Brndsted-Rockafellar property of subdifferentials of prox-bounded - - PowerPoint PPT Presentation

Conference ADGO 2013 October 16 , 2013

Brøndsted-Rockafellar property of subdifferentials

• f prox-bounded functions

Marc Lassonde Universit´ e des 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∗,

• r graph T ⊂ X × X∗, is monotone provided

y∗ − x∗, y − x ≥ 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 ∂f(x) :=

• x∗ ∈ X∗ : x∗, y − x + f(x) ≤ f(y), ∀y ∈ X
• ,

and the duality operator J : X ⇒ X∗ is J(x) :=

• x∗ ∈ X∗ : x∗, x = x2 = x∗2

. It is easily verified that J(x) = ∂j(x) where j(x) = (1/2)x2. 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.

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PROOF WHEN X = H IS HILBERT (taken from Brezis, 1973) By Hahn-Banach, f ≥ ℓ + α for some ℓ ∈ X∗ and α ∈ R, and j + ℓ is coercive (j(x) + ℓ(x) = (1/2)x2 + ℓ(x) → +∞ as x → +∞), so f + j is coercive. Hence f + j attains its minimum at some ¯ x ∈ 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).)

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PROOF IN THE GENERAL BANACH CASE: STEP 1 Claim: 0 ∈ R(∂f + J). (1) First, f ≥ ℓ+α for some ℓ ∈ X∗ and α ∈ R, and j +ℓ bounded below, so f + j is bounded below. Next, let ε > 0 arbitrary and let yε ∈ dom f such that (f + j)(yε) ≤ (f + j)(y) + ε2, ∀y ∈ X. By Brøndsted-Rockafellar approximation theorem (1965), ∃x∗

ε ∈ X∗ with x∗ ε ≤ ε and zε ∈ X such that x∗ ε ∈ ∂(f + j)(zε).

By Rockafellar’s sum rule (1966), x∗

ε ∈ ∂f(zε) + J(zε).

Conclusion: ∃x∗

ε ∈ R(∂f + J) with x∗ ε ≤ ε, proving the claim.

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PROOF: STEP 2 Let (x, x∗) ∈ X × X∗ such that y∗ − x∗, y − x ≥ 0, ∀(y, y∗) ∈ ∂f. (2) Applying (1) to f(x + .) − x∗, we get x∗ ∈ R(∂f(x + .) + J). Thus, there are (x∗

n) ⊂ X∗ with x∗ n → x∗ and (hn) ⊂ X such that

x∗

n ∈ ∂f(x + hn) + J(hn). Let (y∗ n) ⊂ X∗ such that

y∗

n ∈ ∂f(x + hn)

and x∗

n − y∗ n ∈ J(hn).

By definition of J, we have x∗

n − y∗ n, hn = x∗ n − y∗ n2 = hn2.

(3) From (2) and y∗

n ∈ ∂f(x + hn), we get x∗ − y∗ n, x + hn − x ≤ 0, so

hn2 = x∗

n−x∗, hn+x∗−y∗ n, x+hn−x ≤ x∗ n−x∗, hn ≤ x∗ n−x∗hn.

Hence, hn → 0, so, by (3), x∗

n−y∗ n → 0, therefore y∗ n → x∗. Since ∂f

has closed graph and y∗

n ∈ ∂f(x + hn), we conclude that x∗ ∈ ∂f(x).

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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-

• ther 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.

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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. xλ − ¯ x ≤ λ, f(xλ) ≤ f(¯ x), and x → f(x) + (ε/λ)x − xλ has a minimum at xλ. Subdifferential Separation Principle. For any lsc functions f, ϕ on X with ϕ convex Lipschitz near ¯ x ∈ dom f ∩ dom ϕ, f + ϕ has a local minimum at ¯ x = ⇒ 0 ∈ ∂f(¯ x) + ∂ϕ(¯ x).

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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´

echet subdifferential ∂F in Asplund spaces,

• the limiting Hadamard subdifferential

∂H in separable spaces,

• the limiting proximal subdifferential

∂P in Hilbert spaces. For more details, see, e.g., Jules-Lassonde (2013, 2013b).

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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 is ¯

x ∈ 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 → f(x)+ϕ(x)+−x∗, x+εx−xε attains its minimum at xε. Now, applying the Separation Principle with the convex locally Lipschitz ψ : x → ϕ(x) + −x∗, x + εx − xε we obtain x∗

ε ∈ ∂f(xε) such that

−x∗

ε ∈ ∂ψ(xε) = ∂ϕ(xε) − x∗ + εBX∗. So, there is y∗ ε ∈ ∂ϕ(xε) such

that x∗ − y∗

ε − x∗ ε ≤ ε.

Thus, for every ε > 0 the ball B(x∗, ε) contains x∗

ε + y∗ ε ∈ ∂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 δ∗

C of all those functionals which are bounded above on C (Bishop-Phelps).

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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∗.

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GOING FURTHER: MONOTONE ABSORPTION Given T : X ⇒ X∗, or T ⊂ X × X∗, and ε ≥ 0, we let T ε := { (x, x∗) ∈ X × X∗ : y∗ − x∗, y − x ≥ −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

• ne has

∀ε ≥ 0, T ε ⊂

• T +
• λ−1ε BX ×

√ λε BX∗

• .

Equivalently: ∀ε ≥ 0, (x, x∗) ∈ T ε ⇒ ∃(xn, x∗

n) ⊂ T : limn x − xn ≤

√ λ−1ε and limn x∗ − x∗

n ≤

√ λε.

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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: ∀ε ≥ 0, T ε ⊂ cl

• T +
• λ−1εBX ×

√ λεBX∗

• .

(5) Proof. Let ε ≥ 0 and let (x, x∗) ∈ T ε. Since T(x + .) + λJ has a dense range, we can find (x∗

n) ⊂ X∗ with x∗ n → x∗ and (yn) ⊂ X such

that x∗

n ∈ T(x + yn) + λJyn. Let (y∗ n) ⊂ X∗ such that

y∗

n ∈ T(x + yn)

and x∗

n − y∗ n ∈ λJyn.

By definition of J, we have λ−1x∗

n − y∗ n, yn = λ−1(x∗ n − y∗ n)2 = yn2.

(6) But x∗ ∈ T εx and y∗

n ∈ T(x + yn), so x∗ − y∗ n, yn ≤ ε, hence

λyn2 = x∗

n−x∗, yn+x∗−y∗ n, yn ≤ x∗ n−x∗, yn+ε ≤ x∗ n−x∗yn+ε.

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Therefore, λyn2 − x∗

n − x∗yn − ε ≤ 0, so we must have

yn ≤ (x∗

n − x∗ +

• x∗

n − x∗2 + 4ελ)/2λ.

(7) From (7) we derive that lim supn yn ≤ √ λ−1ε, so, by (6), lim sup

n

x∗

n − y∗ n = lim sup n

λyn ≤ √ λε. In conclusion we have (x + yn, y∗

n) ∈ T with

lim sup

n

x − (x + yn) ≤

• λ−1ε,

lim sup

n

x∗ − y∗

n ≤

√ λε, 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 λf ≥ 0. Then: ∀λ > λf, ∀ε ≥ 0, (∂f)ε ⊂ cl

• ∂f +

√ λ−1εBX × √ λεBX∗

• .

Equivalently: for all λ > λf and ε ≥ 0, (x∗, x) ∈ (∂f)ε ⇒ ∃((x∗

n, xn))n ⊂ ∂f : limn x − xn ≤

√ λ−1ε & limn x∗ − x∗

n ≤

√ 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.

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