In this work, we introduce and study the $p$-$α$-closest-center problem ($pα$CCP), which generalizes the $p$-second-center problem, a recently emerged variant of the classical $p$-center problem. In the $pα$CCP, we are given sets of customers and potential facility locations, distances between each customer and potential facility location as well as two integers $p$ and $α$. The goal is to open facilities at $p$ of the potential facility locations, such that the maximum $α$-distance between each customer and the open facilities is minimized. The $α$-distance of a customer is defined as the sum of distances from the customer to its $α$ closest open facilities. If $α$ is one, the $pα$CCP is the $p$-center problem, and for $α$ being two, the $p$-second-center problem is obtained, for which the only existing algorithm in literature is a variable neighborhood search (VNS). We present four mixed-integer programming (MIP) formulations for the $pα$CCP, strengthen them by adding valid and optimality-preserving inequalities and conduct a polyhedral study to prove relationships between their linear programming relaxations. Moreover, we present iterative procedures for lifting some valid inequalities to improve initial lower bounds on the optimal objective function value of the $pα$CCP and characterize the best lower bounds obtainable by this iterative lifting approach. Based on our theoretical findings, we develop a branch-and-cut algorithm (B&C) to solve the $pα$CCP exactly. We improve its performance by a starting and a primal heuristic, variable fixings and separating inequalities. In our computational study, we investigate the effect of the various ingredients of our B&C on benchmark instances from related literature. Our B&C is able to prove optimality for 17 of the 40 instances from the work on the VNS heuristic.

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