# zbMATH — the first resource for mathematics

An $$RQP$$ algorithm using a differentiable exact penalty function for inequality constrained problems. (English) Zbl 0767.90060
The authors propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function. This algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an Armijo-type line search procedure that avoids the need to evaluate second order derivatives of the problem functions. One proves that the algorithm possesses global and superlinear convergence properties. Numerical results are presented.

##### MSC:
 90C20 Quadratic programming 90-08 Computational methods for problems pertaining to operations research and mathematical programming 49M30 Other numerical methods in calculus of variations (MSC2010) 49M37 Numerical methods based on nonlinear programming
Full Text:
##### References:
 [1] B. Bank, J. Guddatt, D. Klatte, B. Kummer and K. Tammer,Non-linear Parametric Optimization (Birkhäuser, Basel, 1983). · Zbl 0502.49002 [2] M.C. Bartholomew-Biggs, ”ALRQP for inequality constraints,” Technical Report No. 150, Numerical Optimisation Centre, Hatfield Polytechnic (Hatfield, 1984). [3] M.C. Biggs, ”On the convergence of some constrained minimization algorithms based on recursive quadratic programming,”Journal Institute of Mathematics and its Applications 21 (1978) 67–82. · Zbl 0373.90056 · doi:10.1093/imamat/21.1.67 [4] D.P. Bertsekas,Constrained Optimization and Lagrange Multiplier Methods (Academic Press, New York, 1982). · Zbl 0572.90067 [5] P.T. Boggs, J.W. Tolle and P. Wang, ”On the local convergence of quasi-Newton methods for constrained optimization,”SIAM Journal on Control and Optimization 20 (1982) 161–171. · Zbl 0494.65036 · doi:10.1137/0320014 [6] R.M. Chamberlain, ”Some examples of cycling in variable metric methods for constrained minimization,”Mathematical Programming 16 (1979) 378–383. · Zbl 0402.90088 · doi:10.1007/BF01582123 [7] R.M. Chamberlain, C. Lemarechal, H.C. Pedersen, and M.J.D. Powell, ”The watch-dog technique for forcing convergence in algorithms for constrained optimization,”Mathematical Programming Study 16 (1982) 1–17. · Zbl 0477.90072 · doi:10.1007/BFb0120945 [8] T.F. Coleman and A.R. Conn, ”Nonlinear programming via an exact penalty function method: asymptotic analysis,”Mathematical Programming 24 (1982) 123–136. · Zbl 0501.90078 · doi:10.1007/BF01585100 [9] T.F. Coleman and A.R. Conn, ”Nonlinear programming via an exact penalty function method: global analysis,”Mathematical Programming 24 (1982) 137–161. · Zbl 0501.90077 · doi:10.1007/BF01585101 [10] J.W. Daniel, ”Stability of the solution of definite quadratic programs,”Mathematical Programming 5 (1973) 41–53. · Zbl 0269.90037 · doi:10.1007/BF01580110 [11] G. Di Pillo and L. Grippo, ”A continuously differentiable exact penalty function for nonlinear programming problems with inequality constraints,”SIAM Journal on Control and Optimization 23 (1985) 72–84. · Zbl 0569.90072 · doi:10.1137/0323007 [12] G. Di Pillo and L. Grippo, ”An exact penalty method with global convergence properties for nonlinear programming problems,”Mathematical Programming 36 (1986) 1–18. · Zbl 0631.90061 · doi:10.1007/BF02591986 [13] L.C.W. Dixon, ”Exact penalty function methods in nonlinear programming,” Technical Report No. 103, Numerical Optimisation Centre, Hatfield Polytechnic (Hatfield, 1979). [14] L.C.W. Dixon, ”On the convergence properties of variable metric recursive quadratic programming methods,” Technical Report No. 110, Numerical Optimisation Centre, Hatfield Polytechnic (Hatfield, 1980). · Zbl 0491.49025 [15] R.E. Fletcher, ”Second order corrections for nondifferentiable optimization,” in: G.A. Watson, ed.,Numerical Analysis Dundee 1981 (Springer, Berlin, 1982) pp. 144–157. [16] R. Fontecilla, T. Steihaug and R.A. Tapia, ”A convergence theory for a