On the use of piecewise linear models in nonlinear programming

• Published in: Mathematical programming Vol. 137; no. 1-2; pp. 289 - 324
• Main Authors: Byrd, Richard H., Nocedal, Jorge, Waltz, Richard A., Wu, Yuchen
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.02.2013; Springer Nature B.V
• Subjects: Algorithms; Analysis; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Computer engineering; Computer science; Convergence; Cost control; Derivatives; Full Length Paper; Interpolation; Lagrange multiplier; Linear programming; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Methods; Nonlinear programming; Numerical Analysis; Optimization; Optimization algorithms; Quadratic programming; Studies; Theoretical; 90C06; 65K05; 90C30; 49M37; 90C55
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-011-0492-9

This paper presents an active-set algorithm for large-scale optimization that occupies the middle ground between sequential quadratic programming and sequential linear-quadratic programming methods. It consists of two phases. The algorithm first minimizes a piecewise linear approximation of the Lagrangian, subject to a linearization of the constraints, to determine a working set. Then, an equality constrained subproblem based on this working set and using second derivative information is solved in order to promote fast convergence. A study of the local and global convergence properties of the algorithm highlights the importance of the placement of the interpolation points that determine the piecewise linear model of the Lagrangian.