Most efficient computational method to numerically minimize a 8 variables constrained system

You need to give importance multiplier to constraints and the optimization method should not be greedy.

A genetic algorithm combined with multiple-starting points (or simulated annealing for diminishing mutations) tends to converge to global minima (hence not greedy) with more time given to it but there is not guarantee that the heuristic will complete X function in Y time. The more time given to it, the better it converges to global minima.

In genetic algorithm, you can add big constraint penalties like this:

fitness_minima = some_function_output_between_1_and_10 + 
                 constraints_breached?1000.0f:0;

so that the DNAs with no contraint-violations will be favored for the crossover part of GA.

"As efficient as possible" depends on your algorithm. If you can parallelize the algorithm and run it on multiple GPUs, it should give substantial speedup over CPU. Compared to some hours of Mona-Lisa painting by CPU, a parallelized version running on 3 low-end GPUs complete within 10 minutes (https://www.youtube.com/watch?v=QRZqBLJ6brQ). At least some OpenCL/CUDA supporting libraries/frameworks (like Tensorflow) should be able to accelerate your algorithm if you don't want to do the work distribution yourself.