Journal of Convex Analysis Volume 3 (1996), No. 2, 367–370

A Resolution of Simons’ Maximal Monotonicity Problem

Dinh The Luc^{1}

Institute of Mathematics, P.O.Box 631 Boho, 10000 Hanoi, Vietnam. e-mail: dtluc@thevinh.ac.vn

Received July 26, 1995 Revised manuscript received February 27, 1996

We present a complete answer to Simons’ question about strong maximal monotonicity of subdifferentials of convex functions.

Keywords : Convex function, maximal monotonicity, mean value theorem, subdifferential.

1991 Mathematics Subject Classification: 47H05, 49J52

1. Introduction

Rockafellar’s maximal monotonicity theorem [5], [4] is one of the fundamental results of convex analysis. It says that if f is a proper convex lower semicontinuous function on a real Banach space X with topological dual X′, then subdifferential of f is maximal monotone, that is if (y, y^{∗}) ∈ X × X′ and for every x ∈ X, x^{∗} ∈ ∂f(x) one has ⟨y^{∗} − x^{∗}, y − x⟩ ≥ 0, then y^{∗} ∈ ∂f(y). By our knowledge there exist at least five different proofs of this fact in the literature, the original one of Rockafellar [5] and the others by Taylor [8], Borwein [1], Simons [6], Luc [2]. The proof given in the latter paper via Zagrodny’s approximate mean value theorem [9] seems to be the simplest.

In [6], [7] Simons generalized maximal monotonicity by replacing y or y^{∗} above by appropriate convex sets. Specifically, he proved the following important results which subsume maximal monotonicity:

(i) Let B be a nonempty weakly compact convex subset of X and b^{∗} ∈ X′ \∂f(B). Then there exist y ∈ X and y^{∗} ∈ ∂f(y) such that ⟨y^{∗} − b^{∗}, b − y⟩ > 0, for all b ∈ B;

(ii) Let B^{∗} be a nonempty weakly^{∗} compact convex subset of X′ and b ∈ X with ∂f(b) ∩ B^{∗} = ∅. Then there exist y ∈ X and y^{∗} ∈ ∂f(y) such that ⟨y^{∗} − b^{∗}, b − y⟩ > 0, for all b^{∗} ∈ B^{∗};

Simple proofs of these results were given in [3] by using a “region type” generalization of Zagrodny’s mean value theorem.

368 Dinh The Luc / Maximal monotonicity

The following question was asked by Simons in his lecture delivered at the CIRM Conference held from 22 to 26 June 1992 at Marseille, France (see [7]). If B is a nonempty weakly compact convex subset of X and B^{∗} is a nonempty weakly^{∗} compact convex subset of X′ with ∂f(B)∩B^{∗} = ∅, does there exist y ∈ X and y^{∗} ∈ ∂f(y) such that ⟨y^{∗} −b^{∗}, b−y⟩ ≥ 0, for all b ∈ B, b^{∗} ∈ B^{∗}.

1 This paper was written during the author’s stay at the University of Avignon, France.

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

The aim of the present note is to give a complete answer to this question. Namely, we show that the answer is yes in the case where X is one-dimensional and the answer is no in higher dimensions. We thank an anonymous referee for useful suggestions which lead to a shortened proof of Theorem 2.2.

2. The one-dimensional case

In this section we suppose that f is a proper convex lower semicontinuous function on IR with values in the extended real line IR∪{+∞}. We shall need the following result, which is a one-dimensional version of Lemma 2.1 of [2].

Lemma 2.1. Assume that g is a weakly lower semicontinuous function on IR with values in IR ∪ {+∞}, x_{0} < a and g(x_{0}) < g(a). Then there exist y < a and z^{∗} ∈ ∂g(y) such that z^{∗} > 0.

In the above lemma the subdifferential can be any one which satisfies certain natural requirements (see [3]). It is sufficient to note that when g is convex, the subdifferential is understood in the sense of convex analysis, that is ∂g(x) = {x^{∗} ∈ X′ : ⟨x^{∗}, v⟩ ≤ g(x + v) − g(x) for all v ∈ X}.

Now we present the main result of this section.

Theorem 2.2. Let f be a proper convex lower semicontinuous function on IR with values in the extended real line. Let [a, b] and [c, d] be two segments in IR such that ∂f(x) ∩ [c, d] = ∅ for all x ∈ [a, b]. Then there exist y ∈ IR \ [a, b] and y^{∗} ∈ ∂f(y) such that (y^{∗} − x^{∗})(x − y) > 0 for all x ∈ [a, b], x^{∗} ∈ [c, d].

