Journal of the Korea Convergence Society (한국융합학회논문지)

Korea Convergence Society (한국융합학회)
권호 표지
DOI QR코드

DOI QR Code

An Algorithm for the Singly Linearly Constrained Concave Minimization Problem with Upper Convergent Bounded Variables

상한 융합 변수를 갖는 단선형제약 오목함수 최소화 문제의 해법

  • Oh, Se-Ho (Dept. of Industrial Engineering, Cheongju University)
  • Received : 2016.09.09
  • Accepted : 2016.10.20
  • Published : 2016.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

This paper presents a branch-and-bound algorithm for solving the concave minimization problem with upper bounded variables whose single constraint is linear. The algorithm uses simplex as partition element. Because the convex envelope which most tightly underestimates the concave function on the simplex is uniquely determined by solving the related linear equations. Every branching process generates two subsimplices one lower dimensional than the candidate simplex by adding 0 and upper bound constraints. Subsequently the feasible points are partitioned into two sets. During the bounding process, the linear programming problems defined over subsimplices are minimized to calculate the lower bound and to update the incumbent. Consequently the simplices which do certainly not contain the global minimum are excluded from consideration. The major advantage of the algorithm is that the subproblems are defined on the one less dimensinal space. It means that the amount of work required for the subproblem decreases whenever the branching occurs. Our approach can be applied to solving the concave minimization problems under knapsack type constraints.

본 논문에서는 한 개의 선형 제약식 하에서 의사결정변수가 상한 값을 갖는 오목 함수 최소화 문제를 다룬다. 제시된 분지 한계 해법은 단체를 분할 단위로 사용하였다. 오목함수를 가장 단단하게 하한추정하는 볼록덮개함수를 단체 상에서 유일하게 구할 수 있기 때문이다. 분지가 일어날 때마다 후보 단체로부터 1 차원 낮은 2 개의 하위 단체들이 생성된다. 이 때 후보 단체에 포함되어 있던 가능해 집합은 각각의 하위 단체로 분할된다. 한계 연산 절차는 선형인 볼록 덮개 함수를 목적 함수로 하는 선형계획법을 부문제로 정의하고 해를 구한다. 부문제의 최적 목적함수 값으로부터 최적 오목목적함수의 하한과 상한을 갱신하고, 원문제의 최적해를 포함하지 않는 단체들을 고려 대상에서 제외시킨다. 본 해법의 최대 장점은 하위 단체로 분할될수록 부문제들의 크기가 점점 작아진다는데 있다. 이것은 한계 연산의 계산량이 줄어든다는 것을 의미한다. 본 연구의 결과는 배낭 제약식 유형의 제약식 하에서의 오목 함수 최소화 문제의 해법을 개발하는데 응용될 수 있을 것이다.

