  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.
