11.4 Finite Differences and Complex-Step Derivatives

Core idea¶

Finite Differences and Complex-Step Derivatives must be treated as a system-level decision rather than an isolated technique. For nested and simultaneous suspension formulations using identical meshes, derivatives, tolerances, and starts, state what is fixed, what is optimized, what information is available, and what equations define feasibility.

The relevant quantities are outer $p$, trajectory $w=[X,U]$, and sparse residuals $R(p,w)$. The chapter-level formulation is

For this section, trace how the choice changes derivatives and scaling, the active constraints, and the implementable engineering design. A method is useful only when its assumptions are explicit and its result answers the same system question as the baseline.

Engineering interpretation¶

Ask three questions:

1. Which physical, informational, computational, or economic resource changed?

2. Which objective component or active constraint made the change valuable?

3. Does the conclusion survive model, disturbance, initialization, uncertainty, and implementation checks?

A practical action is to match tolerances. Record units and assumptions before optimization, report component objectives and margins afterward, and verify the result using an independent calculation or higher-fidelity model.

Activity 11.4: quantify finite differences and complex-step derivatives¶