Local Convergence Analysis of a Primal–Dual Method for Bound-Constrained Optimization Without SOSC

Paul Armand () and Ngoc Nguyen Tran ()
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Paul Armand: Université de Limoges - Laboratoire XLIM
Ngoc Nguyen Tran: Quy Nhon University

Journal of Optimization Theory and Applications, 2021, vol. 189, issue 1, No 5, 96-116

Abstract: Abstract 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.

Keywords: Primal–dual method; Interior point method; Regularization; Singular solutions; Local error bounds; Nonlinear optimization; 49M15; 65K05; 90C26; 90C30; 90C33; 90C51 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10957-021-01822-1

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