PROX-BOUNDED FUNCTIONS

A function f is called prox-bounded if there exists λ > 0 such that the function f + λj is bounded below; the infimum λf of the set of all such λ is called the threshold of prox-boundedness for f:

λf := inf{λ > 0 : inf(f + λj) > −∞}. Any convex lsc function g is prox-bounded with threshold λg = 0, the sum f + g of a prox-bounded f and of a convex lsc g is prox- ∗ ∗ bounded with λf+g ≤ λf , for every x ∈ X , λf+x∗ = λf , and for every x ∈ X, f(x + .) + λj is bounded below for any λ > λf (see Rockafellar-Wets book (1998)).

Consequence: if f is prox-bounded, then for every λ > λf , ∀x ∈ X, dom (f(x + .) + λj)∗ = X∗.

From this and the previous result we get:

Proposition Let X Banach and let f : X → ]−∞, +∞] be lsc and prox-bounded with threshold λf . Then, for every λ > λf , ∀x ∈ X, cl (R(∂f(x + .) + λJ)) = X∗.

10 GOING FURTHER: MONOTONE ABSORPTION

∗ ∗ Given T : X ⇒ X , or T ⊂ X × X , and ε ≥ 0, we let T ε := { (x, x∗) ∈ X × X∗ : hy∗ − x∗, y − xi ≥ −ε, ∀(y, y∗) ∈ T } be the set of pairs (x, x∗) ε-monotonically related to T .

An operator T is monotone provided T ⊂ T 0 and monotone maximal provided T = T 0.

A non necessarily monotone operator T is declared to be monotone absorbing provided T 0 ⊂ T ( norm-closure).

A non necessarily monotone operator T is declared to be widely monotone absorbing with threshold λT ≥ 0 provided for every λ > λT one has  q √  ε −1 ∀ε ≥ 0,T ⊂ T + λ ε BX × λε BX∗ . Equivalently: 0, ( ∗) ε ∀ε ≥ x, x ∈ T √⇒ √ ∗ −1 ∗ ∗ ∃(xn, xn) ⊂ T : limn kx − xnk ≤ λ ε and limn kx − xnk ≤ λε. 11 SUFFICIENT CONDITION FOR WIDE MONOTONE ABSORPTION

∗ Proposition Let T : X ⇒ X and λ > 0. Assume that ∀x ∈ X, cl (R(T (x + .) + λJ) = X∗. (4) Then:  q √  ε −1 ∀ε ≥ 0,T ⊂ cl T + λ εBX × λεBX∗ . (5)

Proof. Let ε ≥ 0 and let (x, x∗) ∈ T ε. Since T (x + .) + λJ has a ∗ ∗ ∗ ∗ dense range, we can find (xn) ⊂ X with xn → x and (yn) ⊂ X such ∗ ∗ ∗ that xn ∈ T (x + yn) + λJyn. Let (yn) ⊂ X such that ∗ ∗ ∗ yn ∈ T (x + yn) and xn − yn ∈ λJyn. By definition of J, we have −1 ∗ ∗ −1 ∗ ∗ 2 2 λ hxn − yn, yni = kλ (xn − yn)k = kynk . (6) ∗ ε ∗ ∗ ∗ But x ∈ T x and yn ∈ T (x + yn), so hx − yn, yni ≤ ε, hence 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ λkynk = hxn−x , yni+hx −yn, yni ≤ hxn−x , yni+ε ≤ kxn−x kkynk+ε. 12 2 ∗ ∗ Therefore, λkynk − kxn − x kkynk − ε ≤ 0, so we must have q ∗ ∗ ∗ ∗ 2 kynk ≤ (kxn − x k + kxn − x k + 4ελ)/2λ. (7) √ −1 From (7) we derive that lim supn kynk ≤ λ ε, so, by (6), √ ∗ ∗ lim sup kx − y k = lim sup λkynk ≤ λε. n n n n ∗ In conclusion we have (x + yn, yn) ∈ T with q √ −1 ∗ ∗ lim sup kx − (x + yn)k ≤ λ ε, lim sup kx − y k ≤ λε, n n n and without loss of generality we can replace lim supn by limn.

