An Adaptive Vector Barrier Interior-Point Method Using Majorant Functions

This paper proposes an adaptive interior-point method for solving convex optimization problems subject to inequality constraints. The approach is based on a logarithmic barrier formulation in which the barrier parameter is taken as a vector rather than a single scalar. In contrast to classical interior-point methods that rely on uniform updates of the barrier parameter, the proposed algorithm introduces a componentwise adaptive strategy, allowing each constraint to be treated with an individual level of penalization. This flexibility improves the algorithm’s ability to handle heterogeneous constraints and enhances numerical performance, particularly for large-scale problems.

A key feature of the method is the computation of the step size using a carefully constructed majorant function. This strategy eliminates the need for traditional line search procedures, thereby reducing computational overhead while maintaining robustness. The algorithm is designed to ensure that all iterates remain strictly within the feasible region, guaranteeing feasibility preservation throughout the optimization process. Furthermore, it is shown that the objective function value decreases monotonically along the iterations.

Rigorous theoretical analysis is provided to establish the descent property, feasibility preservation, and global convergence of the proposed method under standard assumptions for convex optimization. These results demonstrate that the algorithm converges to an optimal solution of the original constrained problem.

To evaluate the practical performance of the method, numerical experiments are conducted on a set of large-scale convex optimization problems. The results indicate that the proposed adaptive interior-point method outperforms classical interior-point approaches in terms of efficiency and robustness, highlighting its potential for solving high-dimensional constrained optimization problems.