Usually it is undesirable to exactly minimize the function in the generic linesearch algorithm. One way to inexactly minimize is by finding an that gives a sufficient decrease in the objective function (assumed smooth), in the sense of the Armijo condition holding. This condition, when used appropriately as part of a backtracking linesearch, is enough to generate an acceptable step length. (It is not sufficient on its own to ensure that a reasonable value is generated, since all small enough will satisfy the Armijo condition. To avoid the selection of steps that are too short, the additional curvature condition is usually imposed.)
In other words, reduce geometrically, with rate , until the Armijo condition holds.