Abstract
In this paper, we present a new homotopy method which is a non-interior point homotopy method for solving semi-infinite programming problems. Under suitable assumptions, we prove that the method determines a smooth path from a given point. The new homotopy method generalizes the existing combined homotopy interior point method for semi-infinite programming problems to unbounded set, moreover, it is more convenient in that it enlarges the choice scope of the initial point. Some numerical examples are given to show its efficiency.
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Leon, T., Sanmatias, S., Vercher, E.: On the numerical treatment of linearly constrained semi-infinite optimization problems. Eur. J. Oper. Res. 121(1), 78–91 (2000)
Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, Berlin (2006)
Still, G.: Discretization in semi-infinite programming: the rate of convergence. Math. Program. 91, 53–69 (2001)
Bhattacharjee, B., Lemonidis, P., Green Jr., W.H., Barton, P.I.: Global solution of semi-infinite programs. Math. Program. 103, 283–307 (2005)
Cánovas, M.J., Hantoute, A., López, M.A., Parra, J.: Stability of indices in the KKT conditions and metric regularity in convex semi-infinite optimization. J. Optim. Theory Appl. 139, 485–500 (2008)
López, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491–518 (2007)
Reemtsen, R., Ruckmann, J.: Semi-Infinite Programming. Kluwer Academic Publishers, Boston (1998)
Stein, O., Still, G.: Solving semi-infinite optimization problems with interior point techniques. SIAM J. Control Optim. 42, 769–788 (2003)
Zhang, L.P., Wu, S.-Y., López, M.A.: A new exchange method for convex semi-infinite programming. SIAM J. Optim. 20, 2959–2977 (2010)
Okuno, T., Hayashi, S., Yamashita, N., Gomoto, K.: An exchange method with refined subproblems for convex semi-infinite programming problems. Optim. Methods Softw. 31, 1–20 (2016)
Li, D.H., Qi, L., Tam, J., Wu, S.Y.: A smoothing Newton method for semi-infinite programming. J. Glob. Optim. 30, 169–194 (2004)
Ling, C., Ni, Q., Qi, L., Wu, S.Y.: A new smoothing Newton-type algorithm for semi-infinite programming. J. Glob. Optim. 47(1), 133–159 (2010)
Qi, L., Wu, S.Y., Zhou, G.L.: Semismooth Newton methods for solving semi-infinite programming problems. J. Glob. Optim. 27, 215–232 (2003)
Xu, M., Wu, S.Y., Ye, J.J.: Solving semi-infinite programs by smoothing projected gradient method. Comput. Optim. Appl. 59(3), 591–616 (2014)
Liu, G.: A homotopy interior point method of semi-infinite programming problems. J. Glob. Optim. 37(4), 631–646 (2007)
Shang, Y., Yu, B.: Boundary moving combined homotopy method for nonconvex nonlinear programming and its convergence. J. Jinlin Univ.: Sci. Ed. 44(3), 357–361 (2006)
Allgower, E.L., Georg, K.: Numerical Continuation Methods: An Introduction. Springer, Berlin (1990)
Naber, G.L.: Topological Method in Euclidean Space. Cambridge University Press, London (1980)
Makela, M.M., Neittaanmaki, P.: Nonsmooth Optimization. World Scientific, Singapore (1992)
Acknowledgments
The authors are grateful to the referees for valuable comments which help us to improve the paper. This work is supported by the Natural Sciences Foundation of China under Grant No. 11201240.
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Fan, X., Li, M. & Gao, F. A noninterior point homotopy method for semi-infinite programming problems. J. Appl. Math. Comput. 56, 179–194 (2018). https://doi.org/10.1007/s12190-016-1067-y
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DOI: https://doi.org/10.1007/s12190-016-1067-y