A Compendium of NP Optimization Problems by Pierluigi Crescenzi, Viggo Kann.

By Pierluigi Crescenzi, Viggo Kann.

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0 0 0 0 is nonsingular in the so that /(σ^,σ, ,,^ ,^ ,) = 0. ; 0°) and (σ^; 0°), respectively, so that for every (σ π ; θη) in V2 there is a unique (σχ; 0J = (^(σ^ 0Π); Λ π (σ π , 0Π)) in Vx for which /( σ ι> σΐΡ 0ρ ^π) — 0. Furthermore, /(^Ι,σΙΙ,βΙ,βΙΙ) = + da[ de'u + #0;i/ ffl? 5) 36 Chapter 2. Identification Since p(hIV6u) = p(0°), we have (ΛΠ;0Π) G 7ί. Due to the regularity assumption we find for points (h^a^h^O^, where (σ π ;0 π ) G V2, that the columns of df /δθ'η are linearly dependent on the columns of is nonsingular.

Now, if we let 0f = 0? and differentiate f2 with respect to the remaining elements of 0, we get the Jacobian matrix

The new restriction may either reduce the dimension of the null-set, which holds if rank{J r+1 (0 0 )} = rank{J(0 0 )} + 1, or it may reduce the dimension of the range-set, which holds if rank{J r+1 (0 0 )} = rank{J(0 0 )}. Only in the former case we might say that the new restriction is helpful for the identification of 0°. If 0° is locally identified without the additional restriction, the restriction reduces the dimension of the range-set. , if it is not already identified, or it restricts the range-set, if 0?

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