This paper studies the cost-minimizing serial asset replacement problem under changing operating and replacement costs. Technological improvements affect these costs in different ways, which leads to different duration of sequential replacement cycles of the same asset. Therefore, a firm should optimize a chain of consecutive asset replacements using available information about future costs. We offer a novel multi-cycle algorithm for making such replacement decisions and prove that it converges to the infinite-horizon solution when the number of cycles increases. Important qualitative properties of the obtained optimal non-equal-life replacement strategies are derived, and practical recommendations are discussed.