Incremental bundle methods using upper models

van Ackooij, Wim and Frangioni, Antonio (2016) Incremental bundle methods using upper models. Technical Report del Dipartimento di Informatica, TR . University of Pisa, Pisa, IT. (Submitted)

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Abstract

We propose a family of proximal bundle methods for minimizing sum-structured convex nondifferentiable functions which require two slightly uncommon assumptions, that are satisfied in many relevant applications: Lipschitz continuity of the functions and oracles which also produce upper estimates on the function values. In exchange, the methods: i) use upper models of the functions that allow to estimate function values at points where the oracle has not been called; ii) provide the oracles with more information about when the function computation can be interrupted, possibly diminishing their cost; iii) allow to skip oracle calls entirely for some of the component functions, not only at ``null steps'' but also at ``serious steps''; iv) provide explicit and reliable a-posteriori estimates of the quality of the obtained solutions; v) work with all possible combinations of different assumptions on the oracles. We also discuss introduction of constraints (or, more generally, of easy components) and use of (partly) aggregated models.

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