An interior point algorithm for global optimal solutions and KKT points

In this paper some theorems which characterize the global optimal solutions of nonlinear programming problems are proved. Two algorithms are derived using these results. The first is a path following algorithm to approximate the Karush Kuhn Tucker points of linearly constrained optimization problems and the second is an algorithm to solve linearly constrained global optimization problems. The convergence of these algorithms is proved under suitable assumptions. Numerical results obtained on several test problems are shown.

[1]  M. Todd,et al.  Mathematical Developments Arising from Linear Programming , 1990 .

[2]  Susan W. Palocsay,et al.  Optimizing the sum of linear fractional functions , 1992 .

[3]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[4]  T Talaky,et al.  Interior Point Methods of Mathematical Programming , 1997 .

[5]  Y. Almogy,et al.  A Class of Fractional Programming Problems , 1971, Oper. Res..

[6]  Solomon Lefschetz,et al.  Stability by Liapunov's Direct Method With Applications , 1962 .

[7]  M. C. Recchioni,et al.  A Quadratically Convergent Method for Linear Programming , 1991 .

[8]  P. Wolfe.  On the convergence of gradient methods under constraint , 1972 .

[9]  On the convergence of the projected gradient method , 1993 .

[10]  D. Bayer,et al.  The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories , 1989 .

[11]  Hiroshi Konno,et al.  Minimizing and maximizing the product of linear fractional functions , 1992 .

[12]  Renato D. C. Monteiro,et al.  Interior path following primal-dual algorithms. part II: Convex quadratic programming , 1989, Math. Program..

[13]  Singiresu S. Rao,et al.  Optimization Theory and Applications , 1980, IEEE Transactions on Systems, Man, and Cybernetics.

[14]  Yinyu Ye,et al.  An extension of Karmarkar's projective algorithm for convex quadratic programming , 1989, Math. Program..

[15]  K. Swarup.  Letter to the Editor-Linear Fractional Functionals Programming , 1965 .

[16]  A. Cambini,et al.  On Maximizing a Sum of Ratios , 1989 .

[17]  John N. Hooker,et al.  Karmarkar’s Linear Programming Algorithm , 1986 .

[18]  R. C. Monteiro,et al.  Interior path following primal-dual algorithms , 1988 .

[19]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .

[20]  D. Bayer,et al.  The Non-Linear Geometry of Linear Pro-gramming I: A?ne and projective scaling trajectories , 1989 .

[21]  Maria Cristina Recchioni,et al.  Monotone Variable–Metric Algorithm for Linearly Constrained Nonlinear Programming , 2000 .