Abstract
In this paper, we propose conjugate gradient path method for solving derivative-free unconstrained optimization. The iterative direction is obtained by constructing and solving quadratic interpolation model of the objective function with conjugate gradient methods. The global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, the numerical results are reported to show the effectiveness of the proposed algorithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrei, N.: An unconstrained optimization test functions collection. Adv. Model. Optim. 10(1), 147–161 (2008)
Bao, J., Zhu, D.: An affine scaling conjugate gradient path mothed for nonlinear optimization subject to bounds. Math. Numer. Sin. 31(1), 37–50 (2009)
Conn, A.R., Scheinberg, K., Toint, Ph.L.: On the convergence of derivative-free methods for unconstrained optimization, in Approximation Theory and Optimization, Tributes to M. J. D. Powell, M. D. Buhmann and A. Iserles, eds., pp 83–108. Cambridge University Press, Cambridge (1997)
Conn, A.R., Scheinberg, K., Toint, Ph.L.: Recent progress in unconstrained nonlinear optimization without derivatives. Math. Program. 79, 397–414 (1997)
Conn, A.R., Scheinberg, K., Vicente, L.N.: Geometry of interpolation sets in derivative free optimization. Math. Program. 111, 141–172 (2008)
Conn, A.R., Scheinberg, K., Vicente, L.N.: Global convergence of general derivative-free trust-region algorithms to first- and second-order critical points. SIAM J. Optim. 20, 387–415 (2009)
Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to Derivative-free Optimization, MPS-SIAM Series on Optimization, SIAM Philadelphia (2009)
Conn, A.R., Toint, Ph.L.: An Algorithm Using Quadratic Interpolation for Unconstrained Derivative Free Optimization. In: Di Pillo, G., Giancssi, E (eds.) Nonlinear Optimization Aml Applications, pp 27–47. Plenum Publishing, New York (1996)
Du, D., Pardalos, P.M., Wu, W.: Mathematical theory of optimization. kluwer academic publishers (2001)
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)
Fasano, G., Morales, J.L., Nocedal, J.: On the geometry phase in model-based algorithms for derivative-free optimization. Optim. Methods Softw. 24(1), 145–154 (2009)
Floudas, C.A., Pardalos, P.M.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers, Dordredit (1999)
Floudas, C.A., Pardalos, P.M.: Encyclopedia of Optimization. Kluwer Academic Publishers, Dordredit (1999)
Gratton, S., Toint, Ph.L., Tröltzsch, A.: An active-set trust-region method for derivative-free nonlinear bound-constrained optimization. Optim. Methods Softw. 26(4–5), 873–894 (2011)
Moré, J.J., Wild, S.M.: Benchmarking derivative-free optimization algorithms. SIAM J. Optim. 20(1), 172–191 (2009)
La Cruz, W., Raydan, M.: Nonmonotone spectral methods for large-scale nonlinear systems. Optim. Methods Softw. 18, 583–599 (2003)
La Cruz, W., Martínez, J.M., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems: Theory and experiments, Technical Report RT-04-08, Dpto.de Computation UCV (2004)
Nocedal, J., Wright, S.: Numerical Optimization. Springer, Berlin (1999)
Powell, M.J.D.: UOBYQA: Unconstrained optimization by quadratic approximation, Technical Report DAMTP NA14, Department of Applied Mathematics and Theoretical Physics. Cambridge University, Cambridge (2000)
Powell, M.J.D.: On the Use of Quadratic Models in Unconstrained Minimization without Derivatives. Technical Report NA2003/03, Department of Applied Mathematics and Theoretical Physics. Cambridge University, Cambridge (2003)
Powell, M.J.D.: The NEWUOA software for unconstrained optimization without derivatives, Technical Report DAMTP NA2004/08, Department of Applied Mathematics and Theoretical Physics. Cambridge University, Cambridge (2004)
Schittkowski, K.: More Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical System. Springer, Berlin (1981)
Wild, S.M., Shoemaker, C.: Global convergence of radial basis function trust region derivative-free algorithms. SIAM J. Optim. 21, 761–781 (2011)
Zhang, H., Conn, A.R.: On the local convergence of a derivative-free algorithm for least-squares minimization. Comput. Optim. Appl. 51, 481–507 (2012)
Zhang, H., Conn, A.R., Scheinberg, K.: A derivative-free algorithm for least-squares minimization. SIAM J. Optim. 20(6), 3555–3576 (2010)
Author information
Authors and Affiliations
Corresponding authors
Additional information
The authors gratefully acknowledge the partial supports of the National Natural Science Foundation (11371253) of China.
Rights and permissions
About this article
Cite this article
Wang, J., Zhu, D. Conjugate gradient path method without line search technique for derivative-free unconstrained optimization. Numer Algor 73, 957–983 (2016). https://doi.org/10.1007/s11075-016-0124-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-016-0124-9