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Basic Feasible Solutions in Linear Programming
en.wikipedia.org/wiki/Basic_feasible_solutionYapay zekadan makale özeti
- Definition and Properties
- Basic feasible solution has minimal non-zero variables
- Each BFS corresponds to a vertex of feasible solutions polyhedron
- BFS depends only on LP constraints, not optimization objective
- BFS has at most m non-zero variables and at least n-m zero variables
- Columns must be linearly independent for BFS to be basic
- Optimal BFS
- Optimal solution implies optimal BFS
- BFS can come from multiple bases
- Simplex algorithm finds optimal BFS by exploring BFS-s
- Any optimal solution can be converted to optimal BFS
- Preliminaries
- Linear program can be converted to equational form with slack variables
- Basis is nonsingular submatrix with m rows and m<n columns
- Feasible solutions are vectors satisfying LP constraints
- BFS can have fewer than m non-zero variables
- Optimal BFS Finding
- Simplex algorithm keeps current basis, BFS, and tableau
- Process explores variables entering and exiting basis
- Megiddo proved strongly polynomial time algorithms for finding optimal BFS