数学代写|凸优化作业代写Convex Optimization代考|Test Functions

Bi-objective problems with one and two variables were chosen for the experiments to enable visual analysis of the results.

Some experiments were performed using objective functions of a single variable. Experimentation with one-dimensional problems was extensive during the development of the single-objective methods based on statistical models; see, e.g., $[139,208,216]$. These test functions have been used also to demonstrate the performance of the Lipschitz model based algorithms in Section 6.2.5: Rastr (6.46). Fo\&Fle (6.47), and Schaf (6.48). The feasible objective regions with the highlighted Pareto front of the considered test functions are shown in Figures $6.5,6.6$, and 6.7.

Two bi-objective test problems of two variables are chosen for the experimentation. The test problems of two variables are chosen similarly to the choice of one-dimensional problems: the first multi-objective test problem is composed using a typical test problem for a single-objective global optimization, and the second one is chosen from the set of functions frequently used for testing multi-objective algorithms. The first test function Shek (1.6) is composed of two Shekel functions which are frequently used for testing global optimization algorithms, see, e.g., [216]. A rather simple multimodal case is intended to be considered, so the number of minimizers of both objectives is selected equal to two. The objective functions are represented by contour lines in Figure 1.3. The second problem Fo\&Fle, (1.5), is especially difficult from the point of view of global minimization, since the functions $f_1(\mathbf{x})$ and $f_2(\mathbf{x})$ in (1.5) are similar to the most difficult objective function whose response surface is comprised of a flat plateau over a large part of the feasible decision region, and of the unknown number of sharp spikes. The estimates of parameters of the statistical model of (1.5) are biased towards the values that represent the “flat” part of response surface. The discrepancy between the statistical model and the modeled functions can negatively influence the efficiency of the statistical models based algorithms.

The selection of test problems can be summarized as follows: two problems, (6.46) and (1.6), are constructed generalizing typical test problems of global optimization, and two other problems, (6.48) and (1.5), are selected from a set of non-convex multi-objective test problems. The former problems are well represented by the considered above statistical model, and the latter ones are not. Both objective functions of problem (1.5) are especially difficult for global optimization, and their properties do not correspond to the properties predictable using the statistical model.

数学代写|凸优化作业代写Convex Optimization代考|Experiments with the P-Algorithm

The MATLAB implementations of the considered algorithms were used for experimentation. The results of minimization obtained by the uniform random search are presented for the comparison. Minimization was stopped after 100 computations of objective function values.

The sites for the first 50 computations of $\mathbf{f}(\mathbf{x})$ were chosen by the $\mathrm{P}$-algorithm randomly with a uniform distribution over the feasible region. That data were used to estimate the parameters of the statistical model as well as in planning of the next 50 observations according to (7.8). The maximization of the improvement probability was performed by a simple version of multistart. The values of the improvement probability (7.9) were computed at 1000 points, generated randomly with uniform distribution over the feasible region. A local descent was performed from the best point using the codes from the MATLAB Optimization Toolbox.

Let us start from the comments about the results of minimization of the functions of one variable. In the experiments with objective functions of a single variable, the algorithms can be assessed with respect to the solutions found in the true Pareto front, while in the case of several variables normally the approximate solutions are solely available for the assessment of the algorithms.

The feasible objective region of problem (6.46) is presented for the visual analysis in Figure 6.5. The (one hundred) points in the feasible objective region, generated by the method of random uniform search (RUS), are shown in Figure 7.1; the non-dominated solutions found (thicker points) do not represent the Pareto front well. The P-algorithm was applied to that problem with two values of the threshold vector. In Figure 7.2, the trial points in the feasible objective region are shown, and non-dominated points are denoted by thicker points. The left-side figure shows the results obtained with the threshold vector equal to $(-1,-1)^T(35$ non-dominated points found), and the right-hand side figure shows the results obtained with the threshold vector equal to $(-0.75,-0.75)^T$ (51 non-dominated points found). In the first case, the threshold is the ideal point; that case is similar to the case of a single-objective minimization, where the threshold is considerably below the current record. Presumably, for such a case the globality of the search strategy prevails and implies the uniformity (over the Pareto front) of the distribution of the non-dominated solutions found. For the threshold closer to the Pareto front, some localization of observations can be expected in the sense of increased density of the non-dominated points closer to the threshold vector. Figure $7.2$ illustrates the realization of the hypothesized properties.

数学代写|凸优化作业代写凸优化代考| P-Algorithm

.实验

$\mathbf{f}(\mathbf{x})$的前50次计算的站点由$\mathrm{P}$ -算法随机选择，并在可行区域内均匀分布。这些数据被用来估计统计模型的参数，以及根据(7.8)规划接下来的50个观察结果。改进概率的最大化是由一个简单版本的multistart执行的。改进概率(7.9)的值在1000点处计算，随机生成，在可行区域内分布均匀。使用MATLAB优化工具箱中的代码从最佳点执行局部下降

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