Open problem: We don’t know whether the converse (5) ⇒ (4) is true. WIDE MONOTONE ABSORPTION PROPERTY OF SUBDIFFERENTIALS OF PROX-BOUNDED FUNCTIONS

Combining the last two propositions gives:

Theorem Let X Banach and f : X → ]−∞, +∞] lsc, prox-bounded with threshold λ ≥ 0. Then: f √ √ ε  −1  ∀λ > λf , ∀ε ≥ 0, (∂f) ⊂ cl ∂f + λ εBX × λεBX∗ . Equivalently: for all λ > λ and ε ≥ 0, (x∗, x) ∈ (∂f)ε ⇒ f √ √ ∗ −1 ∗ ∗ ∃((xn, xn))n ⊂ ∂f : limn kx − xnk ≤ λ ε & limn kx − xnk ≤ λε.

In case λf = 0 (in particular for a convex f), the wide monotone ab- sorption property is equivalent to the so-called maximal monotonic- ity of Brøndsted-Rockafellar type studied in Simons (1999, 2008) and others, hence the above theorem extends known results for con- vex functions to the class of prox-bounded non necessarily convex functions, with a more direct proof.

13 REFERENCES

J. M. Borwein, A note on ε-subgradients and maximal monotonicity, Pacific J. Math. 103 (1982), 307–314

H. Br´ezis, Op´erateursmaximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam, 1973; North-Holland Mathematics Studies, No. 5. Notas de Matem´atica(50)

A. Brøndsted and R. T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), 605–611

I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353

F. Jules and M. Lassonde, Subdifferential estimate of the directional derivative, optimality criterion and separation principles, Optimization 62 (2013), 1267–1288

F. Jules and M. Lassonde, Subdifferential test for optimality, J. Global Optim., in press, doi:10.1007/s10898-013-0078-6

M. Lassonde, Brøndsted-Rockafellar property of subdifferentials of prox-bounded functions, arXiv:1306.5466

D. T. Luc, On the maximal monotonicity of subdifferentials, Acta Math. Vietnam. 18 (1993), 99–106

M. Marques Alves and B. F. Svaiter, A new proof for maximal monotonicity of subdifferential operators, J. Convex Anal. 15 (2008), 345–348

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346

G. J. Minty, On the monotonicity of the gradient of a convex function, Pacific J. Math. 14 (1964), 243–247

J.-J. Moreau, Proximit´eet dualit´edans un espace hilbertien, Bull. Soc. Math. France 935 (1965), 273–299

14 R. R. Phelps, Convex functions, monotone operators and differentiability, volume 1364 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, second edition, 1993.

R. T. Rockafellar, Extension of Fenchel’s duality theorem for convex functions, Duke Math. J. 33 (1966), 81–89

R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216

R. T. Rockafellar and R. J.-B. Wets, Variational analysis, volume 317 of Grundlehren der Math- ematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1998, xiv+733 pp.

S. Simons, The least slope of a convex function and the maximal monotonicity of its subdiffer- ential, J. Optim. Theory Appl. 71 (1991), 127–136

S. Simons, Maximal monotone multifunctions of Brøndsted-Rockafellar type, Set-Valued Anal. 7 (1999), 255–294

S. Simons, From Hahn-Banach to monotonicity, volume 1693 of Lecture Notes in Mathematics, Springer, New York, second edition, 2008

S. Simons, A new proof of the maximal monotonicity of subdifferentials, J. Convex Anal. 16 (2009), 165–168

P. D. Taylor, Subgradients of a convex function obtained from a directional derivative, Pacific J. Math. 44 (1973), 739–747

L. Thibault, Limiting convex subdifferential calculus with applications to integration and maxi- mal monotonicity of subdifferential, Constructive, experimental, and nonlinear analysis (Limoges, 1999), CMS Conf. Proc. 27, pp. 279–289, Amer. Math. Soc., Providence, RI, 2000