Aspects of degeneracy in linear programming

Conference · Sat Dec 31 04:00:00 EST 1994

OSTI ID:36354

Osborne, M

Typically, active set descent methods for the linear programming problem assume that at most p constraints are active at the current point x {element_of} R{sup p} when constructing a descent direction. If this assumption is violated then a selection of exactly p constraints has to be made. But now it is possible for the direction generated to lead immediately to violation of some of the ignored active constraints, and the result can be cycling or stalling in the computational procedure. Here methods for computing a suitable descent direction in the case of degeneracy are reviewed. These include recursive methods with guaranteed termination, and non-recursive methods which terminate with a feasible descent direction with probability 1. In contrast, degeneracy is not a problem for interior point methods based on sequential unconstrained minimization techniques using barrier functions provided the linear program has a unique solution. However, non-uniqueness does lead to increasing ill conditioning of the Hessian matrix in the Newton iteration for minimizing the barrier function as the barrier parameter tends to zero. This makes it preferable to switch to the barrier formulation of the dual which is degenerate and typically is well behaved. These comments apply also to primal-dual algorithms which follow a fixed pattern of variable elimination. Detection of the non-unique case is discussed.

• OSTI ID: 36354
• Report Number(s): CONF-9408161--
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
EFFICIENCY
LINEAR PROGRAMMING
MATRICES
NUMERICAL SOLUTION