Issues of piecewise linearity in dual and primal-dual methods for bounded-variable linear programs

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

OSTI ID:36022

Fourer, R

Large-scale linear programming packages convert their input to a standard form that is convenient for computational purposes. Most popular is the {open_quotes}bounded-variable{close_quotes} form that minimizes some linear objective cx subject to general constraints Ax = b and bound constraints 1 {le} x {le} u. The dual of a bounded-variable linear program is not a similarly bounded LP, however, but rather is {open_quotes}merely piecewise linear{close_quotes} in a precise sense that has been elucidated by Rockafellar`s work on monotropic programming. The piecewise linearity of the dual has important implications for both extreme-path and interior-path algorithms. In the case of the dual simplex method, we are led to an intuitively {open_quotes}geometric{close_quotes} derivation in the space of the dual variables, and to the option of longer steps at some iterations. In the case of popular primal-dual path-following methods, a piecewise linear viewpoint helps us derive better lower bounds at each iterate, and lets us see how current LP implementations can be applied directly to least-absolute-deviations data-fitting problems, {open_quotes}elastic{close_quotes} programming problems, and similar formulations.

• OSTI ID: 36022
• Report Number(s): CONF-9408161--
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
LINEAR PROGRAMMING
MATRICES
NUMERICAL SOLUTION
PARAMETRIC ANALYSIS