An explicit spectral Fletcher–Reeves conjugate gradient method for bi-criteria optimization

In this paper, we propose a spectral Fletcher–Reeves conjugate gradient-like method for solving unconstrained bi-criteria minimization problems without using any technique of scalarization. We suggest an explicit formulae for computing a descent direction common to both criteria. The latter further verifies a sufficient descent property that does not depend on the line search nor on any convexity assumption. After proving the existence of a bi-criteria Armijo-type stepsize, global convergence of the proposed algorithm is established. Finally, some numerical results and comparisons with other methods are reported.