Corrected sequential linear programming for sparse minimax optimization

Abstract

We present a new algorithm for nonlinear minimax optimization which is well suited for large and sparse problems. The method is based on trust regions and sequential linear programming. On each iteration a linear minimax problem is solved for a basic step. If necessary, this is followed by the determination of a minimum norm corrective step based on a first-order Taylor approximation. No Hessian information needs to bc stored. Global convergence is proved. This new method has been extensively tested and compared with other methods, including two well known codes for nonlinear programming. The numerical tests indicate that in many cases the new method can find the solution in just as few iterations as methods based on approximate second-order information. The tests also show that for some problems the corrective steps give much faster convergence than for similar methods which do not employ such steps.

Choice of norms for data fitting and function approximation

“…Indeed, for very large problems, there may be little choice but to use the linearized subproblem as the basis for an algorithm, with, if possible, simple modifications introduced to help speed up convergence. Such ideas for lp roblems are proposed by Jonasson (1993), Jonasson and Madsen (1994). It remains to be seen how effective such methods will become, but, in any event, the basic subproblem will remain a very important and robust tool.…”

Robust design of slow-light tapers in periodic waveguides

“…A similar algorithm for solving nonlinear discrete minimax (i.e. multi-scenario) problems was presented in Madsen and Schjaer-Jacobsen (1978), Jonasson and Madsen (1994), where the authors also give a proof of convergence to stationary points. An algorithm that solves a discrete minimax problem using penalty functions and trust-region methods can also be found in Erdmann and Santosa (2004).…”