We propose a local convergence analysis of a primal–dual interior point algorithm for the solution of a bound-constrained optimization problem. The algorithm includes a regularization technique to prevent singularity of the matrix of the linear system at each iteration, when the second-order sufficient conditions do not hold at the solution. These conditions are replaced by a milder assumption related to a local error-bound condition. This new condition is a generalization of the one used in unconstrained optimization. We show that by an appropriate updating strategy of the barrier parameter and of the regularization parameter, the proposed algorithm owns a superlinear rate of convergence. The analysis is made thanks to a boundedness property of the inverse of the Jacobian matrix arising in interior point algorithms. An illustrative example is given to show the behavior and the gain obtained by this regularization strategy.

我们提出了一种原始对偶内点算法的局部收敛性分析，以解决约束受限的优化问题。该算法包括一种正则化技术，可以在每次迭代中不满足二阶足够条件时，防止线性系统矩阵在每次迭代时发生奇异性。这些条件被与局部错误约束条件有关的更温和的假设所代替。这一新条件是对无约束优化中使用的条件的概括。我们表明，通过对障碍参数和正则化参数的适当更新策略，所提出的算法拥有超线性收敛速率。由于内点算法中产生的雅可比矩阵逆的有界性而进行了分析。