Nonlinear Analysis: Theory, Methods & Applications
Integration formulas via the Fenchel subdifferential of nonconvex functions☆
Introduction
Determining a function from its first-order variations is a fundamental principle in nonlinear analysis. This question becomes more involved when the nominal function fails to be differentiable, so that the first problem to deal with concerns the choice of a convenient concept to quantify the variations of this function. The integration theory for lower semicontinuous (lsc, for short) convex proper functions in the Banach space setting was completely solved in the sixties [1], [2] by using the so-called Fenchel subdifferential operator: This concept of generalized differentiation has been shown to be very useful in the framework of convex analysis so that many classical results of differential calculus and linear operator theory are beneficially extended; see, e.g., [3], [4]. Besides this powerful property, this subdifferential operator behaves very badly outside the convex framework; just think of the (Lipschitz continuous) function defined on the real line, which has an empty subdifferential at each point. This fact has led to the introduction of many tools of nonsmooth analysis which realize useful integration properties, like the Clarke, the Fréchet, the Ioffe, the Michel–Penot, the Mordukhovich subdifferentials, among many others; we refer to the book [5] and the references therein.
Nevertheless, the Fenchel subdifferential can also be useful for many purposes even when dealing with non-necessarily convex functions, namely, those verifying some kind of coercivity as condition (5). In what concerns this paper, we provide some integration results by using the concept of -subdifferential. These results ensure, for a large class of non-necessarily convex functions defined on a Banach space, the coincidence up to an additive constant of the lsc convex envelopes rather than the functions themselves: It is clear that both criteria in (1), (2) are equivalent for proper lsc convex functions. In the setting of locally convex spaces, the above integration formula was established in [6] for proper lsc convex functions. Moreover, a criterion using only the exact subdifferential () will be given by means of an appropriate concept of weak lsc envelope: Hence, in the line of [2] we construct the lsc convex envelope of a function by means of its -subdifferential, or, equivalently, in terms of the exact subdifferential of its weak lsc envelope. Since this approach easily breaks down for general functions we will limit ourselves to the useful and quite large family of epi-pointed functions; i.e., those whose conjugate functions are finite and continuous at some point.
To obtain the above results, we follow a natural idea which consists of passing through the conjugate function, which is by construction a (weak∗) lsc proper convex function, and the classical integration formula of [2]. This explains why the first part of this work is dedicated to expressing the subdifferential of the conjugate function in terms of the subdifferential of the nominal function, when the dual space is endowed with its norm topology. First results giving such expressions, dealing with the conjugate function, have been recently established in [7], [8], [9] for the general setting of two real locally convex (Hausdorff) topological vector spaces paired in duality.
Let us first fix some notations that are needed in the problem formulation below and throughout the paper; other ones will be given progressively. We work on a Banach space whose dual and bidual spaces are denoted by and , respectively. We use and to refer to the weak, the weak∗ and the weak∗∗ topologies, respectively. We shall identify to a subset of , by the canonical embedding, and, unless otherwise expressed, endow with the weak∗∗ topology which makes isomorphic to . A subset (or ) is said to be a -neighborhood if it is a convex symmetric neighborhood of the zero vector . If is a given function and , the -subdifferential of at a point is the (possibly empty) subset of given by where denotes the duality product of and ; we will omit referring to when it equals 0. We use to denote the (effective) domain of . We say that is proper if and . We denote the family of the lsc convex proper functions defined on and are defined similarly. The conjugate of is the function given by Similarly, the conjugate of is the function given by In particular, provided that is proper, the restriction of on coincides with the lsc convex envelope of defined by Equivalently, the -subdifferential mapping is written as where is also used to denote the duality product of and . Then, the -subdifferential mapping is written as (provided that the conjugate function is proper) The indicator and the support functions of a subset () are, respectively, The inf-convolution of two functions is . If is a set-valued operator, for two sets , we denote and . We shall write when .
