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A new concave reformulation and its application in solving DC programming globally under uncertain environment

  • * Corresponding author: Yanjun Wang

    * Corresponding author: Yanjun Wang 

We gratefully acknowledge the valuable cooperation of Prof. R. Tyrrell Rockafellar(University of Washington). This research was supported by NSFC (11271243)

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  • In this paper, a new concave reformulation is proposed on a convex hull of some given points. Based on its properties, we attempt to solve DC Programming problems globally under uncertain environment by using Robust optimization method and CVaR method. A global optimization algorithm is developed for the Robust counterpart and CVaR model with two kinds of special convex hulls: simplex set and box set. The global solution is obtained by solving a sequence of convex relaxation programming on the original constraint sets or divided subsets with branch and bound method. Finally, numerical experiments are given for DC programs under uncertain environment with two kinds of constraints: simplex and box sets. Simulation results show the feasibility and efficiency of the proposed global optimization algorithm.

    Mathematics Subject Classification: Primary: 65K05; Secondary: 90C90.

    Citation:

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  • Figure 1.  h(x,ω0) and l(x,ω0) in box case

    Figure 2.  h(x,ω0) and l(x,ω0) in simplex case

    Figure 3.  Optimal value comparison of Robust Model and CVaR model

    Table 1.   

    α CPU(s) Step Nodes Opt Solution Opt Value Opt*
    0.70 356.11 87 69 (0.0000, 1.1250, 1.8750)T -2.2690 -2.2689
    0.75 956.11 163 101 (0.0000, 1.0547, 1.5703)T -2.1214 -2.1214
    0.80 847.73 163 106 (0.0000, 0.7969, 1.1191)T -1.9628 -1.9628
    0.85 516.92 132 106 (0.0000, 0.6211, 0.8789)T -1.7508 -1.7508
    0.90 518.31 122 107 (0.0000, 0.4569, 0.6797)T -1.5661 -1.5663
    0.95 321.43 78 68 (0.0000, 0.3580, 0.5326)T -1.4453 -1.4452
    0.97 143.17 31 27 (0.0000, 0.3387, 0.5038)T -1.4228 -1.4228
    0.98 160.39 30 24 (0.0104, 0.3437, 0.5156)T -1.4222 -1.4222
    0.99 167.81 31 26 (0.0023, 0.3281, 0.5000)T -1.4221 -1.4221
    Robust 218.18 52 51 (0.0080, 0.3515, 0.5002)T -1.4221 -1.4221
     | Show Table
    DownLoad: CSV

    Table 2.   

    α CPU(s) Step Nodes Opt Solution Opt Value Opt*
    0.70 264.90 24 22 (0.0000, 0.6719, 0.9844)T -2.2410 -2.2410
    0.75 105.19 25 29 (0.0000, 0.6641, 0.9844)T -2.1214 -2.1215
    0.80 217.65 24 19 (0.0000, 0.7187, 0.9844)T -1.9520 -1.9520
    0.85 516.92 55 40 (0.0000, 0.6250, 0.8750)T -1.7506 -1.7505
    0.90 350.26 39 34 (0.0000, 0.4531, 0.6741)T -1.5661 -1.5661
    0.95 208.85 38 32 (0.0000, 0.3580, 0.5326)T -1.4452 -1.4452
    0.97 143.17 31 27 (0.0000, 0.3437, 0.5156)T -1.4228 -1.4228
    0.98 167.81 30 26 (0.0104, 0.3437, 0.5156)T -1.4222 -1.4222
    0.99 160.39 31 24 (0.0023, 0.3281, 0.5000)T -1.4221 -1.4221
    Robust 216.98 35 27 (0.0080, 0.3515, 0.5002)T -1.4221 -1.4221
     | Show Table
    DownLoad: CSV

    Table 3.  Simplex feasible

    CPU Time(s) Nodes SEED
    RDC 91 11 1
    Floudas 227 42 1
    RDC 100 15 2
    Floudas 204 40 2
    RDC 120 25 3
    Floudas 280 75 3
     | Show Table
    DownLoad: CSV

    Table 4.  Box feasible

    CPU Time(s) Nodes SEED
    RDC 88 8 1
    Floudas 220 40 1
    RDC 105 12 2
    Floudas 207 36 2
    RDC 117 20 3
    Floudas 281 68 3
     | Show Table
    DownLoad: CSV
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