For multiple-input multiple-output (MIMO) precoded transmission that has individual constraints on the maximum power of each transmit antenna or a subset of transmit antennas, the transmit power optimization problem is a non-linear convex optimization problem with a high level of computational complexity. In this paper, assuming the use of the interior point method (IPM) to solve this problem, we propose two efficient techniques that reduce the computational complexity of the IPM by appropriately setting its parameters. Based on computer simulation, the achieved reductions in the level of the computational complexity are evaluated using the proposed techniques for both the fairness and the sum-rate maximization criteria assuming i.i.d Rayleigh fading MIMO channels and block diagonalization zero-forcing as a multi-user MIMO (MU-MIMO) precoder.

A method is presented for selecting items asked for new users to input their preference rates on those items in recommendation systems based on the collaborative filtering. Optimal item selection is formulated by an integer programming problem and we solve it by using a kind of the Hopfield-network-like scheme for interior point methods.

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.

It is shown by the derivation of solution methods for an elementary optimization problem that the stochastic relaxation in image analysis, the Potts neural networks for combinatorial optimization and interior point methods for nonlinear programming have common formulation of their dynamics. This unification of these algorithms leads us to possibility for real time solution of these problems with common analog electronic circuits.