Parametric Linear Programming
نویسنده
چکیده
Parametric linear programming is the study of how optimal properties depend on data parametrizations. The study is nearly as old as the field of linear programming itself, and it is important since it highlights how a problem changes as what is often estimated data varies. We present what is a modern perspective on the classical analysis of the objective value’s response to parametrizations in the right-hand side and cost vector. We also mention a few applications and provide citations for further study. The study of parametric linear programming dates back to the work of Gass, Saaty, and Mills [6, 23, 26] in the middle 1950s, see [27] as well. The analysis of how optimal properties depend on a problem’s data is important since it allows a model to be used for its intended purpose of explaining the underlying phenomena. This is because models are often constructed with imperfect information, and the study of parametric and sensitivity analysis relates optimal properties to the problem’s data description. The topic is a mainstay in introductory texts on operations research and linear programming. Here we advance the typical introduction and point to some modern applications. For the sake of brevity we omit proofs and instead cite publications in which proofs can be located. We assume the standard form primal and dual throughout, (LP) min{cx : Ax = b, x ≥ 0} and (LD) max{b y : A y + s = c, s ≥ 0}, where A ∈ Rm×n has full row rank, b ∈ R, and c ∈ R. For any B ⊆ {1, 2, . . . , n} we let AB be the submatrix of A whose column indices are in B. We further let N = {1, 2, . . . , n}\B so that AN contains the columns of A not in AB . Similarly, cB and cN denote the subvectors of c whose components are in the set subscripts. The partition (B,N) is optimal if both ABxB = b, xB ≥ 0 and ABy = cB , ANy + sN = cN , sN ≥ 0 (1) are consistent. One special case is if AB is invertible, and we refer to such a partition as basic or as a basis. If a basis is optimal, the above systems can be re-expresses as A−1 B b ≥ 0 and cN −A T N (A T B) cB ≥ 0. (2) Another special case is if B is maximal, meaning that it is not contained in another optimal B set. There is a unique maximal B for any A, b and c, and
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