Comparison of two sets of first-order conditions as bases of interior-point Newton methods for optimization with simple bounds

• Published in: Journal of optimization theory and applications Vol. 113; no. 1; pp. 21 - 40
• Main Authors: JAMROG, D. C, TAPIA, R. A, ZHANG, Y
• Format: Journal Article
• Language: English
• Published: New York, NY Springer 01.04.2002; Springer Nature B.V
• Subjects: Algorithms; Applied mathematics; Applied sciences; Exact sciences and technology; Lagrange multiplier; Methods; Operational research and scientific management; Operational research. Management science; Optimization; Optimization. Search problems; Necessary condition; First order; Interior point method; Convex function; Numerical method; Non linear programming; Objective function; Newton method; Optimization; Quadratic function
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1023/A:1014801112646

In this paper, we compare the behavior of two Newton interior-point methods derived from two different first-order necessary conditions for the same nonlinear optimization problem with simple bounds. One set of conditions was proposed by Coleman and Li; the other is the standard KKT set of conditions. We discuss a perturbation of the CL conditions for problems with one-sided bounds and the difficulties involved in extending this to problems with general bounds. We study the numerical behavior of the Newton method applied to the systems of equations associated with the unperturbed and perturbed necessary conditions. Preliminary numerical results for convex quadratic objective functions indicate that, for this class of problems, the Newton method based on the perturbed KKT formulation appears to be the more robust.