This directory contains the Global BFGS algorithm, the BFGS-Wolfe algorithm and the Cautious BFGS-Armijo algorithm for solving multiobjective optimization problems described in the paper:

L. F. Prudente and D. R. Souza, Global convergence of a BFGS-type algorithm for nonconvex multiobjective optimization problems, technical report, 2023.

• MOPsolverBFGS.f90: routine containing the Global BFGS and BFGS-Wolfe algorithms
• MOPsolverCautiousBFGS.f90: routine containing the Standard Cautious BFGS-Armijo algorithm

This folder also contains the third-party free codes:

• software Algencan 3.1.1

  • . G. Birgin and J. M. Martı́nez, Practical augmented Lagrangian methods for constrained optimization, SIAM, 2014.
  • https://www.ime.usp.br/~egbirgin/tango/
  • Algencan is used to compute the search directions; see innersolver.f90.

• subroutines dcsrch and dcstep of Moré and Thuente

  • J. J. Moré and D. J. Thuente, Line Search Algorithms with Guaranteed Sufficient Decrease, ACM Trans. Math. Softw., 20 (1994), pp. 286–307.
  • http://ftp.mcs.anl.gov/pub/MINPACK-2/csrch/
  • These subroutines are used as the inner solver of lsvecopt.f90 which computes a step size satisfying the (vector) Wolfe conditions.

Instructions:

File main.f90 contains the main program where you can choose the algorithm to be used. Modify myproblem.f90 routine to solve your own problem. Alternatively, set a test problem in main.f90 routine; see myproblem.f90.

The codes are written in Fortran 90. Users need to install gfortran.

1. In the terminal, go to the folder and type:

   make

2. Run typing:

   ./MOPsolver

and see the output in the screen.

• out: outer iteration number
• |theta|: optimality measure
• LS: flag of the line search routine to compute the step size (0 means success)
• IS: flag of the inner solver routine to compute the search direction (0 means success)
• #evalf: number of function evaluations
• #evalg: number of gradient evaluations