3.2 Optimization methods
3.2.5 Limitations and global optimization
The analytic methods to search the minimum of a functional (in presence or not of constraints) are sufficient to determine the global minimum point only if the functional is monotonic.
Usually, the considered functional in the optimization problems, present a huge number of local minima; the analytic methods tend to a certain local minimum depending on the starting point. This aspect prevents the use of gradient based searching algorithms and the so called Euristic methods are introduced in order to overcome this problem searching the global minimum point. Many practical applications made massive the use of this heuristic methods so that many algorithms are today available: simulated annealing,
particle swarm simulation, genetic algorithms etc... Without entering into the details of the different methods, a common drawback is represented by the speed of the procedure that is slower than the analytic methods. As a consequence their utilization is limited when the number of the design variables is large and at the same time, the convergence is reached only if a lot of coefficients are set properly.
In the present work a particular algorithm for the research of the global minimum point is considered. The Local Smoothing algorithm presents a good compromise between computational speed and capability to find a global minimum in presence of several constraints. The method described in the following subsection, has been implemented in Matlab environment in previous works ([47]) with the aim to use robust SQP algorithms to search local minima and consequently applying to them the algorithms for the de- termination of the global minimum.
LOCSMOOTH algorithm
LOCSMOOTH is an algorithm for global optimization, which deals with the search of global optima and it is due to Addis, Locatelli and Schoen.[2] It is supposed to search the global minimum x∗ of the function represented in continuous line in Fig. 3.3. The algorithm works properly when the starting function has a funnel structure, i.e. a superposition of an underlying structure (the dashed curve represented in the follows figure) and some perturbation around it.
The local optimum x1 depends on the starting point x∗; thus, the local
optimum function L(x*) can be defined and reported in Fig. 3.3 with the tick line: this is a step function and a rapid convergence algorithms cannot be applied. For this reason, a Gaussian filtering smooths the thickest curve and the minimum point of the smoothed function is found. From this point, the local minimum of the objective function is searched with a generic local algorithm. This logic is implemented in the LOCSMOOTH algorithm that is organized in two phases:
• Approximation phase; • Displacement phase.
In the approximation phase, the local minimum L(xh), starting from the
point xh, is found. In particular, a uniform random sample of K points,
inside a sphere of radius R and centered at xh, S(xh, R), is generated and
the local minima L(xk), xk∈ S(xk, R) are calculated. These values are used
Figure 3.3: An example of a function with a funnel structure and its step function.
In the displacement phase, the minimum of ˆLBg(x) is taken as the next starting point and the procedure is iterated. The procedure is stopped when no improvements are observed. Then, the procedure is now described in details, in the following paragraph.
First of all, a reference configuration x0 is considered; moreover, other
parameters are introduced, which are: R, K, and M axN oImp. Then an uniform random sample point x is chosen in a sphere S(x0, R) and a local
minimum point x∗is computed starting from the sample point x; furthermore, it is set current = record = f (x∗).
Secondly the steps procedure is described below.
STEP 1 If N oImp < M axN oImp, i is set to 0; else, the optimization process is ended.
STEP 2 If i < K and record ≤ current, i = i+1 and an uniformly random point yi is sampled inside the sphere S(x∗, R); moreover, a local minimum
point of y∗i is computed starting from yi. It is set current = f (y∗).
STEP 3 If current < record then a new record has been found and record = current is posed; moreover, the center of the sphere is changed y∗. Else, N oImp = N oImp + K is set and the smoothed function is built using the stored local minima yi∗. Then, ˆx, the minimum of the smoothed
function, is found. Furthermore, the local minimum y∗ of the objective function is obtained, starting from ˆx and it is set current = f (y∗).
STEP 4 If current < record, a new record has been found. Therefore, record = current is set and the sphere is moved in yi∗ position (x∗ = yi∗): S(y∗i, R). The parameter N oImp is set equal to zero. Instead, x∗= ˆx, go to STEP 1.
The global optimization ends when N oImp = M axN oImp. The smoothed function ˆLBg(x) is given by the model shown in eq. (3.43).
ˆ LBg(x) = K P i=1 L(yi)g(kyi− xk) K P i=1 g(kyi− xk) (3.43) where: g(z) = e− z2 (2σ)2 (3.44)
is the Gaussian kernel and the standard deviation σ is:
σ = rK−n1 (3.45)
If this standard deviation definition is used, all the volume of the sphere (which is proportional to rn) is covered by the Gaussian weight. Indeed,
in order to obtain equal coverage for different radius values, the number of samples must be K = σrnn. This choice of standard deviation may be less
effective when the variable have different range of variation. In order to avoid this, a different standard deviation is used:
σ = n Q i=1 ri K !1 n (3.46) where ri are the radii along the different variable axes. The parameter n,
which represents the number of points n to be sampled, influences the quality of the approximated function; however, it is time consuming, because of a local optimization must be performed for each sample.
The parameter ri influences the exploration amplitude: a too small ra-
dius limits the search of the solution, whereas a huge radius could give too dispersed data and a deterioration of the approximated function.
The optimization procedure has been validated trough several numerical problems and real application reported in.[47] Two example are reported in
Figures 3.4 and 3.5, related to the minimization of the Rosenbrock’s function and the optimization of a packaging problem respectively. Results prove the effectiveness of the adopted procedure.
Figure 3.4: Optimization procedure applied to the Rosenbrock’s function