Optimization stationarity conditions

V L is equal to the constrained derivatives for the problem, which should be zero at the solution to the problem. Also, these stationarity conditions very neatly provide the necessaiy conditions for optimality of an equality-constrained problem. 

Some times the stationarity condition = 0 of an optimal control problem can be solved to obtain an explicit expression for u in terms of the state y and the costate A. That expression, when substituted into state and costate equations, couples them. Thus, state equations become dependent on A and must be integrated simultaneously with the costate equations. The simultaneous integration constitutes a two point boundary value problem in which... 

While this equation is quite simple in form, it hides the fact that the universal functional F[p] is not available in explicit form. Numerous schemes have been formulated for a direct optimization of the density based on these stationarity conditions, but these methods have not really been competitive. The most efficient approach has been to invoke a quasi-independent-particle approximation, formulated in the Kohn-Sham equations. 

Figure 11. The error threshold of replication and mutation in genotype space. Asexually reproducing populations with sufficiently accurate replication and mutation, approach stationary mutant distributions which cover some region in sequence space. The condition of stationarity leads to a (genotypic) error threshold. In order to sustain a stable population the error rate has to be below an upper limit above which the population starts to drift randomly through sequence space. In case of selective neutrality, i.e. the case of equal replication rate constants, the superiority becomes unity, Om = 1, and then stationarity is bound to zero error rate, pmax = 0. Polynucleotide replication in nature is confined also by a lower physical limit which is the maximum accuracy which can be achieved with the given molecular machinery. As shown in the illustration, the fraction of mutants increases with increasing error rate. More mutants and hence more diversity in the population imply more variability in optimization. The choice of an optimal mutation rate depends on the environment. In constant environments populations with lower mutation rates do better, and hence they will approach the lower limit. In highly variable environments those populations which approach the error threshold as closely as possible have an advantage. This is observed for example with viruses, which have to cope with an immune system or other defence mechanisms of the host.

Thus, the processes taking place in technological reactors can have a multi-stationarity even for relatively simple kinetic schemes (in this case we consider a simple non-reversible first-order reaction). In practice, reactors work usually in the conditions close to stationary. Therefore, a problem of optimal organization of the reaction conditions becomes of great importance. In the discussed example the first stationary state is imdesirable from the efficiency point of view. 

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