Numerical methods to solve nonlinear system of inequalities? I know some methods to solve nonlinear system of equaltites: Relaxation Method, Newton method,  nonlinear Jacobi method, nonlinear Seidel method.
Is it exist some analogous method to solve nonlinear systems of inequaltites?
 A: Evidently, there is.  In Mathematica, for example:
intervals = Reduce[Sin[8 x]/x^2 <= x, x]
NumberLinePlot[intervals, x]

(* Out: 
   Root[{-Sin[8 #1] + #1^3 &, -0.99738748628764884151}] <= x <= 
     Root[{-Sin[8 #1] + #1^3 &, -0.87854419062296271449}] || 
   Root[{-Sin[8 #1] + #1^3 &, -0.38553220443841061998}] <= x < 0 || 
   Root[{-Sin[8 #1] + #1^3 &, 0.38553220443841061998}] <= x <= 
     Root[{-Sin[8 #1] + #1^3 &, 0.87854419062296271449}] || 
   x >= Root[{-Sin[8 #1] + #1^3 &, 0.99738748628764884151}]
*)


The technique, I believe, is based on root isolation followed by a simple interval check.
Systems of inequalities can often be decomposed into sequences of inequalities involving increasing number of variables - the so called cylindrical algebraic decomposition.  This is a fundamental technique in computer algebra and, while it is immediately applied to systems of polynomials, it can be extended in some cases to more general equations.  For example:
Reduce[
  x^2 + y^2 < 1 && Sin[x + y] < 1/2,
  {x, y}
]

(* Out: (-1 < x < Pi/12 - Sqrt[72 - Pi^2]/12 && 
  -Sqrt[1 - x^2] <= y <= Sqrt[1 - x^2]) || 
  (Pi/12 - Sqrt[72 - Pi^2]/12 <= x < Pi/12 + Sqrt[72 - Pi^2]/12 &&
  -Sqrt[1 - x^2] <= y < 1/6 (Pi - 6 x))
*)

Now, for a given $x$ value, the problem reduces to a single variable problem.
A: I'm answering from my question after couple years....
Yes, there are various ways to solve numerically objective plus inequalities.
But what is deeply wrong in question is that word non-linearity should not bring panic, in fact the problem is not in linear/non-linear but in convex/non-convex. In USSR it was realized in 1960, in USA approximately at same years.
And there areas outside math. optimization which are still think that problem is in "non-linearity".
For convex optimization problem there are a bunch of methods (e.g. interior point method, penalty method, projective subgradients).
For non-convex problems precise methods don't exist but there are two ways to handle it:


*

*Forget about non-convexity and convexify problem locally in some way
1.a DCCP(convex concave programming) - represent objective/constraint 
   functions as convex-convex and then remove concave part or 
   represent it via affine approximation at point.
1.b Just use convex methods and feed non-convex inequality or objective
1.c Fit convex approximation for objective and constraint via particle 
  method 
1.d Use penalty method and pull containts into objective
1.d Create you own

*Use global methods like branch-and-bound which are leveraged in convex optimization step, but they are usually computanionally very hard (even for small dimensions).
