  We provide $\widetilde{O}(\epsilon^{-1})$-pass semi-streaming algorithms for
computing $(1-\epsilon)$-approximate maximum cardinality matchings in bipartite
graphs. Our most efficient methods are deterministic and use optimal, $O(n)$,
space, improving upon the space complexity of the previous state-of-the-art
$\widetilde{O}(\epsilon^{-1})$-pass algorithm of Ahn and Guha. To obtain our
results we provide semi-streaming adaptations of more general continuous
optimization tools. Further, we leverage these techniques to obtain
improvements for streaming variants of approximate linear programming, optimal
transport, exact matching, transshipment, and shortest path problems.
