整数规划

整数规划 [编辑]

要求所有的未知量都为整数的线性规划问题叫做整数规划 (integer programming, IP) 或整数线性规划 (integer linear programming, ILP) 问题。 相对于即使在最坏情况下也能有效率地解出的线性规划问题，整数规划问题的最坏情况是不确定的，在某些实际情况中（有约束变量的那些）为NP困难问题。

0-1 整数规划是整数规划的特殊情况，所有的变量都要是0或1（而非任意整数）。这类问题亦被分类为NP困难问题。

只要求当中某几个未知数为整数的线性规划问题叫做混合整数规划 (mixed integer programming, MIP) 问题。这类问题通常亦被分类为NP困难问题。

存在着几类 IP 和 MIP 的子问题，它们可以被有效率地解出，最值得注意的一类是具有完全单位模约束矩阵，和约束条件的右边全为整数的一类。

一个解决大型整数线性规划问题的先进算法为 delayed column generation。

参见 [编辑]

• 单纯形法，一种用于线性规划问题求解的常用方法
• Leonid Kantorovich was one of founders of linear programming
• 影子价格
• MPS file format
• LP example, Job Shop problem

参考 [编辑]

• Alexander Schrijver: "Theory of Linear and Integer Programming". John Wiley and Sons. 1998.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Chapter 29: Linear Programming, pp.770–821.
• Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. 1979. ISBN 0716710455. A6: MP1: INTEGER PROGRAMMING, pg.245.

外部链接 [编辑]

• Guidance on Formulating LP problems
• 0-1 Integer Programming Benchmarks with Hidden Optimum Solutions
• COIN-OR- Open Source Library for linear programming
• Cplex - Commercial library for linear programming
• Xpress-MP - Optimization software free to students
• MOSEK - Optimization software for LP, QP, MIP, SOCP and more
• A Tutorial on Integer Programming
• The linear programming FAQ

求解软件包

• AIMMS — include linear programming in industry solutions (free trial license available);
• COIN-OR — COmputational INfrastructure for Operations Research, open-source library
• Cplex — Commercial library for linear programming
• HOPDM — Higher Order Primal Dual Method
• LINDO — LP, IP, Global solver/modeling language
• LiPS — Free easy-to-use program intended for solving linear, integer and goal programming problems.
• lp_solve
• MOSEK — Optimization software for LP, QP, MIP, SOCP and more
• Premium Solver — Spreadsheet add-in
• What's Best! — Spreadsheet add-in
• Xpress-MP — Optimization software free to students
• GIPALS — Linear programming environment and dynamic link library (DLL)
• DecisionPro Linear Programming Optimization Software
• QSopt Optimization software for LP (free for research purposes).
• Microarray Data Classification Server (MDCS) based on linear programming
• Linear programming and linear goal programming A freeware program for MS-DOS
• Simplex Method Tool A quick-loading web page