This report is a sequel of the publications [1] [3] [2]. We consider the multiobjective optimization problem of the simultaneous minimization of n (n ≥ 2) criteria, {J_i(Y)}(i=1,...,n), assumed to be smooth real-valued functions of the design vector Y ∈ OMEGA ⊂ R^N (n ≤ N) where OMEGA is the (open) admissible domain of R^N over which these functions admit gradients. Given a design point Y^0 ∈ OMEGA that is not Pareto-stationary, we introduce the gradients {J_i'}(i=1,...,n) at Y = Y^0, and assume them to be linearly independent. We also consider the possible "scaling factors", {S_i} (i=1,...,n) (S_i > 0 , ∀i), as specified appropriate normalization constants for the gradients. Then we show that the Gram-Schmidt orthogonalization process, if conducted with a particular calibration of the normalization, yields a new set of orthogonal vectors {u_i} (i=1,..,n) spanning the same subspace as the original gradients; additionally, the minimum-norm element of the convex hull corresponding to this new family, omega, is calculated explicitly, and the Fréchet derivatives of the criteria in the direction of omega are all equal and positive. This direct process simplifies the implementation of the previously-defined Multiple-Gradient Descent Algorithm (MGDA).
MGDA II: A direct method for calculating a descent direction common to several criteria