Evolutionary Principles in Viral Epitopes - Mathematical Methods and Models in Biomedicine - page 75

Solution of the Optimization Problem

Equation ( 16 ) is a non-linear optimization problem and can be solved again using the
KKT (Karush-Kuhn-Tucker) conditions [ 25 , 29 ]. Since it is a convex optimization
problem, the KKT point is the unique minimum [ 36 ]. The Lagrangian of Eq. ( 16 )is

c

, η , ν )= ∑ i

T i 0 A i p i

∑ i

− ∑ i η i p i .

( − ν −

L p p i (17)

For the entire development of KKT see in [ 4 ]. As in the asynchronous case, we can
write p i 0 = c − ∑ i = i 0 p i and formulate a new optimization problem:

p i

i = i 0 T i 0 A i p i + T i 0 0 A i 0 c − ∑ i = i 0

min
p i ≥

(18)

0

with the following solution for the active constrains:

T i 0 0 A i 0 c − ∑ i = i 0

p i lnA i 0

T i 0 A i p i lnA i −

η i = (19)

i 0 . The fraction of active constraints is a function of T i 0 and A i .
For all nonzero solutions, the value of p i is

for 1 ≤ i ≤ n and i =

ln T i 0 0 lnA i 0
T i 0 lnA i

A i 0 p i 0

= / .

p i lnA i (20)

Equation ( 20 ) can be simplified by defining

T i 0 0 lnA i 0
T i 0 lnA i

Z i =

to obtain

lnZ i + c

1
lnA i

lnZ j
lnA j

1
lnA j

− ∑ j

∑ j

p i =

lnZ i + C 1

lnA i ,

= (21)

where C 1 is constant.

Comparison Between Synchronous and Asynchronous Models

A few basic differences exist between the solutions of the two models. Both models
are fully determined by the value of g i =

t budding
0

x i d t . Thus the optimal epitope