Rank-two programs involving linear fractional functions

Abstract

The aim of this paper is to deepen the study of solution methods for rank-two nonconvex problems with polyhedral feasible region, expressed by means of equality, inequality and box constraints, and objective function in the form of ϕcTx+c0,dTx+d0bTx+b0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi \\left( c^Tx+c_0,\\frac{d^Tx+d_0}{b^Tx+b_0}\\right) $$\\end{document} or ϕ¯c¯Ty+c¯0aTy+a0,dTy+d0bTy+b0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{\\phi }\\left( \\frac{\\bar{c}^Ty+\\bar{c}_0}{a^Ty+a_0}, \\frac{d^Ty+d_0}{b^Ty+b_0}\\right) $$\\end{document}. These problems arise in bicriteria programs, quantitative management science, data envelopment analysis, efficiency analysis and performance measurement. Theoretical results are proved and applied to propose a solution algorithm. Computational results are provided, comparing various splitting criteria.