On the formulation and theory of the Newton interior-point method for nonlinear programming

In this work, we first study in detail the formulation of the primal-dual interior-point method for linear programming. We show that, contrary to popular belief, it cannot be viewed as a damped Newton method applied to the Karush-Kuhn-Tucker conditions for the logarithmic barrier function problem. Next, we extend the formulation to general nonlinear programming, and then validate this extension by demonstrating that this algorithm can be implemented so that it is locally and Q-quadratically convergent under only the standard Newton method assumptions. We also establish a global convergence theory for this algorithm and include promising numerical experimentation.

[1]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[2]  M. Hestenes.  Multiplier and gradient methods , 1969 .

[3]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[4]  R. Tapia.  Role of slack variables in quasi-Newton methods for constrained optimization , 1979 .

[5]  Klaus Schittkowski,et al.  Test examples for nonlinear programming codes , 1980 .

[6]  Klaus Schittkowski,et al.  More test examples for nonlinear programming codes , 1981 .

[7]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[8]  J. Nocedal,et al.  A tool for the analysis of Quasi-Newton methods with application to unconstrained minimization , 1989 .

[9]  N. Megiddo.  Pathways to the optimal set in linear programming , 1989 .

[10]  Nimrod Megiddo,et al.  Homotopy Continuation Methods for Nonlinear Complementarity Problems , 1991, Math. Oper. Res..

[11]  Margaret H. Wright,et al.  Interior methods for constrained optimization , 1992, Acta Numerica.

[12]  R. Tapia,et al.  Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited , 1992 .

[13]  Stephen J. Wright.  An Interior-Point Algorithm for Linearly Constrained Optimization , 1992, SIAM J. Optim..

[14]  Roy E. Marsten,et al.  The interior-point method for linear programming , 1992, IEEE Software.

[15]  Yin Zhang,et al.  On the Superlinear and Quadratic Convergence of Primal-Dual Interior Point Linear Programming Algorithms , 1992, SIAM J. Optim..

[16]  Stephen J. Wright,et al.  A globally and superlinearly convergent potential reduction interior point method for convex programming , 1992 .

[17]  Roy E. Marsten,et al.  On Implementing Mehrotra's Predictor-Corrector Interior-Point Method for Linear Programming , 1992, SIAM J. Optim..

[18]  Stephen G. Nash,et al.  A Barrier Method for Large-Scale Constrained Optimization , 1993, INFORMS J. Comput..

[19]  K. Anstreicher,et al.  On the convergence of an infeasible primal-dual interior-point method for convex programming , 1994 .

[20]  R. D. Murphy,et al.  Iterative solution of nonlinear equations , 1994 .

[21]  Yin Zhang,et al.  A Study of Indicators for Identifying Zero Variables in Interior-Point Methods , 1994, SIAM Rev..

[22]  Tao Wang,et al.  A Positive Algorithm for the Nonlinear Complementarity Problem , 1995, SIAM J. Optim..

[23]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[24]  Stephen J. Wright,et al.  A superlinear infeasible-interior-point algorithm for monotone complementarity problems , 1996 .

[25]  Hiroshi Yamashita.  A globally convergent primal-dual interior point method for constrained optimization , 1998 .