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Submodular maximization of concave utility functions composed with a set-union operator with applications to maximal covering location problems

Submodular maximization of concave utility functions composed with a set-union operator with applications to maximal covering location problems
Submodular maximization of concave utility functions composed with a set-union operator with applications to maximal covering location problems

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.

Benders decomposition, Branch-and-Cut, Influence maximization, Stochastic maximal covering location problems, Submodular maximization
0025-5610
9-56
Coniglio, Stefano
03838248-2ce4-4dbc-a6f4-e010d6fdac67
Furini, Fabio
0bf78b98-3255-4fa0-aff7-699d8f8bb292
Ljubić, Ivana
d8dca290-8f9d-4c86-815b-bbdfdd19b2c6
Coniglio, Stefano
03838248-2ce4-4dbc-a6f4-e010d6fdac67
Furini, Fabio
0bf78b98-3255-4fa0-aff7-699d8f8bb292
Ljubić, Ivana
d8dca290-8f9d-4c86-815b-bbdfdd19b2c6

Coniglio, Stefano, Furini, Fabio and Ljubić, Ivana (2022) Submodular maximization of concave utility functions composed with a set-union operator with applications to maximal covering location problems. Mathematical Programming, 196 (1-2), 9-56. (doi:10.1007/s10107-022-01884-7).

Record type: Article

Abstract

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.

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More information

Accepted/In Press date: 5 July 2022
e-pub ahead of print date: 26 October 2022
Published date: 1 November 2022
Keywords: Benders decomposition, Branch-and-Cut, Influence maximization, Stochastic maximal covering location problems, Submodular maximization

Identifiers

Local EPrints ID: 473330
URI: http://eprints.soton.ac.uk/id/eprint/473330
ISSN: 0025-5610
PURE UUID: 10290367-f022-4d02-aee1-f369a3d75e45
ORCID for Stefano Coniglio: ORCID iD orcid.org/0000-0001-9568-4385

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Date deposited: 16 Jan 2023 17:33
Last modified: 17 Mar 2024 03:40

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Contributors

Author: Fabio Furini
Author: Ivana Ljubić

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