Generalized Reduced Jacobian Method

In a recent work, we presented the reduced Jacobian method (RJM) as an extension of Wolfe's reduced gradient method to multicriteria (multiobjective) optimization problems dealing with linear constraints. This approach reveals that using a reduction technique of the Jacobian matrix of the objective avoids scalarization. In the present work, we intend to generalize RJM to handle nonlinear constraints too. In fact, we propose a generalized reduced Jacobian (GRJ) method that extends Abadie-Carpentier's approach for single-objective programs. To this end, we adopt a global reduction strategy based on the fundamental theorem of implicit functions. In this perspective, only a reduced descent direction common to all the criteria is computed by solving a simple convex program. After establishing an Armijo-type line search condition that ensures feasibility, the resulting algorithm is shown to be globally convergent, under mild assumptions, to a Pareto critical (KKT-stationary) point. Finally, experimental results are presented, including comparisons with other deterministic and evolutionary approaches.