class of quasi-Newton methods for constrained optimization,”SIAM Journal on Numerical Analysis 24 (1987) 1133–1151. · Zbl 0634.65045 · doi:10.1137/0724075 [17] M. Fukushima, ”A successive quadratic programming algorithm with global and superlinear convergence properties,”Mathematical Programming 35 (1986) 253–264. · Zbl 0597.90077 · doi:10.1007/BF01580879 [18] D. Gabay, ”Reduced quasi-Newton methods with feasibility improvement for nonlinearly constrained optimization,”Mathematical Programming Study 16 (1982) 18–44. · Zbl 0477.90065 · doi:10.1007/BFb0120946 [19] P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, ”Sequential quadratic programming methods for nonlinear programming,” in: E.J. Haug, ed.,Computer Aided Analysis and Optimization of Mechanical System Dynamics, NATO ASI Series Vol. F9 (Springer, Berlin, 1984). · Zbl 0551.90079 [20] T. Glad and E. Polak, ”A multiplier method with automatic limitation of penalty growth,”Mathematical Programming 17 (1979) 140–155. · Zbl 0414.90078 · doi:10.1007/BF01588240 [21] S.P. Han, ”Superlinearly convergent variable metric algorithm for general nonlinear programming problems,”Mathematical Programming 11 (1976) 263–282. · Zbl 0364.90097 · doi:10.1007/BF01580395 [22] S.P. Han, ”A globally convergent method for nonlinear programming,”Journal of Optimization Theory and Applications 22 (1977) 297–309. · Zbl 0336.90046 · doi:10.1007/BF00932858 [23] W. Hock and K. Schittkowski,Test Examples for Nonlinear Programming Codes (Springer, Berlin, 1981). · Zbl 0452.90038 [24] N. Maratos, ”Exact penalty function algorithms for finite dimensional and control optimization problems,” Ph.D. Thesis, Imperial College of Science and Technology (London, 1978). [25] D.Q. Mayne and E. Polak, ”A superlinearly convergent algorithm for constrained optimization problems,”Mathematical Programming Study 16 (1982) 45–61. · Zbl 0477.90071 · doi:10.1007/BFb0120947 [26] M.J.D. Powell, ”A fast algorithm for nonlinearly constrained optimization calculations,” in: G.A. Watson, ed.,Numerical Analysis Dundee 1977 (Springer, Berlin, 1978) pp. 144–157. [27] M.J.D. Powell, ”The convergence of variable metric methods for nonlinear constrained optimization calculations,” in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming 3 (Academic Press, New York, 1978) pp. 27–63. · Zbl 0464.65042 [28] M.J.D. Powell, ”Variable metric methods for constrained optimization,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming, The State of the Art (Springer, Berlin, 1983) pp. 288–311. [29] M.J.D. Powell, ”Methods for nonlinear constraints in optimization calculations,” Report DAMTP 1986/NA5, University of Cambridge (Cambridge, 1986). · Zbl 0624.90091 [30] M.J.D. Powell and Y. Yuan, ”A recursive quadratic programming algorithm that uses differentiable exact penalty functions,”Mathematical Programming 35 (1986) 265–278. · Zbl 0598.90079 · doi:10.1007/BF01580880 [31] M. Sahba, ”Globally convergent algorithm for nonlinearly constrained optimization problems,”Journal of Optimization Theory and Applications 52 (1987) 291–309. · Zbl 0585.90076 · doi:10.1007/BF00941288 [32] K. Schittkowski, ”The nonlinear programming method of Wilson, Han and Powell with augmented Lagrangian type line search function,”Numerische Mathematik 38 (1981) 83–114. · Zbl 0534.65030 · doi:10.1007/BF01395810 [33] J. Stoer, ”Principles of sequential quadratic programming methods for solving nonlinear programs,” in: K. Schittkowski, ed.,Computational Mathematical Programming, NATO ASI Series Vol. F15 (Springer, Berlin, 1985) pp. 165–207. · Zbl 0581.90067 [34] K. Tone, ”Revisions of constraint approximations in the successive QP method for nonlinear programming problems,”Mathematical Programming 26 (1983) 144–152. · Zbl 0516.90065 · doi:10.1007/BF02592051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.