2.3 Distributions optimization 67

why we cannot use any linear programming tool for optimization, we need a hill-climbing

method. As usually with **LDPC** **codes** optimization, we use the differential evolution algorithm

[16]. The optimization problem has been expressed in the previous section. Several

problems arise when optimizing hybrid **LDPC** **codes** with differential evolution:

• The parameter space. When there is no additional constraint on Π different of those

above mentioned, the number of parameters, which are joint proportions, to be determined

by the optimization method is D = qmax(qmax+1) d

2 vmax d cmax . To get an idea

on how many parameters DE algorithm is able to h**and**le, the authors automatically

limit the number of parameters to 35 in their code available from [16]. This limit

is quickly reached in the case of optimization of hybrid **LDPC** **codes**, leading to

an equivalent high number of population vectors **and** hence very slow convergence

of DE. Therefore we have to make a heuristic reduction of the parameter space by,

e.g., allowing only very small connection degrees for variable nodes (d vmax = 5 to

10), only two different check degrees **and** two different group orders.

• The initialization problem. In spite of the reduction of the dimension of the parameter

space, this space remains too big to allow to r**and**omly initialize the population

vectors, otherwise too few of them fulfill the code rate. That is why we

need another method to well initialize the population vectors. We show that the

initialization problem of finding vectors of proportions which correspond to code

ensembles with target code rate R (see equation (2.16)) can be expressed by a convex

combination problem [59]. This can be seen when one picks at r**and**om the

marginal proportions Π(j, l) for all (j, l), **and** looks for the conditional probabilities

Π(i, k|j, l) satisfying the code rate. To solve this problem, the solution we have

used is the simplex method [60] with r**and**om cost function, which is used when the

cost function **and** the problem constraints are linear in terms of the parameters to be

optimized. However, the solutions found by the simplex algorithm always satisfy

with equality some of the inequality constraints because the cost function is linear,

therefore the solution to the maximization or minimization of the cost function is

on facets of the constraint polytope which is a convex hull. This implies that a nonnegligible

part of proportion vector components will be set to zero or one by the

brute simplex method. Thus, to use simplex for initialization of of the vector population

of DE to non-trivial very bad components, we need to empirically adapt the

lower **and** upper bounds of the vector components from [0, 1] to, e.g., [0.03, 0.95].

• Interpolations. Another difficulty in using DE to optimize hybrid **LDPC** distributions

is the computation time entailed by J v (˙,·, ·) **and** J c (·, ·) functions. Indeed, the

J v (·, ·, ·) **and** J c (·, ·) functions are evaluated by Monte-Carlo simulations offline,

**and** then interpolated. For a given group order q l , J c is the function of only one

parameter, which is the mean of any component of the LM-invariant vectors going

into or out of the check node, **and** hence we use a mono-dimensional polynomial

interpolation to get a functional approximation. For a given group order q k , J v is

the function of three parameters, **and** hence we use a 2-dimensional spline surface

to interpolate J v . Since these functions are used in the multi-dimensional EXIT