Part 1: In the context of probability, the following optimization problem is often called "Multi-Modal" as it is a sum of different Normal Distributions, and each distribution has its own "mode" - therefore it is "multi modal". Furthermore, this is said to be a difficult optimization problem due to its multi-modality (presumably because of multiple local minima/maxima):
\begin{align*} p(x|\theta) &= \sum_{i=1}^{k} \pi_i \mathcal{N}(x|\mu_i, \sigma_i^2) \end{align*}
$$\sum_{i=1}^{k} \pi_i = 1$$
Optimization Problem: \begin{align*} L(\theta|X) &= \prod_{j=1}^{n} p(x_j|\theta) = \prod_{j=1}^{n} \sum_{i=1}^{k} \pi_i \mathcal{N}(x_j|\mu_i, \sigma_i^2) \end{align*}
Part 2: In Machine Learning, the loss function of a neural network is usually "Non-Convex" as there are multiple local minima and local maxima (and saddle points). This too is a difficult optimization problem:
\begin{equation} L(w_1, w_2, ..., w_n) = \frac{1}{N} \sum_{i=1}^{N} L_i(y_i, f(x_i; w_1, w_2, ..., w_n)) \end{equation}
My Question: Given these examples, is it correct to say that "Multi-Modal" and "Non-Convex" are basically synonyms? They both refer to complex and irregular surfaces : a Multi-Modal function is likely Non-Convex and a Non-Convex function is Multi-Modal.
Is my understanding correct? Multi-Modal and Non-Convex are the same thing?
Thanks!