标题： 利用保角映射的测地线和轮廓优化 显示英文标题

标题： Geodesic and Contour Optimization Using Conformal Mapping

摘要： 我们提出了一种用于连续函数的新型优化算法，该算法利用保角映射下的测地线和等高线。该算法可以通过首先沿着测地线曲线找到局部最优解，然后沿着等高线曲线移动到下一个搜索区域来找到多个最优解。为了提高效率，牛顿-拉夫森算法也在局部搜索步骤中被采用。基于实现的测地线提出的跳跃机制使算法能够跳转到附近区域，从而避免陷入局部最优解。保角映射用于解决求解经典测地线方程相关的数值不稳定性。通过使用局部二次逼近，构造了保角映射下的测地线流。算法中的参数自适应选择以反映目标函数的局部几何特征。与许多常用的优化算法（包括梯度法、信任区域法、遗传算法和全局搜索方法）进行比较表明，所提出的算法在几乎所有测试用例中都优于大多数常用方法，仅有少数例外。 显示英文摘要

摘要： We propose a novel optimization algorithm for continuous functions using geodesics and contours under conformal mapping.The algorithm can find multiple optima by first following a geodesic curve to a local optimum then traveling to the next search area by following a contour curve. To improve the efficiency, Newton-Raphson algorithm is also employed in local search steps. A proposed jumping mechanism based on realized geodesics enables the algorithm to jump to a nearby region and consequently avoid trapping in local optima. Conformal mapping is used to resolve numerical instability associated with solving the classical geodesic equations. Geodesic flows under conformal mapping are constructed numerically by using local quadratic approximation. The parameters in the algorithm are adaptively chosen to reflect local geometric features of the objective function. Comparisons with many commonly used optimization algorithms including gradient, trust region, genetic algorithm and global search methods have shown that the proposed algorithm outperforms most widely used methods in almost all test cases with only a couple of exceptions.