Abstract
In this paper, we consider a general class of nonlinear mixed discrete programming problems. By introducing continuous variables to replace the discrete variables, the problem is first transformed into an equivalent nonlinear continuous optimization problem subject to original constraints and additional linear and quadratic constraints. Then, an exact penalty function is employed to construct a sequence of unconstrained optimization problems, each of which can be solved effectively by unconstrained optimization techniques, such as conjugate gradient or quasi-Newton methods. It is shown that any local optimal solution of the unconstrained optimization problem is a local optimal solution of the transformed nonlinear constrained continuous optimization problem when the penalty parameter is sufficiently large. Numerical experiments are carried out to test the efficiency of the proposed method.
Similar content being viewed by others
References
Bremicker M., Papalambros P.Y., Loh H.T.: Solution of mixed-discrete structural optimization problems with a new sequential linearization algorithms. Comput. Struct. 37(4), 451–461 (1990)
Cai J.B., Thierauf G.: Discrete optimization of structures using an improved penalty function method. Eng. Optim. 21(4), 293–306 (1993)
Fang S.C., Gao D.Y., Sheu R.L., Wu S.Y.: Canonical dual approach to solving 0–1 quadratic programming problems. J. Ind. Manage. Optim. 4(1), 125–142 (2008)
Fu J.F., Fenton R.G., Cleghorn W.L.: A mixed integer-discretecontinuous programming method and its application to engineering design optimization. Eng. Optim. 17(4), 263–280 (1991)
Gao D.Y.: Duality principles in nonconvex systems—theory, methods and applications. Kluwer Academic Publishers, Dordrecht (2000)
Gao D.Y., Ruan N.: Solutions to quadratic minimization problems with box and integer constraints. J. Global Optim. 47(3), 463–484 (2010)
Li H.L., Chou C.T.: A global approach for nonlinear mixed discrete programming in design optimization. Eng. Optim. 22(2), 109–122 (1993)
Linderoth J.T., Savelsbergh M.W.P.: A computational study of search strategies for mixed integer programming. INFORMS J. Comput. Spring 11(2), 173–188 (1999)
Loh H.T., Papalambros P.Y.: A sequential linearization approach for solving mixed-discrete nonlinear design optimization problems. J. Mech. Des. 113, 325–334 (1991)
Loh H.T., Papalambros P.Y.: Computational implementation and tests of a sequential linearization algorithm for mixed-discrete nonlinear design optimization. J. Mech. Des. 113(3), 335–344 (1991)
Lucidi S., Rinaldi F.: Exact penalty functions for nonlinear integer programming problems. J. Optim. Theory Appl. 145(3), 479–488 (2010)
Murray W., Ng K.M.: An algorithm for nonlinear optimization problems with binary variables. Comput. Optim. Appl. 47(2), 257–288 (2010)
Ng C., Zhang L.: Discrete filled function method for discrete global optimization. Comput. Optim. Appl. 31(1), 87–115 (2005)
Pardalos, P.M., Rosen, J.B.: Constrained global optimization: algorithms and applications. In: Lecture Notes in Computer Science, vol. 268. Springer, Berlin (1987)
Pardalos, P.M.: Continuous Approaches to Discrete Optimization Problems. In: Di Pillo, G., Giannessi, F. (eds.) Nonlinear Optimization and Applications, pp. 313–328. Plenum Publishing, New York (1996)
Pardalos, P.M., Prokopyev, O., Busygin, S.: Continuous Approaches for Solving Discrete Optimization Problems. In: Appa, G., Pitsoulis, L., Williams, H.P. (eds.) Handbook on Modelling for Discrete Optimization, pp. 39–60. Springer, New York (2006)
Pardalos P.M., Yatsenko V.: Optimization and control of bilinear systems. Springer, New York (2009)
Ringertz U.T.: On methods for discrete structural optimization. Eng. Optim. 13(1), 47–64 (1988)
Sandgren E.: Nonlinear integer and discrete programming in mechanical design optimization. J. Mech. Des. 112(2), 223–229 (1988)
Shin D.K., Gürdal Z., Griffin J.R., Griffin J.R.: A penalty approach for nonlinear optimization with discrete design variables. Eng. Optim. 16(1), 29–42 (1990)
Sun X.L., Li J.L., Luo H.Z.: Convex relaxation and Lagrangian decomposition for indefinite integer quadratic programming. Optimization 59(5), 627–641 (2010)
Tseng C., Arora J.: On implementation of computational algorithms for optimization design. Part I and Part. Int. J. Numer. Methods Eng. 26(6), 1365–1402 (1988)
Wang S., Teo K.L., Lee H.W.J.: A new approach to nonlinear mixed discrete programming problems. Eng. Optim. 30(3), 249–262 (1998)
Yu C.J., Teo K.L., Zhang L.S., Bai Y.Q.: A new exact penalty function method for continuous inequality constrained optimization problems. J. Ind. Manage. Optim. 6(4), 895–910 (2010)
Zheng X.J., Sun X.L., Li D.: Separable relaxation for nonconvex quadratic integer programming: integer diagonalization approach. J. Optim. Theory Appl. 146(2), 463–489 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yu, C., Teo, K.L. & Bai, Y. An exact penalty function method for nonlinear mixed discrete programming problems. Optim Lett 7, 23–38 (2013). https://doi.org/10.1007/s11590-011-0391-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-011-0391-2