Implicit representation of variable upper bounds in linear programming

1975 ◽  
pp. 118-132 ◽  
Author(s):  
Linus Schrage
Keyword(s):  
Linear Programming ◽  
Upper Bounds ◽  
Implicit Representation ◽  
Variable Upper Bounds

Implicit representation of generalized variable upper bounds in linear programming

1978 ◽  
Vol 14 (1) ◽  
pp. 11-20 ◽  
Author(s):  
Linus Schrage
Keyword(s):  
Linear Programming ◽  
Upper Bounds ◽  
Implicit Representation ◽  
Variable Upper Bounds

scholarly journals Verifying a Solver for Linear Mixed Integer Arithmetic in Isabelle/HOL

2020 ◽  
pp. 233-250
Author(s):  
Ralph Bottesch ◽  
Max W. Haslbeck ◽  
Alban Reynaud ◽  
René Thiemann
Keyword(s):  
Linear Programming ◽  
Branch And Bound ◽  
Decision Procedure ◽  
Simplex Algorithm ◽  
Upper Bounds ◽  
Mixed Integer ◽  
Small Solution ◽  
Code Generator ◽  
Integer Arithmetic ◽  
Termination Proofs

AbstractWe implement a decision procedure for linear mixed integer arithmetic and formally verify its soundness in Isabelle/HOL. We further integrate this procedure into one application, namely into , a formally verified certifier to check untrusted termination proofs. This checking involves assertions of unsatisfiability of linear integer inequalities; previously, only a sufficient criterion for such checks was supported. To verify the soundness of the decision procedure, we first formalize the proof that every satisfiable set of linear integer inequalities also has a small solution, and give explicit upper bounds. To this end we mechanize several important theorems on linear programming, including statements on integrality and bounds. The procedure itself is then implemented as a branch-and-bound algorithm, and is available in several languages via Isabelle’s code generator. It internally relies upon an adapted version of an existing verified incremental simplex algorithm.

Chandpur Enterprises Limited, Steel Division

2017 ◽  
pp. 1-4
Author(s):  
Samuel E. Bodily ◽  
Akshay Mittal
Keyword(s):  
Linear Programming ◽  
Linear Model ◽  
Raw Materials ◽  
Nonlinear Models ◽  
Upper Bounds ◽  
Raw Material ◽  
Shadow Prices ◽  
Lower And Upper Bounds ◽  
Optimization Function ◽  
Introductory Class

The managing director of a steel plant faces the decision of how much of each raw material to order for the plant for the following month. Due to lower and upper bounds on the amounts of each raw material in a batch and varying amounts of electricity and time consumed for different raw materials, one can't simply use the cheapest raw material. A linear program and the solver optimization function of Excel will provide the optimal amounts that meet the constraints. Interestingly, the best mixture for a batch is not the best mixture for a monthly plan. Shadow prices indicate the value of relaxing constraints. The typical monthly model from a student will be nonlinear, although it can be written as a linear model. This case provides the basis for an introductory class on linear programming and linear versus nonlinear models.

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