Number of local minima of a energy function Given some particles at positions $x_i$, $i = 1,\dots,n$ attached to each other  by elastic springs of rest lengths $l_{ij}$ and spring constants $k_{ij}$ and to some fixed points at positions $X_j$, $j = 1,\dots,m$ by springs of rest lengths $L_{ij}$ and spring constants $K_{ij}$.
The total potential energy of the system is
$$E(x) = \sum_{i=1}^n\Big(\sum_{j=1}^n k_{ij}\Big|l_{ij}-|x_i - x_j|\big|^2 + \sum_{j=1}^m K_{ij}\Big|L_{ij}-|x_i - X_j|\Big|^2\Big) $$
Depending on the parameters $l_{ij}, k_{ij}, L_{ij}, K_{ij}$ there may be no, one, several or very many stable local minima of $E(x)$.
The exact number $N(l_{ij}, k_{ij}, L_{ij}, K_{ij})$ of stable local minima of $E(x)$ is a well-defined function of the parameters $l_{ij}, k_{ij}, L_{ij}, K_{ij}$, even though not given in closed form but calculable brut force by calculating and counting all solutions of $\nabla E(x) = 0$.
I wonder, if there is a way to guess or estimate $N(l_{ij}, k_{ij}, L_{ij}, K_{ij})$ without going the hard way.
 A: The presence of modules in the initial formula theoretically requires the use of the interval method in which the estimation of the level of total potential energy is carried out over an ensemble of all possible combinations of signs under these modules (each sign corresponds to the compressed or extended state of the spring). For each combination of signs, the resulting quadratic function admits no more than one minimum. At the same time, this problem determines the gradient of the potential energy for each point, which allows the replacement of all bonds of a given point by one equivalent.
The problem that arises is analogous to the "knapsack problem"б which can be solved by a search method. In this case, the sequence of circuit of connections must be determined by the level of influence of this connection on the change in the potential energy of the system as a whole ("greedy" strategy). Greedy strategy leads to a reduction in the search through options that can not affect the optimum.
