# Calculation of special functions: gamma function, by van der Laan C.G., Temme N.M. By van der Laan C.G., Temme N.M.

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Extra resources for Calculation of special functions: gamma function, exponential integrals, error-like functions

Example text

Let PD := conv(D). Projection and Lifting in Combinatorial Optimization 43 Deﬁne P 0 (= P ) := {x ∈ Rn : Ax ≥ b}, and for j = 1, . . , t, P j := conv (P j−1 ∩ {x : ∨ (dh x ≥ dh0 )}). h∈Qj Then P t = PD . While faciality is a suﬃcient condition for sequential convexiﬁability, it is not necessary. A necessary and suﬃcient condition is given in . The most important class of facial disjunctive programs are mixed 0-1 programs, and for that case Theorem 11 asserts that if we denote PD := conv {x ∈ Rn+ : Ax ≥ b, xj ∈ {0, 1}, j = 1, .

63. A. Schrijver. On cutting planes. Annals of Discrete Mathematics, 9:291 – 296, 1980. 64. Ramesh Sharda. Linear programming solver software for personal computers: 1995 report. OR/MS Today, 22(5):49 – 57, 1995. 65. Uwe H. Suhl and R. Szymanski. Supernode processing of mixed-integer models. Computational Optimization and Applications, 3:317 – 331, 1994. 66. Stefan Thienel. ABACUS A Branch-And-CUt System. PhD thesis, Universit¨ at zu K¨ oln, 1995. 67. J. A. Tomlin and J. S. Welsh. Finding duplicate rows in a linear program.

To do this, for a row of the form xi = ai0 + aij (−xj ), j∈J with xj integer-constrained for j ∈ J1 := {1, . . , 4}, continuous for j ∈ J2 := {5, 6}, one deﬁnes fij = aij − [aij ], j ∈ J ∪ {0}, ϕi0 = fi0 , and   fij , ϕij = fij − 1,  aij , j ∈ J1+ = {j ∈ J1 |fi0 ≥ fij }, j ∈ J1− = {j ∈ J1 |fi0 < fij }, j ∈ J2 . Then every x which satisﬁes the above equation and the integrality constraints on xj , j ∈ J1 ∪ {i}, also satisﬁes the condition yi = ϕi0 + ϕij (−xj ), yi integer. 1(−x6 ), y1 integer, y2 integer.