Issues of piecewise linearity in dual and primal-dual methods for bounded-variable linear programs
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-; TRN: 94:009753-0290
- Resource Relation:
- Conference: 15. international symposium on mathematical programming, Ann Arbor, MI (United States), 15-19 Aug 1994; Other Information: PBD: 1994; Related Information: Is Part Of Mathematical programming: State of the art 1994; Birge, J.R.; Murty, K.G. [eds.]; PB: 312 p.
- Country of Publication:
- United States
- Language:
- English
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