A practical anti-cycling procedure for linearly constrained optimization

Abstract

A procedure is described for preventing cycling in active-set methods for linearly constrained optimization, including the simplex method. The key ideas are a limited acceptance ofinfeasibilities in all variables, and maintenance of a "working" feasibility tolerance that increases over a long sequence of iterations. The additional work per iteration is nominal, and "stalling" cannot occur with exact arithmetic. The method appears to be reliable, based on computational results for the first 53 linear programming problems in the Netlib set.

Solving Reduced KKT Systems in Barrier Methods for Linear and Quadratic Programming

“…In most cases it has the effect of making x * ≈ 1, wherex * is the solution of the scaled problem. As noted elsewhere [GMSW89,Marx89], the test problems grow7, grow15 and grow22 are exceptions in that x * ≈ 10 7 . To assist such cases we have implemented an additional scaling that takes effect if v = 0 in (6.1).…”

Newton Methods For Large-Scale Linear Inequality-Constrained Minimization

“…Nonetheless, degeneracy (or near degeneracy) is possible and needs to be dealt with in any practical implementation. In practice degeneracy may be dealt with by techniques that allow the standard iteration to be used; see, e.g., Gill et al [14]. Such a technique is used within the MINOS code, see Murtagh and Saunders [24], which has been used to solve thousands of practical problems.…”

“…In inexact arithmetic precisely what is the active set is not clear. We prefer therefore to rely on the approach adopted by Gill et al [14]. This technique allows infeasibility tolerances on the constraints that are altered at each iteration.…”

“…This is likely to be common on problems where the linearly constrained problem being solved is an approximation to a nonlinearly constrained problem whose Jacobian is rank deficient at the solution. The use of a procedure similar to that in [14] is in any event essential in practice for the purpose of trying to introduce a choice in the definition of A k in an attempt to ensure that the condition number of A k is not too large. For example, if infeasibilities are allowed then nearly dependent active constraints need not be included in the working set.…”

The simplest examples where the simplex method cycles and conditions where expand fails to prevent cycling

“…Gill et al [7] developed the expand method in an attempt to improve on the good features of the Devex method of Harris and also to incorporate some features of Wolfe's method which guarantee finite termination. The performance of minos was significantly improved by the incorporation of expand.…”