Bracketing and zoom terminate with a checked step or a failure

The initial direction check either returns failure or establishes ${\varphi }^{\mathrm{\prime }}(0)<0$. Expansion trial parameters are positive: initially $t=1$, and later they are positive doubles. Before each doubling, $0\le a<t$; hence both possible endpoint orders passed to zoom have distinct nonnegative endpoints, at least one positive.

A midpoint of distinct nonnegative endpoints, at least one positive, is positive and strictly between them. Replacing $b$ by that midpoint preserves distinct nonnegative endpoints. In the other update, replacing $a$ by the midpoint, either retaining $b$ or first replacing $b$ by the old $a$, also preserves those properties. Thus every zoom trial is positive, even with reversed endpoint order.

Before any trial evaluation the domain check either returns failure or verifies feasibility. In expansion, a return of a step occurs only after the Armijo-failure branch was bypassed and the strong curvature test passed. In zoom the same two tests precede every step return. Therefore [F2] and steps 1.1–2.1 prove every returned step has the asserted properties.

Each expansion pass either returns, calls zoom and returns its output, or consumes one of at most $E$ passes. Thus at most one zoom is called. Each zoom pass either returns or consumes one of at most $Z$ passes. All evaluations terminate by hypothesis; after exhaustion the specified failure is returned. There are at most $E+Z$ evaluated trial points, with the objective and directional derivative evaluated at each; domain exits consume no trial evaluation. These finite counts prove termination but imply neither an arithmetic-work bound nor successful discovery of an existing step.