We study a family of discrete optimization problems asking for the maximization of
the expected value of a concave, strictly increasing, and differentiable function composed with a set-union operator. The expected value is computed with respect to a set
of coefficients taking values from a discrete set of scenarios. The function models the
utility function of the decision maker, while the set-union operator models a covering relationship between two ground sets, a set of items and a set of metaitems. This
problem generalizes the problem introduced by Ahmed S, Atamtürk A (Mathematical
programming 128(1-2):149–169, 2011), and it can be modeled as a mixed integer
nonlinear program involving binary decision variables associated with the items and
metaitems. Its goal is to find a subset of metaitems that maximizes the total utility
corresponding to the items it covers. It has applications to, among others, maximal
covering location, and influence maximization problems. In the paper, we propose a
double-hypograph decomposition which allows for projecting out the variables associated with the items by separately exploiting the structural properties of the utility
function and of the set-union operator. Thanks to it, the utility function is linearized via
an exact outer-approximation technique, whereas the set-union operator is linearized in two ways: either (i) via a reformulation based on submodular cuts, or (ii) via a
Benders decomposition. We analyze from a theoretical perspective the strength of the
inequalities of the resulting reformulations, and embed them into two branch-and-cut
algorithms. We also show how to extend our reformulations to the case where the
utility function is not necessarily increasing. We then experimentally compare our
algorithms inter se, to a standard reformulation based on submodular cuts, to a state-of-the-art global-optimization solver, and to the greedy algorithm for the maximization
of a submodular function. The results reveal that, on our testbed, the method based
on combining an outer approximation with Benders cuts significantly outperforms the
other ones.

## Dettaglio pubblicazione

2022, MATHEMATICAL PROGRAMMING, Pages 9-56 (volume: 196)

### Submodular maximization of concave utility functions composed with a set-union operator with applications to maximal covering location problems (*01a Articolo in rivista*)

#### Coniglio S., Furini F., Ljubic I.

**Gruppo di ricerca:**Combinatorial Optimization

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