A simplex-like method with bisection for linear programming¹

Constraint Optimal Selection Techniques (COSTs) for Linear Programming

“…subtracts a term from (14) to give (11) above. For calculation of b was used for all results presented.…”

“…For densities between 0.005 and 0.09, SUB, NRAD, and variation 1 (13) of NRAD performed about the same. Introducing the use of i b in (14), (5), and (11) significantly improved the CPU times over (13). Between (11) and (14), (14) which only considered the j term was faster.…”

“…273-274), for example, suggest that a constraint with a larger cosine value may be more likely to be binding at optimality. Pan [13,14] applied the cosine criterion to pivot rules of the simplex algorithm as the "most-obtuse-angle" rule. The cosine criterion has also been utilized to obtain an initial basis for the simplex algorithm by Trigos et al [15] and Junior et al [16].…”

On an Efficient Implementation of the Face Algorithm for Linear Programming

“…The first milestone is the well-known simplex method founded by Dantzig [2], in which the philosophy is to move on the underlying polyhedron, from vertex to an adjacent vertex, until reaching an optimal vertex. Since then, extensive theoretical analysis, numerical implementations as well as numerous variants (see e.g., [1,3,4,6,7,9,[13][14][15][16][17][18][20][21][22][23][24]) have been developed. To date, the simplex algorithm is accepted as one of the most famous and widely used mathematical tools in the world [13].…”

An affine-scaling pivot algorithm for linear programming

“…Algorithms proposed by Pan [32][33][34][35] can be viewed as variants of the simplex algorithm. Nevertheless, some of the proposed pivot rules lead to non-monotone changes of the objective value in solution process, and produce iterates that are no longer necessarily vertices.…”