This paper introduces a novel mixed precision methodology for mathematical optimisation. It involves the use of reduced precision FPGA optimisers for searching potential regions containing the global optimum, and double precision optimisers on a general purpose processor (GPP) for verifying the results. An empirical method is proposed to determine parameters of the mixed precision methodology running on a reconfigurable accelerator consisting of FPGA and GPP. The effectiveness of our approach is evaluated using a set of optimisation benchmarks. Using our mixed precision methodology and a modern reconfigurable accelerator, we can locate the global optima 1.7 to 6 times faster compared with quadcore optimiser. The mixed precision optimisations search up to 40.3 times more starting vector per unit time compared with quad-core optimisers and only 0.7 to 2.7 % of these searches are refined using GPP double precision optimisers. We also provide performance estimates of mixed precision optimisation using an improved FPGA optimiser and speed gains of up to 21.1 times over quad-core GPP designs can be achieved. The proposed methodology also allows us to accelerate problems with more complicated functions or solve problems involving higher dimensions.