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Do you know of a (possibly heuristic) solver of a system of nonlinear inequalities? WolframAlpha seems to work only for a very limited number of inequalities.
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It will help knowledgeable commenters if you can say more about the structure of the system or systems you need to solve. What is the structure of a typical inequality? Approximately how many are there?

In general, heuristic "hill-climbing" algorithms are the best you are going to do, but for very particular kinds of systems there are special tricks.
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