Keywords

References

  1. H. Tui, "Concave Programming under Linear Constraints", Dok. Akad. Nauk SSSR 159, 32-35, Translated 1964 in Soviet Math. Dokl. Vol. 4, pp. 1437-1440, 1964.
  2. B. Kalantari and A. Bagchi, "An Algorithm for Quadratic Zero-One Programs", Naval Research Logistics Quarterly Vol. 37, pp. 527-538, 1990. https://doi.org/10.1002/1520-6750(199008)37:4<527::AID-NAV3220370407>3.0.CO;2-P
  3. J. J. More and S. A. Vavasis, "On the Solution of Concave Knapsack Problem", Math. Prog. Vol. 49, pp. 397-411, 1991.
  4. J. B. Rosen, "Global Minimization of a Linearly Constrained Concave Function by Partition of Feasible Domain", Math. Opns. Res. Vol. 8, pp. 215-230, 1983. https://doi.org/10.1287/moor.8.2.215
  5. X. L. Sun, F. L. Wang and L. Li, "Exact Algorithm for Concave Knapsack Problems: Linear Underestimation and Partition Method", J. of Global Optimization, Vol. 33, pp.15-30, 2005. https://doi.org/10.1007/s10898-005-1678-6
  6. Young-Jae Park, "The Design of the Container Logistics Information System Reflects the Port Logistics Environment", Journal of digital Convergence, Vol. 13, No. 5, pp. 159-167, 2015. https://doi.org/10.14400/JDC.2015.13.5.159
  7. Dong-Hee Hong and Chang-Gon Kim, "Improvement of wireless communications environment of Web-pad on board Yard tractor in container terminal use convergence technology.", Journal of digital Convergence , Vol. 13, No. 8, pp. 281-288, 2015. https://doi.org/10.14400/JDC.2015.13.8.281
  8. T. V. Tieu, "Convergent Algorithms for Minimizing a Concave Function", Acta Mathematica Vietnamica, Vol. 5, pp. 106-113, 1978.
  9. Soon-Ho Kim and Chi-Su Kim, "An Algorithm of the Minimal Time on the (sLa-Camera-pLb)path", Journal of digital Convergence , Vol. 13, No. 10, pp. 337-342, 2015.
  10. In-Kyoo Park, "A Big Data Analysis by Between-Cluster Information using k-Modes Clustering Algorithm", Journal of digital Convergence , Vol. 13, No. 11, pp. 157-164, 2015. https://doi.org/10.14400/JDC.2015.13.11.157
  11. Yong-Tae Kim, Yoon-Su Jeong, "Optimization Routing Protocol based on the Location, and Distance information of Sensor Nodes ", Journal of digital Convergence , Vol. 13, No. 2, pp. 127-133, 2015.
  12. J. E. Falk and K. R. Hoffman, "A Successive Underestimation Method for Concave Minimization Problems", Math. Opns. Res. Vol. 1, pp. 251-259, 1976. https://doi.org/10.1287/moor.1.3.251
  13. R. M. Soland, "Optimal Facility Location with Concave Costs", Opns. Res. Vol. 22, pp. 373-382, 1974. https://doi.org/10.1287/opre.22.2.373
  14. K. G. Murty, Operations Reasearch: deterministic optimization models, Prentice-Hall, Inc, 1995.
  15. H. P. Benson, "Deterministic Algorithms for Constrained Concave Minimization: A Unified Critical Survey", Naval Research Logistics Quarterly Vol. 43, pp. 756-795, 1996.
  16. B. Kalantari and J. B. Rosen, "An Algorithm for Global Minimization of Linearly Constrained Concave Quadratic Functions", Math. Opns. Res. Vol. 12, pp. 544-560, 1987. https://doi.org/10.1287/moor.12.3.544
  17. H. P. Benson , "A Finite Algorithm for Concave Minimization over a Polyhedron", Naval Research Logistics Quarterly Vol. 32, pp. 165-177, 1985. https://doi.org/10.1002/nav.3800320119
  18. H. P. Benson and S. S. Erenguc, "A Finite Algorithm for Concave Minimization over a Polyhedron", IEEE, 2009.
  19. J. E. Falk and K. R. Hoffman, "Concave Minimization via Collapsing Polytopes", Opns. Res. Vol. 34, pp. 919-929, 1986. https://doi.org/10.1287/opre.34.6.919
  20. R. Horst, "An Algorithm for Nonconvex Programming Problems", Math. Prog. Vol. 10, pp. 312-321, 1976. https://doi.org/10.1007/BF01580678
  21. R. Horst, "A General Class of Branch-and-bound Methods in Global Optimization with Some New Approachs for Concave Minimization", J. Optim. Theory Appl. Vol. 51, pp. 271-291, 1986. https://doi.org/10.1007/BF00939825
  22. Sang Cho, "Blockly webc Programming Convergent Learning System", Journal of the Korea Convergence Society, Vol. 6, No. 1, pp. 23-28, 2015.
  23. Se-Ho Oh, "A Fuzzy Linear Programming Problem with Fuzzy Convergent Equality Constraints", Journal of the Korea Convergence Society, Vol. 6, No. 5, pp. 227-232, 2015. https://doi.org/10.15207/JKCS.2015.6.5.227