Abstract
This paper contributes to the development of the field of augmented Lagrangian multiplier methods for general nonlinear programming by introducing a new update for the multipliers corresponding to inequality constraints. The update maintains naturally the nonnegativity of the multipliers without the need for a positive-orthant projection, as a result of the verification of the first-order necessary conditions for the minimization of a modified augmented Lagrangian penalty function.
In the new multiplier method, the roles of the multipliers are interchanged: the multipliers corresponding to the inequality constraints are updated explicitly, whereas the multipliers corresponding to the equality constraints are approximated implicitly. It is shown that the basic properties of local convergence of the traditional multiplier method are valid also for the proposed method.
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References
Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics, Academic Press, New York, NY, 1982.
Conn, A.R., Gould, N.I.M., and Toint, P.L., A Globally Convergent Augmented Lagrangian Algorithm for Optimization with General Constraints and Simple Bounds, SIAM Journal on Numerical Analysis, Vol. 28, pp. 545-572, 1991.
Vicente, L.N., Local Analysis of a New Multipliers Method, European Journal of Operational Research, Vol. 143, pp. 432-451, 2002.
Dennis, J. E., Heinkenschloss, M., and Vicente, L.N., Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems, SIAM Journal on Control and Optimization, Vol. 36, pp. 1750-1794, 1998.
Conn, A. R., Gould, N. I. M., and Toint, P. L., LANCELOT: A Fortran Package for Large-Scale Nonlinear Optimization (Release A), Springer Verlag, New York, NY, 1992.
Goodman, J., Newton's Method for Constrained Optimization, Mathematical Programming, Vol. 33, pp. 162-171, 1985.
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Avelino, C.P., Vicente, L.N. Updating the Multipliers Associated with Inequality Constraints in an Augmented Lagrangian Multiplier Method. Journal of Optimization Theory and Applications 119, 215–233 (2003). https://doi.org/10.1023/B:JOTA.0000005444.50285.4d
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DOI: https://doi.org/10.1023/B:JOTA.0000005444.50285.4d