Engineering Optimization

4Univariate (Single DV) Search Techniques

4.1 Univariate (Single DV)

The term “line search” is often used to mean that the optimization application has a single decision variable. I will use the term univariate, one variable, to indicate a single DV:

(4.1)

The variable x could be a primary, fundamental variable, such as what might be graphed in an f(x) w.r.t. x presentation.

However, the decision variable might be distance along a line that goes through space, and the objective might be to find the point along the line to maximize proximity (minimize distance) to a planet. Using a parametric notation for a line to calculate the (x, y, z) position as a function of parameter “s” from the origin:

(4.2)

Note: At some position (x, y, z) distance along the line from the origin (x₀, y₀, z₀) is S, not s:

(4.3)

However, here the objective is not to minimize distance along the line, but the distance from the line to the point. Then, with the OF to minimize distance to a planet, a point object, at (x_p, y_p, z_p), the optimization statement becomes

(4.4)

Here the DV is not distance along an axis in ...