Apex Method: A New Scalable Iterative Method for Linear Programming †
Abstract
:1. Introduction
2. Related Work
3. Theoretical Background
4. Description of Apex Method
4.1. Algorithm for Calculating Pseudoprojection
Algorithm 1 Calculating the pseudoprojection . |
Require: , ,
|
Algorithm 2 Parallel calculation of a pseudoprojection. | |
Master | lth Worker |
|
|
- m:
- length of the list ;
- D:
- latency (time taken by the master to send a one one-byte message to a single worker);
- :
- time taken by the master to send the current approximation to a single worker and receive the pair from it (including latency);
- :
- time taken by a single worker to process the higher-order function Map for the entire list ;
- :
- time taken by computing the binary operation ⊕.
- :
- quantity of numbers sent from the master to the lth worker and back within one iteration;
- :
- quantity of arithmetic and comparison operations required to compute the function defined by Equation (101);
- :
- quantity of arithmetic and comparison operations required to compute the binary operation ⊕ .
4.2. Quest Stage
- 1.
- Calculate a feasible point .
- 2.
- Calculate the apex point z.
- 3.
- Calculate the point that is the pseudoprojection of the apex point z onto the feasible polytope M.
4.3. Target Stage
Algorithm 3 Target stage. |
Require: , ,
|
5. Implementation and Computational Experiments
6. Discussion
- 1.
- What is the scientific contribution of this article?
- 2.
- What is the practical significance of the apex method?
- 3.
- What is our confidence that the apex method always converges to the exact solution of the LP problem?
- 4.
- How can we speed up the convergence of the Algorithm 1 calculating a pseudoprojection on the feasible polytope M?
7. Conclusions and Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Notations
real Euclidean space | |
Euclidean norm | |
dot product of two vectors | |
concatenation of two vectors | |
linear objective function | |
c | gradient of objective function |
unit vector parallel to vector c | |
solution of LP problem | |
M | feasible polytope |
set of boundary points of feasible polytope M | |
ith row of matrix A | |
half-space defined by inequality | |
hyperplane defined by equation | |
set of row indices in matrix A | |
set of indices for which the half-space is c-recessive | |
orthogonal projection onto hyperplane | |
pseudoprojection onto feasible polytope M | |
metric projection onto feasible polytope M |
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Parameter | Value |
---|---|
Number of processor nodes | 480 |
Processor | Intel Xeon X5680 (6 cores, 3.33 GHz) |
Processors per node | 2 |
Memory per node | 24 GB DDR3 |
Interconnect | InfiniBand QDR (40 Gbit/s) |
Operating system | Linux CentOS |
No | Problem | Quest Stage | Target Stage | |||
---|---|---|---|---|---|---|
Name | Exact Solution | Rough Solution | Error | Refined Solution | Error | |
1 | adlittle | |||||
2 | afiro | |||||
3 | blend | |||||
4 | fit1d | |||||
5 | kb2 | |||||
6 | recipe | |||||
7 | sc50a | |||||
8 | sc50b | |||||
9 | sc105 | |||||
10 | share2b |
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Sokolinsky, L.B.; Sokolinskaya, I.M. Apex Method: A New Scalable Iterative Method for Linear Programming. Mathematics 2023, 11, 1654. https://doi.org/10.3390/math11071654
Sokolinsky LB, Sokolinskaya IM. Apex Method: A New Scalable Iterative Method for Linear Programming. Mathematics. 2023; 11(7):1654. https://doi.org/10.3390/math11071654
Chicago/Turabian StyleSokolinsky, Leonid B., and Irina M. Sokolinskaya. 2023. "Apex Method: A New Scalable Iterative Method for Linear Programming" Mathematics 11, no. 7: 1654. https://doi.org/10.3390/math11071654
APA StyleSokolinsky, L. B., & Sokolinskaya, I. M. (2023). Apex Method: A New Scalable Iterative Method for Linear Programming. Mathematics, 11(7), 1654. https://doi.org/10.3390/math11071654