Abstract
This short paper describes a boundedness property of the inverse of a regularized Jacobian matrix that arises in optimization algorithms for solving constrained problems. The regularization considered here is to overcome a rank deficiency of the Jacobian matrix of constraints. We show that the norm of the inverse of the regularized matrix is locally bounded by a coefficient inversely proportional to the regularization parameter. We also show how this result can be used for the local convergence analysis of algorithms without a constraint qualification hypothesis.
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Notes
Note that assumption (iii) in Lemma 1 is slightly more general from its counterpart in [1] which is: for all \(k\in \mathbb {N}\) and \(i\in \{1,\ldots ,n\}\), \([x_k]_i [z_k]_i \ge \mu \). But the proof is exactly the same, except that one inequality must be changed. More precisely, it suffices to replace the term \(\frac{1}{\mu }x_j^k\) by \(\frac{1}{\mu }\) in [1, page 590, line 6].
References
Armand, P., Benoist, J.: Uniform boundedness of the inverse of a Jacobian matrix arising in regularized interior-point methods. Math. Program. 137(1), 587–592 (2013)
Armand, P., Omheni, R.: A globally and quadratically convergent primal-dual augmented Lagrangian algorithm for equality constrained optimization. Optim. Methods Softw. 32(1), 1–21 (2017)
Armand, P., Omheni, R.: A mixed logarithmic barrier-augmented Lagrangian method for nonlinear optimization. J. Optim. Theory Appl. 173(2), 523–547 (2017)
Arreckx, S., Orban, D.: A regularized factorization-free method for equality-constrained optimization. SIAM J. Optim. 28(2), 1613–1639 (2018)
Debreu, G.: Definite and semidefinite quadratic forms. Econometrica 20, 295–300 (1952)
Dennis, J.E., Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Classics in Applied Mathematics, vol. 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1996)
Friedlander, M.P., Orban, D.: A primal-dual regularized interior-point method for convex quadratic programs. Math. Program. Comput. 4(1), 71–107 (2012)
Gill, P.E., Kungurtsev, V., Robinson, D.P.: A stabilized SQP method: superlinear convergence. Math. Program. 163(1–2), 369–410 (2017)
Greif, C., Moulding, E., Orban, D.: Bounds on eigenvalues of matrices arising from interior-point methods. SIAM J. Optim. 24(1), 49–83 (2014)
Izmailov, A.F., Solodov, M.V.: Stabilized SQP revisited. Math. Program 133(1–2), 93–120 (2012)
Vanderbei, R.J.: Symmetric quasi-definite matrices. SIAM J. Optim. 5(1), 100–113 (1995)
Wright, S.J.: An algorithm for degenerate nonlinear programming with rapid local convergence. SIAM J. Optim. 15(3), 673–696 (2005)
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Armand, P., Tran, N.N. Boundedness of the inverse of a regularized Jacobian matrix in constrained optimization and applications. Optim Lett 16, 2359–2371 (2022). https://doi.org/10.1007/s11590-021-01829-7
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DOI: https://doi.org/10.1007/s11590-021-01829-7