具有畸形约束极值点问题的优化

Optimization of the problem with constrained monstrosity extremum points

在畸形约束极值点附近,约束边界与目标函数等值线接近于相切,可行适用方向区非常狭小,难以寻得真正的约束极值点。为了使优化方法更好地解决各领域的复杂优化问题,研究具有畸形约束极值点问题的优化。针对该类问题的一个算例,分别采用随机方向方法、复合形法、内点惩罚函数法、外点惩罚函数法进行了优化,并对比了计算结果。随机方向法和复合形法在寻得边界点之后,难以找到可行适用方向,因此给出了伪最优点。而惩罚函数法由于其渐进优化的特点,可寻得最接近于约束极值点的最优点。计算结果验证了基于盲人探路优化思想的改进随机方向法,可减少随机方向的产生次数;验证了基于盲人探路思想的改进复合形法,可减少复合形的构造次数;也验证了加固围墙的内点惩罚函数法不要求初始点一定在可行域之内,也不会因寻优越界而给出伪最优点。对于存在多个约束极值点的优化问题算例,只要适当选取初始点,采用内点法就能寻得所有局部最优点。通过多种优化方法的对比研究,得出了对于畸形约束极值点优化问题,宜选用惩罚函数法求解的结论。

Near the constrained monstrosity extremum point, the objective function isoline nearly parallels with constrain border and the feasible and applicable region is very little. So the optimization point is difficult to approach the extremum point. In order to better solve complex optimization problems in each domain with the optimization method, the optimization problem with constrained monstrosity extremum points was studied. The random direction method, complex shape method, inner point and outer point penalty function method were used to calculate an example of this kind of problem. The comparison of results indicates that the former two methods give the false optimal point due to the difficulty in finding the feasible and applicable region after seeking out the border point. However, with the characteristic of gradual optimization, the penalty function method can approach the extremum point. The calculation result validates that the improved random direction method with blind-walking idea can reduce the number of random directions; the improved complex shape method with blind-walking idea canreduce the number of complex shapes; and the reinforced wall of inner penalty function method does not need the feasible region of the initial point. For the problem with multi-extremum points, all the local optimal points can be obtained by starting at the proper initial point. Thus the penalty function method is the best choice for optimization of the problem with constrained monstrosity extremum points.