Proof. Let h be a convex function on IR defined by h(x) := d(a−x) if x ≤ a and h(x) := c(a − x) if x > a. Consider the function g(x) := f(x) + h(x). This function is proper convex, lower semicontinuous. Moreover, ∂g(x) ⊆ ∂f(x) + ∂h(x) ⊆ ∂f(x) + [−d, −c]. It follows from the condition of the theorem that 0 ∉ ∂g([a, b]). Consequently, the segment [a, b] does not contain any minimum of g, that is there exists a point x_{0} ∈ IR such that g(x_{0}) < inf_{x∈[a,b]} g(x). We assume x_{0} < a. The case x_{0} > b is similar. Then g(x_{0}) < g(a). By Lemma 2.1, there exist y < a and z^{∗} ∈ ∂g(y) such that z^{∗} > 0. For x < a, one has ∂h(x) = {−d}, therefore one can find y^{∗} ∈ ∂f(y) such that z^{∗} = y^{∗} − d. Hence y∗ > d. Now, for x ∈ [a, b] and x^{∗} ∈ [c, d] we have x > y and y^{∗} > x^{∗}. Consequently, (y^{∗} − x^{∗})(x − y) > 0 as requested.

It is worthwhile noticing that Corollary 2.2 of [3] which is a generalization of Lemma 2.1 to the case where g is defined on a general Banach space and a is replaced by a weakly compact convex set can also be used to prove Theorem 2.2 above.

3. A counterexample

We are going to construct a differentiable convex function f on IR^{2} and two convex compact sets B, B^{∗} in IR^{2} for which the answer to Simons’ question is “no”. First we define a function h on IR by

h(x) = \{ \begin{matrix}−1 & if x < −2, \\ (x + 2)^{2} − 1 & if x ∈ [−2, −1], \\ 2(x + 1) & if x ∈ [−1, 1], \\ x^{2} + 3 & if x > 1. \end{matrix}

It is obvious that h is a differentiable convex function on IR. Now define f on IR^{2} by f(x, y) = h(x) + y/2. Furthermore, let B^{∗} be the unit ball in IR^{2} (equiped with the Euclidean norm) and B the rectangle [−1, 1]×[−20, 20]. Since for (x, y) ∈ B the derivative of f is a constant vector equal to (2, 1/2) ∉ B^{∗}, the hypothesis of Simons’ question is satisfied. We wish to show that there exists no point (a, b) ∈ IR^{2} such that

⟨f′(a, b) − (u, v), (x, y) − (a, b)⟩ > 0  (3.1)

for all (x, y) ∈ B, (u, v) ∈ B^{∗}. In fact, let us rewrite the above inequality in the following form

⟨f′(a, b), (x, y) − (a, b)⟩ > ∥(x, y) − (a, b)∥  (3.2)

for every (x, y) ∈ B. Direct calculation shows that

⟨f′(a, b), (x, y) − (a, b)⟩ = \{ \begin{matrix}(y − b)/2 & if a < −2, \\ 2(a + 2)(x − a) + (y − b)/2 & if a ∈ [−2, −1], \\ 2(x − a) + (y − b)/2 & if a ∈ [−1, 1], \\ 2a(x − a) + (y − b)/2 & if a > 1. \end{matrix}

By taking x = −1 and y = 0 in the above formula we see that for a > 1,

⟨f′(a, b), (x, y) − (a, b)⟩ = −2a(1 + a) − b/2 < −b/2,

while

||(x, y) − (a, b)|| = \sqrt{(1 + a)^{2} + b^{2}} > \sqrt{4 + b^{2}},

which shows that (3.2) does not hold. In the other cases we have the following estimate

⟨f′(a, b), (x, y) − (a, b)⟩ < 5 + (y − b)/2

for every (x, y) ∈ B because

max_{a∈[−2,−1],x∈[−1,1]} 2(a + 2)(x − a) ≤ max_{a∈[−2,−1]} 2(a + 2)(1 − a)

≤ max_{a∈IR} 2(a + 2)(1 − a) = 4.5,

max_{a∈[−2,−1],x∈[−1,1]} 2(x − a) = 4.

This and (3.2) imply that

5 + (y − b)/2 > \sqrt{1 + (y − b)^{2}}

for every y ∈ [−20, 20]. Consequently, y − b ∈ [(10 − \sqrt{388})/3, (10 + \sqrt{388})/3] ⊆ [−4, 10] for all y ∈ [−20, 20]. Such a number b does not exist, hence neither does (a, b) satisfying (3.1) exist.

References

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

[2] D. T. Luc: On the maximal monotonicity of subdifferentials, Acta Mathematica Vietnamica 18 (1993) 99–106.

[3] D. T. Luc: A strong mean value theorem and applications, Nonlinear Analysis 26 (1996) 915–923.

[4] R. R. Phelps: “Convex Functions, Monotone Operators and Differentiability”, Springer-Ver-lag, Berlin, 1989.

[5] R. T. Rockafellar: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1971) 209–216.

[6] S. Simons: The least slope of a convex function and the maximal monotonicity of its subdifferential, J. Optim. Theory Applic. 71 (1991) 127–136.

[7] S. Simons: Swimming below icebergs, Set-valued Analysis 2 (1994) 327–337.

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

[9] D. Zagrodny: Approximate mean value theorem for upper subderivatives, Nonlinear Analysis, Theory, Methods and Applications 12 (1988) 1413—1428.