Abstract
Penalty function is an important tool in solving many constrained optimization problems in areas such as industrial design and management. In this paper, we study exactness and algorithm of an objective penalty function for inequality constrained optimization. In terms of exactness, this objective penalty function is at least as good as traditional exact penalty functions. Especially, in the case of a global solution, the exactness of the proposed objective penalty function shows a significant advantage. The sufficient and necessary stability condition used to determine whether the objective penalty function is exact for a global solution is proved. Based on the objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the objective penalty function is proved for a local solution. An algorithm is presented in the paper in finding a local solution, with its convergence proved under some conditions. Finally, numerical experiments show that a satisfactory approximate optimal solution can be obtained by the proposed algorithm.
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References
Du D., Pardalos P.M., Wu W.: Mathematical Theory of Optimization. Kluwer, Dordrecht (2001)
Pardalos P.M., Resende M.G.C.: Handbook of Applied optimization. Oxford University Press, Oxford (2002)
Zangwill W.I.: Nonlinear programming via penalty function. Manang. Sci. 13, 334–358 (1967)
Han S.P., Mangasrian O.L.: Exact penalty function in nonlinear programming. Math. Program. 17, 251–269 (1979)
Rosenberg E.: Globally convergent algorithms for convex programming. Math. Oper. Res. 6, 437–443 (1981)
Rosenberg E.: Exact penalty functions and stability in locally Lipschitz programming. Math. Program. 30, 340–356 (1984)
Di Pillo G., Grippo L.: An exact penalty function method with global conergence properties for nonlinear programming problems. Math. Program. 36, 1–18 (1986)
Di Pillo G., Grippo L.: Exact penalty functions in constrained optimization. SIAM J. Control Optim. 27, 1333–1360 (1989)
Pinar M.C., Zenios S.A.: On smoothing exact penalty functions for convex constraints optimization. SIAM J. Optim. 4, 486–511 (1994)
Mongeau M., Sartenaer A.: Automatic decrease of the penalty parameter in exact penalty functions methods. Eur. J. Oper. Res. 83, 686–699 (1995)
Zaslavski A.J.: Exact penalty property for a class of inequality-constrained minimization problems. Optim. Lett. 2, 287–298 (2008)
Zaslavski A.J.: A sufficient condition for exact penalty functions. Optim. Lett. 3, 593–602 (2009)
Rubinov A.M., Glover B.M., Yang X.Q.: Extended lagrange and penalty functions in continuous optimization. Optimization 46, 327–351 (1999)
Rubinov A.M., Glover B.M., Yang X.Q.: Decreasing functions with applications to penalization. SIAM J. Optim. 10, 289–313 (1999)
Yang X.Q., Huang X.X.: A nonlinear Lagrangian approach to constrained optimization problems. SIAM J. Optim. 11, 1119–1144 (2001)
Huang X.X., Yang X.Q.: Duality and exact penalization for vector optimization via augumentd Lagrangian. J. Optim. Theory Appl. 111, 615–640 (2001)
Morrison D.D.: Optimization by least squares. SIAM J. Numer. Anal. 5, 83–88 (1968)
Fletcher R.: Practical Method of Optimization. Wiley-Interscience, New York (1981)
Fletcher R.: Penalty functions. In: Bachem, A., Grotschel, M., Korte, B. (eds.) Mathematical Programming: The State of the Art. Springer, Berlin (1983)
Burke J.V.: An exact penalization viewpoint of constrained optimization. SIAM J. Control Optim. 29, 968–998 (1991)
Fiacco A.V., McCormick G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)
Mauricio D., Maculan N.: A boolean penalty method for zero-one nonlinear programming. J. Glob. Optim. 16, 343–354 (2000)
Meng Z.Q., Hu Q.Y., Dang C.Y., Yang X.Q.: An objective penalty function method for nonlinear programming. Appl. Math. Lett. 17, 683–689 (2004)
Meng Z.Q., Hu Q.Y., Dang C.Y.: A penalty function algorithm with objective parameters for nonlinear mathematical programming. J. Ind. Manag. Optim. 5, 585–601 (2009)
Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Burke J.V.: Calmness and exact penalization. SIAM J. Control Optim. 29, 493–497 (1991)
Henrion R., Outrata J.: Calmness in optimization problems with equality constraints. J. Math. Anal. Appl. 189, 502–513 (1995)
Huang X.X., Teo K.L., Yang X.Q.: Calmness and exact penalization in vector optimization with cone constraints. Comput. Optim. Appl. 35, 47–67 (2006)
Uderzo A.: Exact penalty functions and calmness for mathematical programming under nonlinear perturbations. Nonlinear Anal. Theory Methods Appl. 73, 1596–1609 (2010)
Meng Z.Q., Dang C.Y., Jiang M., Shen R.: A smoothing objective penalty function algorithm for inequality constrained optimization problems. Numer. Funct. Anal. Optim. 32, 806–820 (2011)
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Meng, Z., Dang, C., Jiang, M. et al. Exactness and algorithm of an objective penalty function. J Glob Optim 56, 691–711 (2013). https://doi.org/10.1007/s10898-012-9900-9
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DOI: https://doi.org/10.1007/s10898-012-9900-9