By Walter J Savitch
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Extra resources for Abstract machines and grammars (Little, Brown computer systems series)
The major novelty in our approach here is the multilinear tensor relaxation, instead of quadratic SDP relaxation methods in [72, 76]. The relaxed multilinear form optimization problems admit polynomialtime approximation algorithms discussed in Sect. 1. After we solve the relaxed problem approximately, the solutions for the tensor model will then be used to produce a feasible solution for the original polynomial optimization model. The remaining task of the section is to illustrate how this can be done.
Obviously, (T PS¯ ) can be relaxed to (T PS¯( 2)), since if √x¯ is feasible for (T PS¯ ) then x¯ 2 = x 2 + xh 2 ≤ 1 + 1 = 2. Consequently, v(T PS¯( 2)) ≥ v(T PS¯ ). Both the objective and the constraints are now homogeneous, and it is obvious that for all t > 0, (T PS¯ (t)) is equivalent (in fact scalable) to each other. Moreover, (T PS¯(1)) is max F(¯x 1 , x¯ 2 , . . t. x¯ k ∈ S¯ n+1 , k = 1, 2, . . , d, which is exactly (TS¯ ) as we discussed in Sect. 1. 5, (T PS¯(1)) admits a polynomial-time approximation d−2 algorithm with approximation ratio (n + 1)− 2 .
X k ∈ Bnk , k = 1, 2, . . , d where n1 ≤ n2 ≤ · · · ≤ nd . Essentially, the approximation method follows a similar flow as we solve the model (TS¯ ) in Sect. 1: we first propose a base algorithm for the case d = 2, and then design a decomposition routine in order to enable a recursive scheme. Unlike (TS¯ ), the case d = 2 for (TB ) is already NP-hard.
Abstract machines and grammars (Little, Brown computer systems series) by Walter J Savitch