Brøndsted-Rockafellar property of subdifferentials of prox-bounded functions. Marc Lassonde Université des Antilles et de la Guyane

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1 Conference ADGO 2013 October 16, 2013 Brøndsted-Rockafellar property of subdifferentials of prox-bounded functions Marc Lassonde Université des Antilles et de la Guyane Playa Blanca, Tongoy, Chile

2 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 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 = x 2 = x 2}. It is easily verified that J(x) = j(x) where j(x) = (1/2) x 2. Theorem (Rockafellar, 1970) Let X be a Banach space. The subdifferential f of a proper convex lower semicontinuous function f : X ], + ] is maximal monotone. 2

3 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) x 2 + 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).) 3

4 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 y ε dom f such that (f + j)(y ε ) (f + j)(y) + ε 2, y X. By Brøndsted-Rockafellar approximation theorem (1965), x ε X with x ε ε and z ε X such that x ε (f + j)(z ε ). By Rockafellar s sum rule (1966), x ε f(z ε ) + J(z ε ). Conclusion: x ε R( f + J) with x ε ε, proving the claim. 4

5 Let (x, x ) X X such that PROOF: STEP 2 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 (h n ) X such that x n f(x + h n ) + J(h n ). Let (y n) X such that By definition of J, we have y n f(x + h n ) and x n y n J(h n ). x n y n, h n = x n y n 2 = h n 2. (3) From (2) and y n f(x + h n ), we get x y n, x + h n x 0, so h n 2 = x n x, h n + x y n, x+h n x x n x, h n x n x h n. Hence, h n 0, so, by (3), x n y n 0, therefore y n x. Since f has closed graph and y n f(x + h n), we conclude that x f(x). 5

6 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 another 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 inequality and link between subderivative and subdifferential, Zagrodny (1988?), Simons (1991), Luc (1993), etc: subdifferential mean value inequality, Thibault (1999): limiting convex subdifferential calculus, Marques Alves-Svaiter (2008), Simons (2009): conjugate functions and Fenchel duality formula or subdifferential sum rule. 6

7 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 x λ X s.t. x λ x λ, f(x λ ) f( x), and x f(x) + (ε/λ) x x λ has a minimum at x λ. 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). 7

8 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 Fréchet 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). 8

9 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) + ε 2, so, by Ekeland s variational principle, there is x ε X such that x f(x)+ϕ(x)+ x, x +ε x x ε attains its minimum at x ε. Now, applying the Separation Principle with the convex locally Lipschitz ψ : x ϕ(x) + x, x + ε x x ε we obtain x ε f(x ε ) such that x ε ψ(x ε ) = ϕ(x ε ) x + εb X. So, there is y ε ϕ(x ε ) such that x y ε x ε ε. Thus, for every ε > 0 the ball B(x, ε) contains x ε + y ε f(x ε ) + ϕ(x ε ) R( f + ϕ). This means that x cl (R( f + ϕ)). The case ϕ = 0 and f = δ C with C nonempty closed convex set says that the set R( δ C ) 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). 9

10 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 λ f of the set of all such λ is called the threshold of prox-boundedness for f: λ f := inf{λ > 0 : inf(f + λj) > }. Any convex lsc function g is prox-bounded with threshold λ g = 0, the sum f + g of a prox-bounded f and of a convex lsc g is proxbounded with λ f+g λ f, for every x X, λ f+x = λ f, and for every x X, f(x +.) + λj is bounded below for any λ > λ f (see Rockafellar-Wets book (1998)). Consequence: if f is prox-bounded, then for every λ > λ f, 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 λ f. Then, for every λ > λ f, x X, cl (R( f(x +.) + λj)) = X. 10

11 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 λ T 0 provided for every λ > λ T one has ε 0, T ε ( T + λ 1 ε B X ) λε B X. Equivalently: ε 0, (x, x ) T ε (x n, x n) T : lim n x x n λ 1 ε and lim n x x n λε. 11

12 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 1 εb X ) λεb X. (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 (y n ) X such that x n T (x + y n ) + λjy n. Let (y n) X such that By definition of J, we have y n T (x + y n) and x n y n λjy n. λ 1 x n y n, y n = λ 1 (x n y n) 2 = y n 2. (6) But x T ε x and y n T (x + y n ), so x y n, y n ε, hence λ y n 2 = x n x, y n + x y n, y n x n x, y n +ε x n x y n +ε. 12

13 Therefore, λ y n 2 x n x y n ε 0, so we must have y n ( x n x + x n x 2 + 4ελ)/2λ. (7) From (7) we derive that lim sup n y n λ 1 ε, so, by (6), lim sup n lim sup n x n y n = lim sup n In conclusion we have (x + y n, yn) T with x (x + y n ) λ 1 ε, λ y n λε. lim sup n x y n λε, and without loss of generality we can replace lim sup n by lim n. Open problem: We don t know whether the converse (5) (4) is true.

14 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 f 0. Then: λ > λ f, ε 0, ( f) ε cl ( f + λ 1 εb X λεb X ). Equivalently: for all λ > λ f and ε 0, (x, x) ( f) ε ((x n, x n )) n f : lim n x x n λ 1 ε & lim n x x n λε. In case λ f = 0 (in particular for a convex f), the wide monotone absorption property is equivalent to the so-called maximal monotonicity of Brøndsted-Rockafellar type studied in Simons (1999, 2008) and others, hence the above theorem extends known results for convex functions to the class of prox-bounded non necessarily convex functions, with a more direct proof. 13

15 REFERENCES J. M. Borwein, A note on ε-subgradients and maximal monotonicity, Pacific J. Math. 103 (1982), H. Brézis, Opérateurs maximaux 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 Matemática (50) A. Brøndsted and R. T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), F. Jules and M. Lassonde, Subdifferential estimate of the directional derivative, optimality criterion and separation principles, Optimization 62 (2013), F. Jules and M. Lassonde, Subdifferential test for optimality, J. Global Optim., in press, doi: /s M. Lassonde, Brøndsted-Rockafellar property of subdifferentials of prox-bounded functions, arxiv: D. T. Luc, On the maximal monotonicity of subdifferentials, Acta Math. Vietnam. 18 (1993), M. Marques Alves and B. F. Svaiter, A new proof for maximal monotonicity of subdifferential operators, J. Convex Anal. 15 (2008), G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), G. J. Minty, On the monotonicity of the gradient of a convex function, Pacific J. Math. 14 (1964), J.-J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 935 (1965),

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