We study a class of nonlinear dynamical systems to develop efficient algorithms. As an efficient algorithm, interior point method based on Newton's method is well-known for solving convex programming problems which include linear, quadratic, semidefinite and lp-programming problems. On the other hand, the geodesic of information geometry is represented by a continuous Newton's method for minimizing a convex function called divergence. Thus, we discuss a relation between information geometry and convex programming in a related family of continuous Newton's method. In particular, we consider the α-projection problem from a given data onto an information geometric submanifold spanned with power-functions. In general, an information geometric structure can be induced from a standard convex programming problem. In contrast, the correspondence from information geometry to convex programming is slightly complicated. We first present there exists a same structure between the α-projection and semidefinite programming problems. The structure is based on the linearities or autoparallelisms in the function space and the space of matrices, respectively. However, the α-projection problem is not a form of convex programming. Thus, we reformulate it to a lp-programming and the related ones. For the reformulated problems, we derive self-concordant barrier functions according to the values of α. The existence of a polynomial time algorithm is theoretically confirmed for the problem. Furthermore, we present the coincidence with the gradient vectors for the divergence and a modified barrier function. These results connect a part of nonlinear and algorithm theories by the discreteness of variables.

From an information geometric viewpoint, we investigate a characteristic of the submanifold of a mixture or exponential family in the manifold of finite discrete distributions. Using the characteristic, we derive a direct calculation method for an em-geodesic in the submanifold. In this method, the value of the primal parameter on the geodesic can be obtained without iterations for a gradient system which represents the geodesic. We also derive the similar algorithms for both problems of parameter estimation and functional extension of the submanifold for a data in the ambient manifold. These theoretical approaches from geometric analysis will contribute to the development of an efficient algorithm in computational complexity.