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A Fast Scaling Algorithm for Minimizing Separable Convex Functions Subject to Chain Constraints

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  • Ravindra K. Ahuja

    (Industrial and Systems Engineering Department, University of Florida, Gainesville, Florida 32611)

  • James B. Orlin

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

Abstract

We consider the problem of minimizing (Sigma) j(in)N C j (x j ), subject to the following chain constraints x 1 (le) x 2 (le) x 3 (le) ... (le) x n , where C j (x j ) is a convex function of x j for each j (in) N = {1,2, ... ,n}. This problem is a generalization of the isotonic regression problems with complete order, an important class of problems in regression analysis that has been examined extensively in the literature. We refer to this problem as the generalized isotonic regression problem . In this paper, we focus on developing a fast-scaling algorithm to obtain an integer solution of the generalized isotonic regression problem. Let U denote the difference between an upper bound on an optimal value of x n and a lower bound on an optimal value of x 1 . Under the assumption that the evaluation of any function C j (x j ) takes O (1) time, we show that the generalized isotonic regression problem can be solved in O (n log U ) time. This improves by a factor of n the previous best running time of O (n 2 log U ) to solve the same problem. In addition, when our algorithm is specialized to the isotonic median regression problem (where C j (x j ) = c j |x j -a j |) for specified values of c j s and a j s , the algorithm obtains a real-valued optimal solution in O (n log n ) time. This time bound matches the best available time bound to solve the isotonic median regression problem, but our algorithm uses simpler data structures and may be easier to implement.

Suggested Citation

  • Ravindra K. Ahuja & James B. Orlin, 2001. "A Fast Scaling Algorithm for Minimizing Separable Convex Functions Subject to Chain Constraints," Operations Research, INFORMS, vol. 49(5), pages 784-789, October.
  • Handle: RePEc:inm:oropre:v:49:y:2001:i:5:p:784-789
    DOI: 10.1287/opre.49.5.784.10601
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    References listed on IDEAS

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    1. Menendez, J. A. & Salvador, B., 1987. "An algorithm for isotonic median regression," Computational Statistics & Data Analysis, Elsevier, vol. 5(4), pages 399-406, September.
    2. Stromberg, Ulf, 1991. "An algorithm for isotonic regression with arbitrary convex distance function," Computational Statistics & Data Analysis, Elsevier, vol. 11(2), pages 205-219, March.
    3. Nilotpal Chakravarti, 1992. "Isotonic median regression for orders representable by rooted trees," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(5), pages 599-611, August.
    4. Nilotpal Chakravarti, 1989. "Isotonic Median Regression: A Linear Programming Approach," Mathematics of Operations Research, INFORMS, vol. 14(2), pages 303-308, May.
    5. Robin Roundy, 1986. "A 98%-Effective Lot-Sizing Rule for a Multi-Product, Multi-Stage Production / Inventory System," Mathematics of Operations Research, INFORMS, vol. 11(4), pages 699-727, November.
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    Cited by:

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    2. Cui, Zhenyu & Lee, Chihoon & Zhu, Lingjiong & Zhu, Yunfan, 2021. "Non-convex isotonic regression via the Myersonian approach," Statistics & Probability Letters, Elsevier, vol. 179(C).
    3. T. Ibaraki & S. Imahori & M. Kubo & T. Masuda & T. Uno & M. Yagiura, 2005. "Effective Local Search Algorithms for Routing and Scheduling Problems with General Time-Window Constraints," Transportation Science, INFORMS, vol. 39(2), pages 206-232, May.
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    6. Stout, Quentin F., 2008. "Unimodal regression via prefix isotonic regression," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 289-297, December.
    7. Wojciech Gamrot, 2013. "Maximum likelihood estimation for ordered expectations of correlated binary variables," Statistical Papers, Springer, vol. 54(3), pages 727-739, August.
    8. Nguyen, Kien Trung & Hung, Nguyen Thanh, 2021. "The minmax regret inverse maximum weight problem," Applied Mathematics and Computation, Elsevier, vol. 407(C).
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    11. Ravindra K. Ahuja & James B. Orlin, 2001. "Inverse Optimization," Operations Research, INFORMS, vol. 49(5), pages 771-783, October.

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