We consider unconstrained and constrained optimization problems in the framework of the tropical (idempotent) mathematics, which focuses on the theory and applications of semirings with idempotent addition. The problems are to minimize or maximize functions defined on vectors over idempotent semifields (semirings with multiplicative inverses). We examine several problems of rating alternatives from pairwise comparisons, including problems with constraints on the final scores of alternatives, and multi-criteria rating problems. We reduce the problems to the log-Chebyshev approximation of pairwise comparison matrices by reciprocal matrices of unit rank, and then represent the approximation problems as optimization problems in the tropical mathematics setting. To solve these problems, methods of tropical optimization are used to provide direct, complete solutions in a compact vector form. We discuss the applicability of the results to real-world problems, and provide numerical examples.