My goal is to minimize functions such as
$$f_1(\mathbf{p})=p_1p_3p_7+p_1p_4p_7+p_2p_3p_7+p_2p_4p_7-p_1p_3p_5p_6-p_1p_4p_5p_6-p_2p_3p_5p_6-p_2p_4p_5p_6$$
and
$$f_2(\mathbf{p})=p_1p_3p_7+p_1p_4p_7+p_2p_3p_7+p_2p_4p_7-p_1p_3p_5p_6-p_1p_4p_5p_6-p_2p_3p_5p_6-p_2p_4p_5p_6$$
over the interval $p_i\in [0.01,0.03] \forall i=1,2,3,4,5,6,7$. You can see there trilinear and quad-linear functions but there can higher degree linear functions. My first ideas contained things such as the Lagrangian multiplier method and Simplex but if you look multilinear optimisation algorithms such as this branch-and-bound recursive algorithm and the modified LP approach here -- I am starting to feel a bit ignorant.
What are different ways to find the minimum with a multilinear function? Which method should be used and when? Their pros and cons?