Reverse Bridge Theorem under Constraint Partition

Reverse bridge theorem (RBTH) has been proved to be both a necessary and sufficient condition for solving Nonlinear programming problems. In this paper, we first propose three algorithms for finding constraint minimum points of continuous, discrete, and mixed-integer nonlinear programming problems based on the reverse bridge theorem. Moreover, we prove that RBTH under constraint partition is also a necessary and sufficient condition for solving nonlinear programming problems. This property can help us to develop an algorithm using RBTH under constraints. Specifically, the algorithm first partitions mixed-integer nonlinear programming problems (MINLPs) by their constraints into some subproblems in similar forms, then solves each subproblem by using RBTH directly, and finally resolves those unsatisfied global constraints by choosing appropriate penalties. Finally, we prove the soundness and completeness of our algorithm. Experimental results also show that our algorithm is effective and sound.

[1]  George B. Dantzig,et al.  Decomposition Principle for Linear Programs , 1960 .

[2]  Jeremy F. Shapiro,et al.  Generalized Lagrange Multipliers in Integer Programming , 2011, Oper. Res..

[3]  Ignacio E. Grossmann,et al.  An outer-approximation algorithm for a class of mixed-integer nonlinear programs , 1986, Math. Program..

[4]  Ignacio E. Grossmann,et al.  An outer-approximation algorithm for a class of mixed-integer nonlinear programs , 1987, Math. Program..

[5]  Nicholas I. M. Gould,et al.  CUTE: constrained and unconstrained testing environment , 1995, TOMS.

[6]  Nicholas I. M. Gould,et al.  Numerical experiments with the LANCELOT package (release A) for large-scale nonlinear optimization , 1996, Math. Program..

[7]  Nikolaos V. Sahinidis,et al.  BARON: A general purpose global optimization software package , 1996, J. Glob. Optim..

[8]  Zhe Wu,et al.  The Theory of Discrete Lagrange Multipliers for Nonlinear Discrete Optimization , 1999, CP.

[9]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[10]  Carlos F. Torres,et al.  Global Optimization of Gas Allocation to a Group of Wells in Artificial Lift Using Nonlinear Constrained Programming , 2002 .

[11]  Yixin Chen,et al.  Partitioning of temporal planning problems in mixed space using the theory of extended saddle points , 2003, Proceedings. 15th IEEE International Conference on Tools with Artificial Intelligence.

[12]  Yixin Chen,et al.  Subgoal Partitioning and Global Search for Solving Temporal Planning Problems in Mixed Space , 2004, Int. J. Artif. Intell. Tools.

[13]  Yixin Chen,et al.  SGPlan: Subgoal Partitioning and Resolution in Planning , 2004 .

[14]  Yixin Chen,et al.  Solving Large-Scale Nonlinear Programming Problems by Constraint Partitioning , 2005, CP.

[15]  B. Wah,et al.  Solving nonlinear constrained optimization problems through constraint partitioning , 2005 .

[16]  Yixin Chen,et al.  Constrained Global Optimization by Constraint Partitioning and Simulated Annealing , 2006, 2006 18th IEEE International Conference on Tools with Artificial Intelligence (ICTAI'06).

[17]  Yixin Chen,et al.  Constraint partitioning in penalty formulations for solving temporal planning problems , 2006, Artif. Intell..

[18]  Yixin Chen,et al.  Temporal Planning using Subgoal Partitioning and Resolution in SGPlan , 2006, J. Artif. Intell. Res..

[19]  Yixin Chen,et al.  Simulated annealing with asymptotic convergence for nonlinear constrained optimization , 2007, J. Glob. Optim..

[20]  Yixin Chen,et al.  Constraint Partitioning for Solving Planning Problems with Trajectory Constraints and Goal Preferences , 2007, IJCAI.

[21]  Bing Li,et al.  Effective method for constrained minimum - reverse bridge theorem , 2008, Comput. Math. Appl..

[22]  Benjamin W. Wah,et al.  Finding Good Starting Points for Solving Structured and Unstructured Nonlinear Constrained Optimization Problems , 2008, 2008 20th IEEE International Conference on Tools with Artificial Intelligence.

[23]  Arthur M. Geoffrion,et al.  Lagrangian Relaxation for Integer Programming , 2010, 50 Years of Integer Programming.