A global convergence theory for an active-trust-region algorithm for solving the general nonlinear programing problem. (English) Zbl 1110.65049
This paper describes a new trust-region algorithm for solving general nonlinear programming problems. Based on an active set strategy, it uses a projected Hassian to form the trial step. Global convergence of the algorithm is established.
Reviewer: Ping-Qi Pan (Nanjing)
MSC:
| 65K05 | Numerical mathematical programming methods |
| 90C30 | Nonlinear programming |
| 90C51 | Interior-point methods |
Keywords:
trust region algorithm; active set strategy; global convergence; constrained optimization; Fritz-John points; stationary pointsReferences:
| [1] | N. Alexandrov, J. Dennis, Multi-level algorithms for nonlinear optimization. Technical Report 94-24, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77251, June (1994); N. Alexandrov, J. Dennis, Multi-level algorithms for nonlinear optimization. Technical Report 94-24, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77251, June (1994) |
| [2] | R. Byrd, E. Omojokun, Robust trust-region methods for nonlinearly constrained optimization, A talk presented at the Second SIAM Conference on Optimization, Houston, TX (1987); R. Byrd, E. Omojokun, Robust trust-region methods for nonlinearly constrained optimization, A talk presented at the Second SIAM Conference on Optimization, Houston, TX (1987) |
| [3] | M. Celis, J. Dennis, R. Tapia, A trust-region strategy for nonlinear equality constrained optimization, In Numerical Optimization 1984. SIAM Philadelphia, Pennsylvania (1985); M. Celis, J. Dennis, R. Tapia, A trust-region strategy for nonlinear equality constrained optimization, In Numerical Optimization 1984. SIAM Philadelphia, Pennsylvania (1985) · Zbl 0566.65048 |
| [4] | Dennis, J.; El-Alem, M.; Maciel, M., A global convergence theory for general trust-region-based algorithms for equality constrained optimization, SIAM J. Optim., 7, 1, 177-207 (1997) · Zbl 0867.65031 |
| [5] | J. Dennis, M. El-Alem, K. Williamson, A trust-region approach to nonlinear systems of equalities and inequalities, Technical Report TR94-38, Dept. of Computational and Applied Mathematics, Rice University, Houston, TX 77251 (1994); J. Dennis, M. El-Alem, K. Williamson, A trust-region approach to nonlinear systems of equalities and inequalities, Technical Report TR94-38, Dept. of Computational and Applied Mathematics, Rice University, Houston, TX 77251 (1994) · Zbl 0957.65058 |
| [6] | M. El-Alem, A global convergence theory for a class of trust-region algorithms for constrained optimization. Ph.D. thesis, Department of Mathematical Sciences, Rice University, Houston, Texas (1988); M. El-Alem, A global convergence theory for a class of trust-region algorithms for constrained optimization. Ph.D. thesis, Department of Mathematical Sciences, Rice University, Houston, Texas (1988) |
| [7] | El-Alem, M., A global convergence theory for the Celis-Dennis-Tapia trust region algorithm for constrained optimization, SIAM J. Numer. Anal., 28, 266-290 (1991) · Zbl 0725.65061 |
| [8] | El-Alem, M., A robust trust-region algorithm with a non-monotonic penalty parameter scheme for constrained optimization, SIAM J. Optim., 5, 2, 348-378 (1995) · Zbl 0828.65063 |
| [9] | M. El-Alem, A Global convergence theory for Dennis, El-Alem, and Maciel’s class of trust-region algorithms for constrained optimization without assuming regularity (1999); M. El-Alem, A Global convergence theory for Dennis, El-Alem, and Maciel’s class of trust-region algorithms for constrained optimization without assuming regularity (1999) · Zbl 0957.65059 |
| [10] | Fiacco, A.; McCormick, G., Nonlinear Programming: Sequential Unconstrained Minimization Techniques (1968), John Wiley and Sons: John Wiley and Sons New York · Zbl 0193.18805 |
| [11] | Fletcher, R., Practical methods of optimization (1987), John Wiley and Sons: John Wiley and Sons New York · Zbl 0905.65002 |
| [12] | P. Gill, W. Murray, M. Saunders, M. Wright, Some theoretical properties of an augmented Lagrangian merit function. Report SOL 86-6, Stanford University (1986); P. Gill, W. Murray, M. Saunders, M. Wright, Some theoretical properties of an augmented Lagrangian merit function. Report SOL 86-6, Stanford University (1986) · Zbl 0814.90094 |
| [13] | Han, S., A globally convergent method for nonlinear programming, JOTA, 22, 297-309 (1977) · Zbl 0336.90046 |
| [14] | Levenberg, K., A method for the solution, of certain problems in least-squares, Quart. J. Appl. Math., 2, 164-168 (1944) · Zbl 0063.03501 |
| [15] | Marquardt, D., An algorithm for least-squares estimation of nonlinear parameters, Siam J. Appl. Math., 11, 431-441 (1963) · Zbl 0112.10505 |
| [16] | Mangasarian, O., Nonlinear Programming (1969), McGraw-Hill Book Company: McGraw-Hill Book Company New York · Zbl 0127.36803 |
| [17] | E. Omojokun, Trust-region strategies for optimization with nonlinear equality and inequality constraints. Ph.D. thesis, Department of Computer Science, University of Colorado, Boulder, Colorado (1989); E. Omojokun, Trust-region strategies for optimization with nonlinear equality and inequality constraints. Ph.D. thesis, Department of Computer Science, University of Colorado, Boulder, Colorado (1989) |
| [18] | T. Plantenga, Large-Scale nonlinear constrained optimization using trust region. Ph.D. thesis, Department of Electrical and Comp. Eng., Northwestern University, Evanston, IL (1995); T. Plantenga, Large-Scale nonlinear constrained optimization using trust region. Ph.D. thesis, Department of Electrical and Comp. Eng., Northwestern University, Evanston, IL (1995) |
| [19] | Powell, M., A method for nonlinear constraints in minimization problems, (Fletcher, R., Optimization (1970), Academic Press: Academic Press New York), 283-298 · Zbl 0194.47701 |
| [20] | M. Powell, Algorithms for nonlinear constraints that used Lagrangian functions, Mathematical Programming, North-Holland Publishing Company, 14 (1978) 224-248; M. Powell, Algorithms for nonlinear constraints that used Lagrangian functions, Mathematical Programming, North-Holland Publishing Company, 14 (1978) 224-248 · Zbl 0383.90092 |
| [21] | Schittkowski, K., On the convergence of a sequential quadratic programming method with an augmented Lagrangian line search function, Math. Operations Forchung U. Statistic Ser., Optim., 14, 197-216 (1983) · Zbl 0523.90075 |
| [22] | L. Vicente, Trust-region interior-point algorithms for a class of nonlinear programming problems. Ph.D. thesis, Department of Computational and Applied Mathematics. Rice University, Houston, Texas (1996); L. Vicente, Trust-region interior-point algorithms for a class of nonlinear programming problems. Ph.D. thesis, Department of Computational and Applied Mathematics. Rice University, Houston, Texas (1996) |
| [23] | Yuan, Y., On the convergence of a new trust region algorithm, Numer. Math., 70, 515-539 (1995) · Zbl 0828.65062 |
| [24] | Zhang, J.; Zhu, D., Projected quasi-Newton algorithm with trust-region for constrained optimization, J. Optim. Theory App., 67, 369-393 (1990) · Zbl 0696.90050 |
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