Sufficient global optimality conditions for general mixed integer nonlinear programming problems

Document Type: Research Paper

Authors

1 Department of Mathematics, Yibin University, Yibin, Sichuan, 644007, China.

2 School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China.

Abstract

‎In this paper‎, ‎some KKT type sufficient global optimality conditions‎ ‎for general mixed integer nonlinear programming problems with‎ ‎equality and inequality constraints (MINPP) are established‎. ‎We achieve‎ ‎this by employing a Lagrange function for MINPP‎. ‎In addition‎, ‎verifiable sufficient global optimality conditions for general mixed‎ ‎integer quadratic programming problems are derived easily‎. ‎Numerical‎ ‎examples are also presented.

Keywords

Main Subjects


A. Beck and M. Teboulle, Global optimality conditions for quadratic optimization problems with binary constraints, SIAM J. Optim. 11 (2000), no. 1, 179--188.

B. Borchers and J. E. Mitchell, An improved branch and bound algorithm for mixed integer nonlinear programs, Comput. Oper. Res. 21 (1994), no. 4, 359--367.

C. D'Ambrosio and A. Lod, Mixed integer nonlinear programming tools: an updated practical overview, Ann. Oper. Res. 204 (2013) 301--320.

M. A. Duran and I. E. Grossmann, An outer-approximation algorithm for a class of mixed integer nonlinear programs, Math. Program. 36 (1986), no. 3, 307--339.

R. Fletcher and S. Leyffer, Solving mixed integer nonlinear programs by outer approximation, Math. Program. 66 (1994), no. 3, 327--349.

C. A. Floudas, Deterministic Global Optimization: Theory, Algorithms and Applicaions, Kluwer Academic Publishers, Dordrecht, 2000.

C. A. Floudas, Nonlinear and Mixed Integer Optimization: Fundamentals and Applications, Oxford University Press, Oxford, 1995.

A. M. Geoffrion, Lagrangean relaxation for integer programming, Math. Programming Stud. 2 (1974) 82--114.

I. E. Grossmann and N. Sahinidis, Special Issue on Mixed Integer Programming and its Application to Engineering, Part I. Optim. Eng., 3 , no. 3, Kluwer Academic Publishers, Norwell, 2002.

I. E. Grossmann and N. Sahinidis, Special Issue on Mixed Integer Programming and its Application to Engineering, Part II, Optim. Eng., 4 no. 1-2, Kluwer Academic Publishers, Norwell, 2003.

I. E. Grossmann and Z. Kravanja, Mixed integer nonlinear programming: A survey of algorithms and applications, Large-scale optimization with applications, Part II (Minneapolis, 1995), 73--100, Math. Appl., 93, Springer, New York, 1997.

V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions, Math. Program. Ser. A 110 (2007), no. 3, 521--541.

V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Sufficient global optimality conditions for non-convex quadratic minimization problems with box constraints, J. Global Optim. 36 (2006), no. 3, 471--481.

V. Jeyakuma, G. Li and S. Srisatkunarajah, Global optimality principles for polynomial optimization over box or bivalent constraints by separable polynomial approximations, J. Global Optim. 58 (2014), no. 1, 31--50.

 V. Jeyakumar, S. Srisatkunrajah and N. Q. Huy, Kuhn-Tucker sufficiency for global minimum of multi-extremal mathematical programming problems, J. Math. Anal. Appl. 335 (2007), no. 2, 779--788.

S. Leyffer, Integrating SQP and branch-and-bound for mixed integer nonlinear programming, Comput. Optim. Appl. 18 (2001), no. 3, 295--309.

T. Mohit and N. V. Sahinidis, A polyhedral branch-and-cut approach to global optimization, Math. Program. 103 (2005), no. 2, 225--249.

M. Ruth and C. A. Floudas, ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations, J. Global Optim. 59 (2014), no. 2-3, 1--24.

M. Tawarmalani and N. V. Sahinidis, Convexi_cation and Global Optimization in Continuous and Mixed Integer Nonlinear Programming Theory, Algorithms, Software, and Applications, Kluwer Academic Publishers, Dordrecht, 2002.

Z. Y. Wu, Sufficient global optimality conditions for weakly convex minimization problems, J. Global Optim. 39 (2007), no. 3, 427--440.

Z. Y.Wu, V. Jeyakumar and A. M. Rubinov, Sufficient Conditions for Global Optimality of Bivalent Nonconvex Quadratic Programs with Inequality Constraints, J. Optim. Theo. Appl. 133 (2007), no. 1, 123--130.

Z. Y. Wu and F. S. Bai, Global optimality conditions for mixed nonconvex quadratic programs, Optimization 58 (2009), no. 1, 39--47.

Z. Wu, J. Tian, J. Quan and J. Ugon, Optimality conditions and optimization methods for quartic polynomial optimization,  Appl. Math. Comput. 232 (2014) 968--982.

J. Zhang, P. Liu, Z. Zhou, L. Ma, Z. Li and W. Ni, A mixed integer nonlinear programming approach to the optimal design of heat network in a polygeneration energy system, Appl. Ener. 114 (2014) 146--154.


Volume 42, Issue 5
September and October 2016
Pages 1237-1246
  • Receive Date: 29 April 2015
  • Revise Date: 10 June 2015
  • Accept Date: 11 August 2015