  We propose a multi-precision extension of the Quadratic Regularization (R2)
algorithm that enables it to take advantage of low-precision computations, and
by extension to decrease energy consumption during the solve. The lower the
precision in which computations occur, the larger the errors induced in the
objective value and gradient, as well as in all other computations that occur
in the course of the iterations. The Multi-Precision R2 (MPR2) algorithm
monitors the accumulation of rounding errors with two aims: to provide
guarantees on the result of all computations, and to permit evaluations of the
objective and its gradient in the lowest precision possible while preserving
convergence properties. MPR2's numerical results show that single precision
offers enough accuracy for more than half of objective evaluations and most of
gradient evaluations during the algorithm's execution. However, MPR2 fails to
converge on several problems of the test set for which double precision does
not offer enough precision to ensure the convergence conditions before reaching
a first order critical point. That is why we propose a practical version of
MPR2 with relaxed conditions which converges for almost as many problems as R2
and potentially enables to save about 50 % time and 60% energy for objective
evaluation and 50 % time and 70% energy for gradient evaluation.

Optimization and Control