Fast (approximate?) linear programming solver for few variables and many constraints

I'm writing some code in C++, and it turns out I need to solve an optimization problem which may be formulated using m variables and n linear constraints (i.e. constraint i is of the form \sum_j a_{i,j} x_j >= b_i ; although if you don't know what linear programming is you can probably not answer this question).

I need the solution to be fast - at least in expectation (hopefully no more than one pass over most constraints) - and I'm willing to pay in distance from the optimum for that.

Is there a library for this 'niche' of linear programming? If not, what would be generally fast linear programming libraries?

Requirements:

• Code available
• Works on Linux
• Works on x86_64
• Fast
• Gratis

Desired features:

• Really fast
• Can trade off accuracy for speed
• Multi-platform
• C++ rather than C
• Modern C++ rather than C++98/03
• Well-documented
• If C++ is indeed not an absolute requirement, you can also look at open source Java Solvers, such as OptaPlanner and Choco, fwiw. – Geoffrey De Smet Jul 17 at 7:50

BTW, another way to make the tradeoff, regardless of the specific technique, is performing the computation in a reduced-precision datatype, e.g. float32 and not float64.