software for numerical constraint satisfaction problems

Question

Let $m$ be an even integer greater than $8$. Is there any software I can use to determine for some small $m$ whether the following constraints on $t_0,\ldots,t_{m-1}$ and $w$ have solutions? \begin{cases} &-\pi\leq t_r\leq\pi,\ r=0,\ldots,m-1,\\ &0<w\leq\frac{1}{m},\\ &\sum\limits_{j=1}^{2r-1}\cos(t_j+t_{2r-j})+\sum\limits_{j=2r+1}^{m-1}\cos(t_j+t_{2r+m-j})=w\cos t_{2r},\ r=1,\ldots,\frac{m}{2}-1,\\ &\sum\limits_{j=1}^{2r-1}\sin(t_j+t_{2r-j})+\sum\limits_{j=2r+1}^{m-1}\sin(t_j+t_{2r+m-j})=w\sin t_{2r},\ r=1,\ldots,\frac{m}{2}-1,\\ &t_r+t_{m-r}=0,\ r=1,\ldots,\frac{m}{2},\\ &\cos(t_r+t_{\frac{m}{2}+r})=\cos(t_{2r}+rt_0),\ r=1,\ldots,\frac{m}{2}-1,\\ &\sin(t_r+t_{\frac{m}{2}+r})=\sin(t_{2r}+rt_0),\ r=1,\ldots,\frac{m}{2}-1,\\ &\cos(\frac{mt_0}{2})=1. \end{cases}

Answer 1

The suggestions certainly will depend on the form of the problems.

If your constraints are linear inequalities then there are simple methods (linear programming) which will check feasibility. If they are polynomial (in)equalities, there are general methods for quantifier elimination like cylindrical algebraic decomposition, but these methods are usually only tractable for very small problems. For certain intermediate cases you may be able to cast your problem as a semidefinite program. For more complicated problems, like inequalities that involve polynomials but also the sine function, the problem quickly becomes provably undecidable without additional structure.

If your problem is one of the former types, a google search of the mentioned methods will find you free software to solve it.