Problem formulation. The classical integration formula [2] states that two lsc convex proper functions satisfying coincide up to an additive constant , This result readily breaks down outside the convex framework; for instance, the function recalled above satisfies (3) independently of the choice of the function . Nevertheless, the convexity assumption on the second function is not necessary and can be easily overcome by using the lsc convex envelope of . Indeed, by observing that the inclusion always holds, (3) implies that for all . Therefore, provided that is proper, lsc and convex, by the integration formula for convex functions we get which obviously covers (3). The question is then to what extent do formulas similar to this last one hold? In this paper, we will be interested in criteria like (3) which imply the validity of the following expression But, what kind of assumption would one introduce toward this aim? In view of the example of cited above, it follows that the worst situation occurs when is often empty-valued. So, a reasonable condition to guaranty (4) for non-necessarily convex functions would be that is nonempty “for many points”; for instance, the conjugate function satisfies The functions satisfying this dual condition, very recurrent in the literature [3], [10], are referred to as the (asymptotically) epi-pointed functions in [11] (see, also, [12] for an extension of this property). This condition (5) is also related to the behavior at infinity of the initial function and, due to the current Banach space setting, it is equivalent to being finite and continuous on . It is also worth recalling that, from a primal point of view, (5) reflects the strong coercivity of a linear translation of the initial function ; that is (see, e.g., [13]), there exists such that hence, if and only if is strongly coercive.
We shall prove in this paper (Section 3) that under a slight modification of (3), namely the following variant of (4) holds hence, one can deduce (4) in many practical cases. Another criterion using the exact subdifferential will be given by means of the weak∗∗ lsc envelopes and (Definition 1), namely Conditions (6), (8), together with (7), are somewhat natural since they are implicitly included in the integration statement given in the convex framework; see Remark 3.
We shall apply the previous results in the construction of the lsc convex envelope of epi-pointed functions. Namely, we show in Section 4 that for any function the lsc convex envelope of , is obtained in the following way, for any given , where the supremum is taken over , and . Equivalently, we obtain a relaxed formula which uses the -subdifferential, for any given and , where the supremum is taken over , and . When the space has the Radon–Nikodym property [4, Section 5], the last two formulas above are written by only calling to the pairs of , giving a different proof of a similar result given in [13].
Our main tools are formulas for expressing the subdifferential of the conjugate function with respect to the pair ; i.e., is seen as a subset of and, so, may contain points that are not in (with the abuse of language). Thus, we will have to adapt to our current setting, when is endowed with its norm topology, some similar formulas established in [7], [8], [9] for the duality pair .
The summary of the remainder of the paper is as follows. In Section 2, we gather the main tools of our analysis. Namely, we provide formulas for the subdifferential of the conjugate function with respect to the pair : Proposition 3 uses an enlargement of the Fenchel subdifferential; Proposition 4 uses the -subdifferential of the initial function; Proposition 5 concerns positively homogeneous functions; and Proposition 6 investigates the case when the conjugate is Fréchet-differentiable at the nominal point. In Section 3, the main integration formula is presented in Theorem 9 using the weak∗∗ lsc envelope of the associated functions. A version of this result using the -subdifferential is given in Corollary 10. Finally, in Section 4 we provide the construction of the lsc convex envelope either by means of the weak∗∗ lsc envelope (Theorem 14) or by the -subdifferential (Theorem 13).
Section snippets
Subdifferential of the conjugate function
In this section, we express the subdifferential set of the conjugate function in the Banach space . In our setting, the recent results of [7] (and [9]) cannot be immediately applied unless the dual space is associated with a topology which is compatible with the duality pair ; for instance, the weak∗ topology . Nevertheless, our analysis makes use of these results to overcome the current difficulty which occurs outside of reflexive spaces.
Here, and hereafter, we use
Integration formulas using the Fenchel subdifferential
In this section, we apply the results of the previous section to establish the desired integration formulas using the Fenchel subdifferential of non-necessarily convex functions, defined on the Banach space .
We recall that the bidual space is endowed with the weak∗∗ topology so that is isomorphic to ; hence, we write for the subdifferential of the weak∗∗ lsc envelope of ; that is, the function given in Definition 2. Theorem 9 Let be a Banach space, and
Application to the construction of the lsc convex envelope
In this section, we apply the results of Section 3 to give explicit constructive formulas for the lsc convex envelope of functions defined on the Banach space .
We begin by the following theorem which provides a formula by means of the -subdifferential of the initial function. Theorem 13 Let be a Banach space, be a given function, and . We assume that . Then, for every we have thatwhere the supremum is taken
Acknowledgment
We would like to thank the referee for the suggested improvements.
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