Zhang et al. [10] first introduce B-stationary, C-stationary, M-stationary, S-stationary point, SOCMPCC-linear independence constraint qualification, second-order cone upper level strict complementarity condition at a feasible point of a SOCMPCC problem and discuss the convergence properties of a smoothing approach for solving SOCMPCCs.
Nevertheless, the fact that strict complementarity is not a diffuse factor behind the adoption of all environmental innovation indeed does not come as a surprise.
Remark Theorem 4.4 shows that under the strict complementarity conditions and linear independence constraint qualifications, a local optimal solution is a M-stationary point which is introduced for mathematical programming governed by second-order cone constrained generalized equations in [14].
(10) (b) We say the strict complementarity ((mathcal{SC})) condition holds at ((bar{x},bar{y},bar{v})), if lambdainoperatorname{ri} N_{mathcal{K}^{p}}bigl(A(bar{x} bar{y}+bbigr) for all λ satisfying (lambdain N_{mathcal{K}^{p}}(A(bar{x} bar{y}+b)) and (A(bar{x})^{T}lambda =bar{v}). .
We say the strict complementarity ((mathcal{SC})) condition holds at ((bar{x},bar{y},bar{v})), if lambdainoperatorname{ri} N_{mathcal{K}^{p}}bigl(A(bar{x} bar{y}+bbigr) for all λ satisfying (lambdain N_{mathcal{K}^{p}}(A(bar{x} bar{y}+b)) and (A(bar{x})^{T}lambda =bar{v}).
The only case in which strict complementarity is observed in organisational change concerns CO2 abatement, a relatively complex type of EI, but this is true only when the sample is restricted to more polluting (and regulated) sectors.
(A7) The componentwise strict complementarity condition holds at ( u ∗, b 2 − A 2 x ∗ − B 2 y ∗ ), i.e., u ∗ + ( b 2 − A 2 x ∗ − B 2 y ∗ ) ∈ int K q, or in other words, N ˆ ′ = ∅.
(A3) The componentwise strict complementarity condition holds at ( u ∗, b 2 − A 2 x ∗ − B 2 y ∗ ), i.e., u ∗ + ( b 2 − A 2 x ∗ − B 2 y ∗ ) ∈ int R + q, or, in other words, N ′ = ∅.
In this subsection, we reformulate problem (21) as a single level optimization problem, and then we discuss its optimization conditions under strict complementarity and linear independence constraint qualification assumptions.
We prove that, under the strict complementarity and the strong second order sufficient conditions with the sigma term, the rate of local convergence is superlinear.
