Summary: A bipartite graph \(G=(U,V,E)\) is convex if the vertices in \(V\) can be linearly ordered such that for each vertex \(u\in U\), the neighbors of \(u\) are consecutive in the ordering of \(V\). An induced matching \(H\) of \(G\) is a matching for which no edge of \(E\) connects endpoints of two different edges of \(H\). We show that in a convex bipartite graph with \(n\) vertices and \(m\) weighted edges, an induced matching of maximum total weight can be computed in \(O(n+m)\) time. An unweighted convex bipartite graph has a representation of size \(O(n)\) that records for each vertex \(u\in U\) the first and last neighbor in the ordering of \(V\). Given such a compact representation, we compute an induced matching of maximum cardinality in \(O(n)\) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in \(O(n)\) time. If no compact representation is given, the cover can be computed in \(O(n+m)\) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of \(O(n^2)\) [A. Brandstädt et al., Theor. Comput. Sci. 381, No. 1–3, 260–265 (2007; Zbl 1188.68209)]. The weighted case has not been considered before.