Quadratic programming, a type of convex optimization, is used to solve the minimum variance portfolio weights $$w = \arg \min_w \sigma_P^2 = w^\top \Sigma w$$
because the objective function coincides with quadratic programming, which takes the form: $$x = \arg \min_x x^\top A x$$
The maximum skewness and maximum kurtosis portfolios, on the other hand, are tensors that look like they would require a type of optimization of higher order (order-3 and order-4) than quadratic programming (which is order-2):
$$\arg \max_w \enspace s_P = w M_3 (w^\top\otimes w^\top)$$ $$\arg \max_w \enspace k_P = w M_4 (w^\top\otimes w^\top \otimes w^\top)$$ where $M_3$ and $M_4$ are the co-skewness and co-kurtosis matrices respectively. Would these two objective functions comply with the quadratic programming formula (second from the top)? If not, what is an appropriate optimizer? Or would quadratic programming work as long as the tensors $s_P$ and $k_P$ are flattened into 2-dimensional matrices?
Someone followed up the answers to this question with: