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Generalized Reduced Jacobian Method

M. El Maghri, Y. Elboulqe

TL;DR

The paper introduces the Generalized Reduced Jacobian (GRJ) method for multicriteria optimization with nonlinear constraints, built on reducing the Jacobian via the implicit function theorem to obtain a common descent direction for all criteria. It defines a convex subproblem ($P_x$) to compute a direction in the nonbasic space and employs an Armijo-type line search to preserve feasibility, establishing global convergence to Pareto-KKT-stationary points under mild assumptions. The GRJ framework is complemented by a detailed convergence analysis, a practical algorithm with basis management, and an extensive numerical study against ZMO, MOSQP, and NSGA-II on 30 problems, including Disc Brake and Welded Beam, showing competitive Pareto-front quality and robustness. Overall, GRJ offers a principled, feasible-descent approach to nonlinear constrained multiobjective optimization with strong theoretical guarantees and favorable empirical performance.

Abstract

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.

Generalized Reduced Jacobian Method

TL;DR

The paper introduces the Generalized Reduced Jacobian (GRJ) method for multicriteria optimization with nonlinear constraints, built on reducing the Jacobian via the implicit function theorem to obtain a common descent direction for all criteria. It defines a convex subproblem () to compute a direction in the nonbasic space and employs an Armijo-type line search to preserve feasibility, establishing global convergence to Pareto-KKT-stationary points under mild assumptions. The GRJ framework is complemented by a detailed convergence analysis, a practical algorithm with basis management, and an extensive numerical study against ZMO, MOSQP, and NSGA-II on 30 problems, including Disc Brake and Welded Beam, showing competitive Pareto-front quality and robustness. Overall, GRJ offers a principled, feasible-descent approach to nonlinear constrained multiobjective optimization with strong theoretical guarantees and favorable empirical performance.

Abstract

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.
Paper Structure (13 sections, 65 equations, 3 figures, 2 tables)

This paper contains 13 sections, 65 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Efficiency and multiobjective descent
  3. GRJ strategy
  4. GRJ search direction
  5. GRJ algorithm and its convergence
  6. The algorithm
  7. Convergence analysis
  8. Numerical experiments
  9. Test problems and implementation
  10. Performance measures and interpretation
  11. Real-world applications
  12. 'Disc Brake' design problem
  13. 'Welded Beam' design problem

Figures (3)

  • Figure 1: Performance profiles
  • Figure 2: Best Pareto front approximations for the 'Disc Brake' design problem by GRJ, ZMO, MOSQP and NSGA-II
  • Figure 3: Best Pareto front approximations for the 'Welded Beam' design problem by GRJ, ZMO, MOSQP and NSGA-II