[Paper Review] Sequential and exact formulae for the subdifferential of nonconvex integral functionals

Rafael Corrêa, Abderrahim Hantoute+1 more|arXiv (Cornell University)|Mar 14, 2018

Optimization and Variational Analysis40 references3 citations

TL;DR

This paper establishes sequential and exact calculus rules for the Fréchet and limiting subdifferentials of nonconvex integral functionals $ E_f(x) = \int_T f(t,x)\,d\mu(t) $, where $ f $ is a normal integrand on a $ \sigma $-finite measure space with values in a separable Asplund space. By leveraging robust infima and variational principles, the authors derive exact representations of subdifferentials via limits of integrals of measurable selections of the subdifferential of $ f $, extending classical results to nonconvex settings without qualification conditions.

ABSTRACT

This work concerns the study of the subdifferential of the integral functional $$ E_f(x)=\int_{T} f(t,x)dμ(t), $$ where $f$ is a (not necessarily convex) normal integrand, $({T},\mathcal{A},μ)$ is a $σ$-finite measure space, while the decision variables vary in a separable Asplund space. First, using techniques of variational analysis we establish sequential approximate formulae for the Fréchet subdifferential of $E_f$. Secondly, we introduce a Lipschitz-like condition, which allows us to give an upper-estimation for the limiting subdifferential of $E_{f}$ even when this functional is non-Lipschitz.

Motivation & Objective

• To develop sequential and exact formulae for the Fréchet and limiting subdifferentials of nonconvex integral functionals $ E_f(x) = \int_T f(t,x)\,d\mu(t) $.
• To extend classical subdifferential calculus results—previously limited to convex integrands—to the nonconvex case using variational analysis techniques.
• To establish conditions under which the subdifferential of $ E_f $ can be represented as the limit of integrals of measurable selections of the subdifferential of $ f $, without requiring qualification conditions.
• To provide upper estimates for the limiting and Clarke-Rockafellar subdifferentials using a generalized Lipschitz-like condition.
• To unify and improve upon prior results by Ioffe [28] and Lopez-Thibault [34] in the nonconvex setting.

Proposed method

• Adapting the concept of robust local minima (Definition 3.1) to enable application of the Borwein-Preiss variational principle in the context of integral functionals.
• Using sequential limits of measurable selections $ x_n^*(t) \in \hat{\partial}f(t, x_n(t)) $ to characterize the Fréchet subdifferential of $ E_f $, with convergence conditions on $ x_n \to x $ in $ L^\infty $ and integral convergence of $ f(t, x_n(t)) \to f(t,x) $.
• Applying variational principles in both the primal space $ X $ and the functional space of $ p $-integrable functions to derive exact calculus rules.
• Introducing a Lipschitz-like condition on the integrand $ f $, generalizing classical Lipschitz continuity, to bound the limiting subdifferential of $ E_f $.
• Employing the bipolar theorem and Castaing’s representation to relate integrable compactness in the dual space to primal space properties via polar cones.
• Using weak* sequential upper limits $ \operatorname{Ls}^{w^*} \{ x_n^*(t) \} $ to characterize the limiting subdifferential and proving inclusion in $ \partial f(t,x) $ almost everywhere.

Experimental results

Research questions

• RQ1 Can exact and sequential formulae for the Fréchet subdifferential of a nonconvex integral functional $ E_f $ be derived without qualification conditions?
• RQ2 How can the limiting subdifferential of $ E_f $ be estimated when the integrand $ f $ is non-Lipschitz?
• RQ3 To what extent can the classical calculus rules for convex integral functionals be extended to nonconvex settings?
• RQ4 What is the role of robust infima in enabling variational principles for nonconvex integral functionals?
• RQ5 Under what conditions does the limiting subdifferential of $ E_f $ coincide with the sequential limit of integrals of subgradients of $ f $?

Key findings

• The Fréchet subdifferential of $ E_f $ at $ x $ is characterized as the limit of integrals of measurable selections $ x_n^*(t) \in \hat{\partial}f(t, x_n(t)) $, provided $ \|x - x_n(\cdot)\|_\infty \to 0 $, $ \int_T f(t, x_n(t))\,d\mu \to \int_T f(t,x)\,d\mu $, and $ \rho(\lambda_n \int_T x_n^*(t)\,d\mu - y^*) \to 0 $.
• For the limiting subdifferential, an upper estimate is provided via a generalized Lipschitz-like condition, which extends classical results to non-Lipschitz integrands.
• The limiting subdifferential of $ E_f $ is contained in the set $ \int_T \operatorname{Ls}^{w^*} \{ x_n^*(t) \} \,d\mu + C(x_n^*)^- + W^\perp $, where $ W $ is a closed subspace and $ C(x_n^*)^- $ is the negative polar cone.
• It is shown that $ \operatorname{Ls}^{w^*} \{ x_n^*(t) \} \subseteq \partial f(t,x) $ for almost every $ t $, ensuring the subdifferential limit is consistent with the pointwise subdifferential of $ f $.
• A characterization is established: the ball in $ L^\infty(T,X) $ of radius $ \delta $ around $ e $ is contained in $ \{ x(\cdot) \in L^\infty(T,X) : x(t) \in C(t) \text{ a.e.} \} $ if and only if $ \delta^{-1} \langle c^*(t), -e \rangle \geq \|c^*(t)\| $ for all measurable selections $ c^* $ of $ C^-(t) $.
• The results are valid for separable Asplund spaces with separable duals, and the framework applies to both Lipschitz and non-Lipschitz integrands, generalizing prior work by Ioffe [28] and Lopez-Thibault [34].