Elements of Numerical Mathematical Economics with Excel

5.4. Numerical steepest descent method applied to the unconstrained optimization with VBA

The steepest descent is a numerical method applied to the unconstrained optimization that can be both applied to the case of multivariate and univariate functions.

The algorithm works by iterations through the following steps (e.g., for a minimum):

1. • Step 1. Start with a first approximation x ₀ to the minimum x∗.
2. • Step 2. Evaluate in x ₀ the following function:

g ( t ) = F ( x − t ∇ F ( x ) )

such that it turns out:

g ( t 0 ) = F ( x 0 − t 0 ∇ F ( x 0 ) )

1. • Step 3. Find t 0 ∗