I have to minimize a target function
T(x) = T(x1 ... xn)
under vectorial constraints of equality:
∀ i Ei(x) = 0
and inequality:
∀ i Ii(x) > 0
I'm looking for only one minimum, not necessarily the entire set of minima. Also, approximations are possible (I'm not looking for utter precision, or a complete formula for the answer: just the numerical value of each
xi).
The function and constraints are all expressed in terms of sums, products, powers and other continuous differentiable functions (sin, cos, tan, exp, log...). I have access to an entire formal calculus system for differentiating, simplyfing and extracting various properties from any mathematical expression.
If no constraints are present, minimization can be done with least-squares (linear if applicable, non-linear otherwise). Conversely, if the target function is constant, there are no inequality constraints, but the equality constraints are linear, I can solve it as a system of linear equations. This is also possible with a few polynomials. In fact, even if the equality or inequality constraints are arbitrary, I can use a numerical method to approximate the solution. But as soon as the problem gets more complex, I'm lost.
What general method can I use when the target function is not constant, and there are both equality and inequality constraints (assumed non-trivial) ?