Search for Common Minima in Joint Optimization of Multiple Cost Functions

Research and Services Division of Materials Data and Integrated System, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan We present a novel optimization method, named the Combined Optimization Method (COM), for the joint optimization of two or more cost functions. Unlike the conventional joint optimization schemes, which try to find minima in a weighted sum of cost functions, the COM explores search space for common minima shared by all the cost functions. Given a set of multiple cost functions that have qualitatively different distributions of local minima with each other, the proposed method finds the common minima with a high success rate without the help of any metaheuristics. As a demonstration, we apply the COM to the crystal structure prediction in materials science. By introducing the concept of data assimilation, i.e., adopting the theoretical potential energy of the crystal and the crystallinity, which characterizes the agreement with the theoretical and experimental X-ray diffraction patterns, as cost functions, we show that the correct crystal structures of Si diamond, low quartz, and low cristobalite can be predicted with significantly higher success rates than the previous methods. Continuous optimization, i.e., finding a global minimum of a continuous nonlinear cost function, is one of the most fundamental and important problems encountered in almost all the fields of science and engineering. For solving the optimization problem, a variety of classical optimization algorithms, such as the gradient descent, conjugate gradient, Newton and quasi-Newton methods, downhill simplex method, etc. have been developed so far and long been used widely [1]. If the cost function has a rugged landscape in a high-dimensional search space, however, such classical algorithms easily fail to find the global optimal point, and get trapped by local minima, or metastable states.