On Approximate KKT Condition and its Extension to Continuous Variational Inequalities

In this work, we introduce a necessary sequential Approximate-Karush-Kuhn-Tucker (AKKT) condition for a point to be a solution of a continuous variational inequality, and we prove its relation with the Approximate Gradient Projection condition (AGP) of Garciga-Otero and Svaiter. We also prove that a slight variation of the AKKT condition is sufficient for a convex problem, either for variational inequalities or optimization. Sequential necessary conditions are more suitable to iterative methods than usual punctual conditions relying on constraint qualifications. The AKKT property holds at a solution independently of the fulfillment of a constraint qualification, but when a weak one holds, we can guarantee the validity of the KKT conditions.

Intrachain Energy Migration to Weak Charge-Transfer State in Polyfluorene End-Capped with Naphthalimide Derivative

Polyfluorene end-capped with N-(2-benzothiazole)-1 8-naphthalimide (PF-BNI) is a highly fluorescent material with fluorescence emission modulated by solvent polarity Its low energy excited state is assigned as a mixed configuration state between the singlet S(1) of the fluorene backbone (F) with the charge transfer (CI) of the end group BNI The triexponential fluorescence decays of PF-BNI were associated with fast energy migration to form an intrachain charge-transfer (ICCT) state polyfluorene backbone decay and ICCT deactivation Time-resolved fluorescence anisotropy exhibited biexponential relaxation with a fast component of 12-16 ps in addition to a slow one in the range 0 8-1 4 ns depending on the solvent showing that depolarization occurs from two different processes energy migration to form the ICCT state and slow rotational diffusion motion of end segments at a longer time Results from femtosecond transient absorption measurements agreed with anisotropy decay and showed a decay component of about 16 ps at 605 nm in PF BNI ascribed to the conversion of S(1) to the ICCT excited state From the ratio of asymptotic and initial amplitudes of the transient absorption measurement the efficiency of intrachain ICCT formation is estimated in 0 5 which means that on average, half of the excited state formed in a BNI-(F)(n)-BNI chain with n = 32 is converted to its low energy intrachain charge-transfer (ICCT) state