Integration formulas via the Fenchel subdifferential of nonconvex functions

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Abstract

Starting from explicit expressions for the subdifferential of the conjugate function, we establish in the Banach space setting some integration results for the so-called epi-pointed functions. These results use the ε-subdifferential and the Fenchel subdifferential of an appropriate weak lower semicontinuous (lsc) envelope of the initial function. We apply these integration results to the construction of the lsc convex envelope either in terms of the ε-subdifferential of the nominal function or of the subdifferential of its weak lsc envelope.

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: f(x)g(x)for all xXf=g+constant . 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 f(x)=|x| 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 ε-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: εf(x)εg(x)for all xX and ε>0 small enough co¯f=co¯g+constant . 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 (ε=0) will be given by means of an appropriate concept of weak lsc envelope: f̄w(x)ḡw(x)for all xXco¯f=co¯g+constant . Hence, in the line of [2] we construct the lsc convex envelope of a function by means of its ε-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 X 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 (X,) whose dual and bidual spaces are denoted by X and X, respectively. We use σ(X,X),σ(X,X) and σ(X,X) to refer to the weak, the weak and the weak∗∗ topologies, respectively. We shall identify X to a subset of X, by the canonical embedding, and, unless otherwise expressed, endow X with the weak∗∗ topology σ(X,X) which makes (X) isomorphic to X. A subset UX (or X) is said to be a θ-neighborhood if it is a convex symmetric neighborhood of the zero vector θ. If f:XR{+} is a given function and ε0, the ε-subdifferential of f at a point xX is the (possibly empty) subset of X given by εf(x){xXx,yxf(y)f(x)+ε for all yX}, where , denotes the duality product of X and X; we will omit referring to ε when it equals 0. We use domf to denote the (effective) domain of f,domf{xXf(x)<+}. We say that f is proper if domf and f>. We denote Γ0(X) the family of the lsc convex proper functions defined on X;Γ0(X) and Γ0(X) are defined similarly. The conjugate of f:XR¯ is the function f:XR¯ given by f(x)supX(xf). Similarly, the conjugate of f is the function f:XR¯ given by f(x)=supX(xf). In particular, provided that f is proper, the restriction of f on X coincides with the lsc convex envelope of f,co¯f:XR¯ defined by co¯f(x)sup{g(x)gΓ0(X),gf}. Equivalently, the ε-subdifferential mapping εf:XX is written as εf(x)={xXf(x)+f(x)x,x+ε}, where , is also used to denote the duality product of X and X. Then, the ε-subdifferential mapping εf:XX is written as (provided that the conjugate function is proper) εf(x)={xXf(x)+f(x)x,x+ε}. The indicator and the support functions of a subset A (X,X) are, respectively, IA(x)0if xA;+if xA,σAIA. The inf-convolution of two functions f,g:XR¯ is fginfxX{f(x)+g(x)}. If M:YZ is a set-valued operator, for two sets Y,Z, we denote M1(z){yYzMy},ImM{MyyY} and domM{yYMy}. We shall write (y,z)M when zMy.

Problem formulation. The classical integration formula [2] states that two lsc convex proper functions f,g:XR{+} satisfying f(x)g(x)for all xX, coincide up to an additive constant c, f=g+c. This result readily breaks down outside the convex framework; for instance, the function f(x)=|x| recalled above satisfies (3) independently of the choice of the function g. Nevertheless, the convexity assumption on the second function g is not necessary and can be easily overcome by using the lsc convex envelope of g. Indeed, by observing that the inclusion g(x)(co¯g)(x) always holds, (3) implies that f(x)(co¯g)(x) for all xX. Therefore, provided that f is proper, lsc and convex, by the integration formula for convex functions we get f=co¯g+c, 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 co¯f=co¯g+c. But, what kind of assumption would one introduce toward this aim? In view of the example of f(x)=|x| cited above, it follows that the worst situation occurs when f is often empty-valued. So, a reasonable condition to guaranty (4) for non-necessarily convex functions would be that f(x) is nonempty “for many points”; for instance, the conjugate function f satisfies int(domf). The functions f 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 f being finite and continuous on int(domf). 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 f; that is (see, e.g., [13]), there exists xX such that lim infx+f(x)x,xx>0; hence, θint(domf) if and only if f is strongly coercive.

We shall prove in this paper (Section 3) that under a slight modification of (3), namely εf(x)εg(x)for all xX and all ε>0 small enough, the following variant of (4) holds co¯f=(co¯g)σdomf+c; 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 f̄w and ḡw (Definition 1), namely f̄w(x)ḡw(x)for all xX. 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 f the lsc convex envelope of f,co¯f, is obtained in the following way, for any given x0dom(f), co¯f(x)=f(x0)+sup{i=0n1xi,xi+1xi+xn,xxn}, where the supremum is taken over nN,(xi,xi)f̄w,i=1,,n, and x0f(x0). Equivalently, we obtain a relaxed formula which uses the ε-subdifferential, for any given x0dom(f) and δ>0, co¯f(x)=f(x0)+sup{i=0n1xi,xi+1xi+xn,xxni=1nεi}, where the supremum is taken over nN,εi(0,δ),(xi,xi)εif,i=1,,n, and x0f(x0). When the space X has the Radon–Nikodym property [4, Section 5], the last two formulas above are written by only calling to the pairs of f, 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 (X,X); i.e., f(x) is seen as a subset of X and, so, may contain points that are not in X (with the abuse of language). Thus, we will have to adapt to our current setting, when X is endowed with its norm topology, some similar formulas established in [7], [8], [9] for the duality pair (X,X).

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 (X,X): Proposition 3 uses an enlargement of the Fenchel subdifferential; Proposition 4 uses the ε-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 ε-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 ε-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 X,f:XX. In our setting, the recent results of [7] (and [9]) cannot be immediately applied unless the dual space X is associated with a topology which is compatible with the duality pair (X,X); for instance, the weak topology σ(X,X). 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 X.

We recall that the bidual space X is endowed with the weak∗∗ topology σ(X,X) so that (X) is isomorphic to X; hence, we write f̄w:XX for the subdifferential of the weak∗∗ lsc envelope of f; that is, the function f̄w given in Definition 2.

Theorem 9

Let X be a Banach space, and f,g:X

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 X.

We begin by the following theorem which provides a formula by means of the ε-subdifferential of the initial function.

Theorem 13

Let X be a Banach space, f:XR¯ be a given function, and x0dom(f) . We assume that int(domf) . Then, for every δ>0 we have thatco¯f(x)=f(x0)+sup{i=0n1xi,xi+1xi+xn,xxni=1nεi},where the supremum is taken

Acknowledgment

We would like to thank the referee for the suggested improvements.

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    The research of the first author was supported by Project Fondecyt no. 1080173 and the research of the first and third authors was supported by Project ECOS-CONICYT no. C10E08 and Project Fondecyt no. 1110019.

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