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### Solving Integer and Mixed Integer Linear Problems with ABS Method

2013 Acta Polytechnica Hungarica
Solving mixed integer linear programming (MILP) problems is a difficult task due to the parallel use of both integer and non-integer values.  ...  Here we provide for the first time a complete ABS-based algorithm for MILP problems by adaptation of the ABS approach to Gomory's cuttingplane algorithm.  ...  Acknowledgement We thank Ferenc Forgó for helpful discussions and encouraging notes.  ...

### OPTIMAL PRODUCTION OF BRA BLOCK FACTORY BY USING SIMPLEX METHOD

2018 Academic Journal of Nawroz University
In this paper simplex method is used to obtain the optimal solution to maximize the production profit for three different block types in BRA BLOCK FACTORY for heat insulating light construction pomic blocks  ...  The three different model that we have taken to investigat the maximum profit for the factory are BB6H.10, BBS.15 and BB2H.20.  ...  The simplex method provides an algorithm (a rule of procedure usually involving repetitive application of a prescribed operation) which is based on the fundamental theorem of linear programming.  ...

### Verifying a Solver for Linear Mixed Integer Arithmetic in Isabelle/HOL [chapter]

Ralph Bottesch, Max W. Haslbeck, Alban Reynaud, René Thiemann
2020 Lecture Notes in Computer Science
We implement a decision procedure for linear mixed integer arithmetic and formally verify its soundness in Isabelle/HOL.  ...  It internally relies upon an adapted version of an existing verified incremental simplex algorithm.  ...  A system of linear inequalities is a mixed integer system if, for some I ⊆ {1, . . . , n}, it is required that x i ∈ Z for all i ∈ I.  ...

### Page 357 of Mathematical Reviews Vol. 49, Issue 1 [page]

1975 Mathematical Reviews
The author studies a number of combinatorial algorithms for minimizing a nonlinear nonseparable function over a set of (01) matrices X =(x,,) satisfying the condition >i %j=1, j=1,---,n.  ...  .; Sinha, Prabhakant 2090 Improved penalty calculations for a mixed integer branch-and-bound algorithm. Math. Programming 6 (1974), 212-223.  ...

### Discrete optimisation and real world problems [chapter]

Josef Kallrath, Anna Schreieck
1995 Lecture Notes in Computer Science
of the approach based on mixed-integer optimisation.  ...  Computer-based optimisation techniques are the best means of obtaining viable solutions, but until now the mixed integer programs developed have been able to deal only with simple problems.  ...  4 , an integer (ILP) or a mixed-integer linear programming problem (MILP) results.  ...

### Traces of the XII Aussois Workshop on Combinatorial Optimization

Michael Jünger, Thomas M. Liebling, Denis Naddef, William R. Pulleyblank, Gerhard Reinelt, Giovanni Rinaldi, Laurence A. Wolsey
2010 Mathematical programming
"Speeding up IP-based Algorithms for Constrained Quadratic 0-1 Optimization" by Christoph Buchheim, Frauke Liers, and Marcus Oswald presents a general framework for solving constrained quadratic 0-1 problems  ...  Letchford studies valid inequalities for 0-1 knapsack polytopes, in particular separation algorithms for them.  ...

### Page 2910 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews
This is done initially for a certain family of mixed- integer 0-1 linear programs which arise from a linearization of polynomial programs [see H. D. Sherali and W. P. Adams, SIAM J.  ...  It is very common, given a mixed integer 0-1 linear program, to relax the binary restrictions and obtain a linear programming problem whose solution provides bounds for the original problem.  ...

### Page 5953 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews
The authors present a calculation of penalties and stronger penalties for the branch-and-bound algorithm of mixed integer linear pro- gramming, based on the revised dual simplex method for bounded variables  ...  A software package coded by the authors is presented of mixed integer linear programming using the revised dual simplex algorithm for bounded variables and the stronger penalties.  ...

### Nonlinear Control Allocation Using Piecewise Linear Functions: A Linear Programming Approach

Michael A. Bolender, David B. Doman
2005 Journal of Guidance Control and Dynamics
Therefore, as an alternative to the mixed-integer linear programming formulation given in Ref. 1, we will enforce Eq. (28) by modifying the the simplex algorithm such that we only admit a A) into the basis  ...  The result- ing optimization problem is then a mixed-integer linear program.  ...

### Primal cutting plane algorithms revisited

2002 Mathematical Methods of Operations Research
We describe a new primal algorithm for pure 0-1 problems based on strong valid inequalities and give some encouraging computational results.  ...  Possible extensions to the case of general mixed-integer programs are also discussed.  ...  Acknowledgement: The authors would like to thank both Robert Weismantel and the anonymous referee for helpful comments.  ...

### An algorithm for mixed integer optimization

Matthias K�ppe, Robert Weismantel
2003 Mathematical programming
This paper introduces a new algorithm for solving mixed integer programs.  ...  The core of the method is an iterative technique for changing the representation of the original mixed integer optimization problem.  ...  Algorithm 2.2 (Construction of a mixed-integer standard tableau). Input: (z 0 , y 0 ) ∈ Z I + × R C + feasible for (2.1). Output: A mixed-integer tableau of the form (2.5). 0.  ...

### MIP: Theory and Practice — Closing the Gap [chapter]

E. Robert Bixby, Mary Fenelon, Zonghao Gu, Ed Rothberg, Roland Wunderling
2000 IFIP Advances in Information and Communication Technology
Acknowledgments The authors would like to thank John Gregory and Irv Lustig, both of ILOG, for carefully reading this manuscript and contributing signi cantly to improving the exposition.  ...  The standard technique for solving mixed-integer programming problems is a version of divide-and-conquer known as linear-programming based branch-and-bound, or, what is now a more correct name, branchand-cut  ...  INTRODUCTION For many years the principal solution technique used in the practice of mixed-integer programming has remained largely unchanged: Linear programming based branch-and-bound, introduced by Land  ...

### George Dantzig's impact on the theory of computation

Richard M. Karp
2008 Discrete Optimization
George Dantzig created the simplex algorithm for linear programming, perhaps the most important algorithm developed in the 20th century.  ...  of the simplex algorithm and the intrinsic complexity of linear programming and combinatorial optimization.  ...  Acknowledgement Many thanks to Ilan Adler for general advice and in particular for recreating for me his experiences as George Dantzig's Ph.D. student at Stanford.  ...

### Page 667 of Mathematical Reviews Vol. 57, Issue 2 [page]

1979 Mathematical Reviews
Thomas Gal (Aachen) Lauriére, Marion 5083 An algorithm for the 0/1 knapsack problem. Math. Programming 14 (1978), no. 1, 1-10.  ...  A new algorithm is presented for solving the 0-1 knapsack problem max cx subject to ax<b, where c and a are n-vectors whose elements c, and a, (i € J) are integers, x, (i¢ 7) are 0, 1 variables and 5 is  ...

### Finitely Convergent Decomposition Algorithms for Two-Stage Stochastic Pure Integer Programs

Minjiao Zhang, Si̇mge Küçükyavuz
2014 SIAM Journal on Optimization
We study a class of two-stage stochastic integer programs with general integer variables in both stages and finitely many realizations of the uncertain parameters.  ...  Based on Benders' method, we propose a decomposition algorithm that utilizes Gomory cuts in both stages.  ...  We also acknowledge Suvrajeet Sen for the beneficial discussions.  ...
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