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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Convex Optimization代考|MATH620

数学代写|凸优化作业代写凸优化代考|测试函数


选择一个和两个变量的双目标问题进行实验,以便对结果进行可视化分析


有些实验使用单一变量的目标函数进行。在发展基于统计模型的单目标方法期间,对一维问题进行了广泛的实验;例如,$[139,208,216]$。这些测试函数还被用于演示第6.2.5节:Rastr(6.46)中基于Lipschitz模型的算法的性能。Fo&Fle(6.47)和Schaf (6.48)考虑的测试函数的帕累托前突出显示的可行目标区域如图$6.5,6.6$和6.7所示


选择两个变量的双目标检验问题进行实验。两个变量的测试问题的选择与一维问题的选择类似:第一个多目标测试问题是由单目标全局优化的典型测试问题组成的,第二个多目标测试问题是从测试多目标算法的常用函数集中选择的。第一个测试函数Shek(1.6)由两个经常用于测试全局优化算法的Shekel函数组成,如[216]。我们打算考虑一个相当简单的多模态情况,因此选择两个目标的最小化器的数量为2。目标函数用图1.3中的等高线表示。第二个问题Fo&Fle(1.5),从全局最小化的角度来看尤其困难,因为(1.5)中的函数$f_1(\mathbf{x})$和$f_2(\mathbf{x})$类似于最困难的目标函数,其响应面由可行决策区域的大部分上的平坦平台和数量未知的尖峰组成。统计模型(1.5)的参数估计倾向于表示响应面“平坦”部分的值。统计模型与被建模函数之间的差异会对基于统计模型的算法的效率产生负面影响


测试问题的选择可以总结为:两个问题(6.46)和(1.6)是由全局优化的典型测试问题泛化构造的,另外两个问题(6.48)和(1.5)是从一组非凸多目标测试问题中选取的。上述统计模型很好地反映了前者的问题,而后者则不然。问题(1.5)的两个目标函数对全局优化特别困难,其性质与统计模型预测的性质不一致

数学代写|凸优化作业代写凸优化代考| P-Algorithm

.实验


所考虑的算法的MATLAB实现被用于实验。用均匀随机搜索得到的最小值结果进行了比较。在目标函数值计算100次后停止最小化


$\mathbf{f}(\mathbf{x})$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


让我们从单变量函数最小化结果的注释开始。在单变量目标函数的实验中,算法可以根据在真帕累托前沿找到的解来进行评估,而在有多个变量的情况下,通常近似解只能用于评估算法


图6.5给出了问题(6.46)的可行目标区域,用于可视化分析。通过随机均匀搜索(RUS)方法生成的可行目标区域内(100)个点如图7.1所示;发现的非支配解(较厚的点)不能很好地代表帕累托前沿。对该问题采用了p算法,并给出了两个阈值向量。图7.2显示了可行目标区域内的试验点,非支配点用较粗的点表示。左边的图显示了阈值向量等于发现的非主导点$(-1,-1)^T(35$时获得的结果),右边的图显示了阈值向量等于$(-0.75,-0.75)^T$时获得的结果(发现的51个非主导点)。在第一种情况下,阈值是理想点;这种情况类似于单一目标最小化的情况,其中阈值大大低于当前记录。假设,在这种情况下,搜索策略的全局性占主导地位,并意味着所找到的非支配解的分布的一致性(在帕累托前沿)。对于更接近帕累托前沿的阈值,在更接近阈值向量的非支配点的密度增加的意义上,可以预期一些观测的局部化。图$7.2$说明了假设属性的实现

数学代写|凸优化作业代写Convex Optimization代考

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