The methods for solving numerical optimization problems are nonlinear in essence
iterative routines: a routine cannot find the exact solution in finite time.
In fact the method, as treated by Delbos and Gilbert (2005),
generates an infinite sequence (xt) of approximate solutions. The
next iteration xt +1 is formed according to certain rules, based
on local information of the problem, collected on the previous iteration which
is a vector consisting of values of the objective, constraints and possibly
the gradient or even higher derivatives of these functions. The optimization
methods can be classified not only according to the types of problems they solve,
but by type of local information they employ. From this point of view of information,
the methods are divided into:
||Routines zero-order, using only the values of the objective
and constraints but not their derivatives
||Routines-order, using the values and gradients of the objective
||Routines second order, using the values, gradients and Hessians
of the objective and constraints
Optimizing methods of quadratic functions such as the ones presented by Simonnard
(1973) transform the initial quadratic problem into an equivalent linear
program in order and apply the simplex. Achmanov (1984)
only gave approximate solutions by building convergent series which approaches
the real solution. De Werra et al. (2003) studied
feasible direction method with the difficult problem to choose the direction
and the parameter. Hillier and Lieberman (2005) transformed
the initial problem into a linear one but by adding an important number of constraints.
Bierlaire (2006) has treated a simple example needs at
least ten variables. In the method presented here, without any introduction
of any variable, the exact solution is reached in the first attempt.
This analytical technique determines an exact optimal solution to the problem.
It has to be stressed here that the new method is general to any quadratic optimization
problem. The principle of the technique is to project the critical point to
a new convex built from the set of feasible solutions. the critical point x*
is transformed to the point y*; then y* is projected on
one hyper plan separator An algorithm for finding an optimal solution is given
in this study.
SPLITTING THE OBJECTIVE FUNCTION
Consider the following problem:
with a mxn matrix A and vector b of The
coefficients αi and βi are any real numbers,
so that the function f(x) is neither concave nor convex.
By so doing a decomposition for the function f(x) holds as: f(x) = φ(x)+Ψ(x).
where, the summation runs over all the indices i for βi<0.
where, the sum is extended for the indices i for βi≥0.
In the present study, we study the concave function φ(x). We give, here,
the exact optimal solution by projecting the critical point onto
a new convex Ω'. If the critical point x* is inside the convex Ω,
then the optimal solution holds exactly at this point.
We show in theorem 1, that ,
the projection of the point x* on the convex Ω'.
Let T: Ω⊂Rn→Ω'⊂Rn be the application
that transforms the convex Ω into the convex Ω'. is
non zero, because the βi<0, for all i. Then it is conform.
OPTIMIZATION OF THE CONCAVE FUNCTION φ
Let Ω be a closed, bounded convex set of Rn. Lets put:
Theorem: There exists a closed bounded convex set Ω' of Rn and a vector y0 = (y0i)i, such that the following statements are satisfied:
Proof: For every x∈Ω let ,
can be written in the following form:
for every y∈Ω'.
This implies that
for every y∈Ω'. We have therefore .
Because y∈Ω' then ;
Property 2 has demonstrated.
The vector y0 is the projection of the vector y* onto the new convex Ω'.
Hence property 3.
Remark 1: If x*∈Ω, then and
Remark 2: The transformation T: Ω⊂Rn→Ω'⊂Rn,
for each x∈Ω, associating has
as Jacobean matrix.
Its determinant is .
Then it is conform.
Remark 3: The convex Ω is bounded. We can restrict ourselves to the case αi≥0 for each i. In fact, let's assume that we have αixi+βixi2 with αi<0.
Therefore, it is sufficient to replace xi with
Algorithm of computing the optimal solution of the concave function φ.
Algorithm Ahmed Chikhaoui:
In the algorithm we use the supporting hyperplane. Figure 1 shows a separator supporting hyperplane H1.
The following example traits the case x*∈ Ω. Hillier
and Lieberman (2005).
||Supporting hyperplanes H1 and H2. The
supporting hyperplane H2 does not separate x* of Ω while H1 is separate
supporting hyperplane containing
In this case β1 = 9, then Ω <> Ω'.
The critical point x* = (3, 3) ∈ Ω .
Note the rapid method.
The following example traits the case Ω = Ω. Example treats
by Dozzi (2004).
In this case β1 = β2 = 1, then Ω <> Ω'.
The critical point x* = (5/2, 5/2) ∉ Ω because we
compute the projection of x*
where, a = (3, 1), b = 9
The following example traits the case Ω <> Ω.
Hillier and Lieberman (2005) has treated the following
example. We can compare the two methods.
in this case β1 = 1 and β = 2, then Ω≠Ω'.
The critical point because
The critical point x* = (5/2, 2) ∉ Ω because we
compute the projection of y* (y* is the translation of x*).
where, a = (3, 2), b = 6.
projection of the point y* onto Ω' gives the following point y0:
Note that the projection of point x* onto Ω gives the point
|| The points: x*, y*, y0 and
The point of
convex Ω is the nearest point of the critical point x* but it is not the
optimal solution of φ.
Figure 2 shows the transformation of point x* in y*, the
projection of y* in y0 there and finally the calculation of the point
RESULTS AND DISCUSSION
After the decomposition of the objective function into two functions, one concave φ and the other convex Ψ where if the critical point belongs to omega (feasible constraint set), the optimal solution of f is the optimal solution of φ.
If the critical point does not belong to omega then the optimal solution of f is the projection of the critical point onto Ω' (new convex obtained by the translation T).
Convex Ω is replaced by a closed and lower bounded convex in Rn .
Unlike the previous techniques either analytical producing exact solution by
going through linearization process or numerical producing approached solutions
by Bierlaire (2006), Hillier and Lieberman
(2005), Dong (2006) and De Klerk
and Pasechnik (2007), the technique discussed here, produces the exact solution
without going through linearization, thus it has enlarged the field of applications.