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We propose a multi-precision extension of the Quadratic Regularization (R2)algorithm that enables it to take advantage of low-precision computations, andby extension to decrease energy consumption during the solve. The lower theprecision in which computations occur, the larger the errors induced in theobjective value and gradient, as well as in all other computations that occurin the course of the iterations. The Multi-Precision R2 (MPR2) algorithmmonitors the accumulation of rounding errors with two aims: to provideguarantees on the result of all computations, and to permit evaluations of theobjective and its gradient in the lowest precision possible while preservingconvergence properties. MPR2's numerical results show that single precisionoffers enough accuracy for more than half of objective evaluations and most ofgradient evaluations during the algorithm's execution. However, MPR2 fails toconverge on several problems of the test set for which double precision doesnot offer enough precision to ensure the convergence conditions before reachinga first order critical point. That is why we propose a practical version ofMPR2 with relaxed conditions which converges for almost as many problems as R2and potentially enables to save about 50 % time and 60% energy for objectiveevaluation and 50 % time and 70% energy for gradient evaluation.

A Multi-Precision Quadratic Regularization Method for Unconstrained Optimization with Rounding Error Analysis