Abstract
A new, infeasible QP-free algorithm for nonlinear constrained optimization problems is proposed. The algorithm is based on a continuously differentiable exact penalty function and on active-set strategy. After a finite number of iterations, the algorithm requires only the solution of two linear systems at each iteration. We prove that the algorithm is globally convergent toward the KKT points and that, if the second-order sufficiency condition and the strict complementarity condition hold, then the rate of convergence is superlinear or even quadratic. Moreover, we incorporate two automatic adjustment rules for the choice of the penalty parameter and make use of an approximated direction as derivative of the merit function so that only first-order derivatives of the objective and constraint functions are used.
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BOGGS, P. T., and TOLLE, J. W., Sequential Quadratic Programming, Acta Numerica, Vol. 45, pp. 1-51, 1995.
BURKE, J. V., and HAN, S. P., A Robust Sequential Quadratic Programming Method, Mathematical Programming, Vol. 43, pp. 277-303, 1989.
FACCHINEI, F., Robust Recursive Quadratic Programming Algorithm Model with Global and Superlinear Convergence Properties, Journal of Optimization Theory and Applications, Vol. 92, pp. 543-579, 1997.
FACCHINEI, F., and LUCIDI, S., Quadratically and Superlinearly Convergent Algorithms for the Solution of Inequality Constrained Minimization Problems, Journal of Optimization Theory and Applications, Vol. 85, pp. 265-289, 1995.
PANIER, E. R., and TITS, A. L., A Superlinearly Convergent Feasible Method for the Solution of Inequality Constrained Optimization Problems, SIAM Journal on Control and Optimization, Vol. 25, pp. 934-950, 1987.
BIGGS, M. C., On the Convergence of Some Constrained Minimization Algorithms Based on Recursive Quadratic Programming, Journal of the Institute of Mathematics and Its Applications, Vol. 21, pp. 67-81, 1978.
BIGGS, M. C., Recursive Quadratic Programming Methods Based on the Augmented Lagrangian, Mathematical Programming Studies, Vol. 31, pp. 21-41, 1987.
BERTSEKAS, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, NY, 1982.
PANIER, E. R., TITS, A. L., and HERSKOVITS, J. N., A QP-Free Globally Convergent, Locally Superlinearly Convergent Algorithm for Inequality Constrained Optimization, SIAM Journal on Control and Optimization, Vol. 26, pp. 788-811, 1988.
QI, H. D., and QI, L., A New QP-Free Globally Convergent, Locally Superlinearly Convergent Algorithm for Inequality Constrained Optimization, SIAM Journal on Optimization, Vol. 11, pp. 113-132, 2000.
LUCIDI, S., New Results on a Continuously Differentiable Penalty Function, SIAM Journal on Optimization, Vol. 2, pp. 558-574, 1992.
GLAD, T., and POLAK, E., A Multiplier Method with Automatic Limitation of Penalty Growth, Mathematical Programming, Vol. 17, pp. 140-155, 1979.
SPELLUCCI, P., An SQP Method for General Nonlinear Programs Using Only Equality Constrained Subproblems, Mathematical Programming, Vol. 82, pp. 413-448, 1998.
QI, L., and SUN, J., A Nonsmooth Version of Newton's Method, Mathematical Programming, Vol. 58, pp. 353-367, 1993.
ROBINSON, S. M., Strongly Regular Generalized Equations, Mathematics of Operations Research, Vol. 5, pp. 43-62, 1980.
MORé, J. J., and SORENSEN, D. C., Computing a Trust-Region Step, SIAM Journal on Scientific and Statistical Computing, Vol. 4, pp. 553-572, 1983.
KANZOW, C., and QI, H. D., A QP-Free Constrained Newton-Type Method for Variational Inequality Problems, Mathematical Programming, Vol. 85, pp. 81-106, 1999.
DENNIS, J. E., and MORé, J. J., A Characterization of Superlinear Convergence and Its Application to Quasi-Newton Methods, Mathematics of Computation, Vol. 28, pp. 549-560, 1974.
BOGGS, P. T., TOLLE, J. W., and WANG, P., On the Local Convergence of Quasi-Newton Methods for Constrained Optimization, SIAM Journal on Control and Optimization, Vol. 20, pp. 161-171, 1982.
STOER, J., and TAPIA, A., On the Characterization of Q-Superlinear Convergence of Quasi-Newton Methods for Constrained Optimization, Mathematics of Computation, Vol. 49, pp. 581-584, 1987.
FACCHINEI, F., Minimization of SC 1-Functions and the Maratos Effect, Operations Research Letters, Vol. 17, pp. 131-137, 1995.
FACCHINEI, F., LUCIDI, S., and PALAGI, L., A Truncated Newton Algorithm for Large-Scale Box-Constrained Optimization, Technical Report 15-99, Dipartimento di Informatica e Sistemistica, Università di Roma ''La Sapienza'', Roma, Italy, 1999.
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Qi, L., Yang, Y. Globally and Superlinearly Convergent QP-Free Algorithm for Nonlinear Constrained Optimization. Journal of Optimization Theory and Applications 113, 297–323 (2002). https://doi.org/10.1023/A:1014882909302
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DOI: https://doi.org/10.1023/A:1014882909302