A weighted logarithmic barrier interior-point method for linearly constrained optimization

In this paper, a weighted logarithmic barrier interior-point method for solving the linearly convex constrained optimization problems is presented. Unlike the classical central-path, the barrier parameter associated with the per- turbed barrier problems, is not a scalar but is a weighted positive vector. This modi cation gives a theoretical exibility on its convergence and its numerical performance. In addition, this method is of a Newton descent direction and the computation of the step-size along this direction is based on a new e cient tech- nique called the tangent method. The practical e ciency of our approach is shown by giving some numerical results.

A Full-Newton Step Interior Point Method for Fractional Programming Problem Involving Second Order Cone Constraint

Some efficient interior-point methods (IPMs) are based on using a self-concordant barrier function related to the feasibility set of the underlying problem.Here, we use IPMs for solving fractional programming problems involving second order cone constraints. We propose a logarithmic barrier function to show the self concordant property and present an algorithm to compute $\varepsilon-$solution of a fractional programming problem. Finally, we provide a numerical example to illustrate the approach.

Motion Planning for Autonomous Driving With Extended Constrained Iterative LQR

Autonomous driving planning is a challenging problem when the environment is complicated. It is difficult for the planner to find a good trajectory that navigates autonomous cars safely with crowded surrounding vehicles. To solve this complicated problem, a fast algorithm that generates a high-quality, safe trajectory is necessary. Constrained Iterative Linear Quadratic Regulator (CILQR) is appropriate for this problem, and it successfully generates the required trajectory in realtime. However, CILQR has some deficiencies. Firstly, CILQR uses logarithmic barrier functions for hard constraints, which will cause numerical problems when the initial trajectory is infeasible. Secondly, the convergence speed is slowed with a bad initial trajectory, which might violate the real-time requirements. To address these problems, we propose the extended CILQR by adding two new features. The first one is using relaxed logarithmic barrier functions instead of the standard logarithmic barrier function to prevent numerical issues. The other one is adding an efficient initial trajectory creator to generate a good initial trajectory. Moreover, this initial trajectory helps CILQR to converge to a desired local optimum. These new features extend CILQR’s usage to more practical autonomous driving applications. Simulation results show that our algorithm is effective in challenging driving environments.

A log-barrier Newton-CG method for bound constrained optimization with complexity guarantees

We describe an algorithm based on a logarithmic barrier function, Newton’s method and linear conjugate gradients that seeks an approximate minimizer of a smooth function over the non-negative orthant. We develop a bound on the complexity of the approach, stated in terms of the required accuracy and the cost of a single gradient evaluation of the objective function and/or a matrix-vector multiplication involving the Hessian of the objective. The approach can be implemented without explicit calculation or storage of the Hessian.