The following optimization problem is considered: Minimize \[ z=\sum^m_{i=1} c_ix_i \] subject to \[ \max_{1\leq i\leq m}x_i+a_{ij} = 2b_j,\quad j =1,\dots,n,\quad 0\leq x_i\leq 1,\;i=1,\dots,m, \] where \(0\leq a_{ij}\leq 1\), \(0\leq b_j\leq 1\) are given real numbers. Properties of the nonconvex set of feasible solutions of this problem are investigated. Using these properties, the authors propose an algorithm for solving the optimization problem mentioned above. The explanations are illustrated by small numerical examples. The proposed algorithm is in general not appropriate for solving problems with large \(m, n\).