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Complexity analysis of an interior-point algorithm for linear optimization based on a new parametric kernel function with a double barrier term

  • Corresponding author: Ayache Benhadid

    Corresponding author: Ayache Benhadid 
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  • Kernel functions play an important role in the complexity analysis of the interior point methods (IPMs) for linear optimization (LO). In this paper, an interior-point algorithm for LO based on a new parametric kernel function is proposed. By means of some simple analysis tools, we prove that the primal-dual interior-point algorithm for solving LO problems meets $ O\left(\sqrt{n} \log(n) \log(\frac{n}{\varepsilon}) \right) $, iteration complexity bound for large-update methods with the special choice of its parameters.

    Mathematics Subject Classification: Primary: 90C05, 90C51; Secondary: 90C31.

    Citation:

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  • Figure 1.  Generic algorithm

    Table 4.  Comparison for $ p = 3, n = 6 $

    Kernel functions Large update Small update $ \theta $ Inner It. Outer It. Reference
    $ \frac{1}{2}(t^{2}-1)-\log(t) $ $ {\bf{O}}\left(n \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 15079 116 [12]
    0.98 929 5
    $ \frac{t^{2}-1-\log(t)}{2}+\frac{t^{1-q}-1}{2(q-1)} $ $ {\bf{O}}\left(qn^{\frac{q+1}{2q}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(q^{2}\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 36917 116 [7]
    0.98 2432 5
    $ \frac{1}{2}(t-\frac{1}{t})^{2} $ $ {\bf{O}}\left(n^{\frac{2}{3}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 11550 116 [17]
    0.98 476 5
    $ \frac{1}{2}(t^{2}-1)+\frac{t^{1-q}-1}{q-1} $ $ {\bf{O}}\left(qn^{\frac{q+1}{2q}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(q^{2}\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 9053 116 [19]
    0.98 553 5
    $ \psi_{m}(t) $ $ {\bf{O}}\left(mn^{\frac{2m+1}{4m}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(m^{2}\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 3475 116
    0.98 64 5
     | Show Table
    DownLoad: CSV

    Table 5.  Comparison for $ p = 5, n = 10 $

    Kernel functions Large update Small update $ \theta $ Inner It. Outer It. Reference
    $ \frac{1}{2}(t^{2}-1)-\log(t) $ $ {\bf{O}}\left(n \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 21696 121 [12]
    0.98 1423 5
    $ \frac{t^{2}-1-\log(t)}{2}+\frac{t^{1-q}-1}{2(q-1)} $ $ {\bf{O}}\left(qn^{\frac{q+1}{2q}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(q^{2}\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 37948 121 [7]
    0.98 2137 5
    $ \frac{1}{2}(t-\frac{1}{t})^{2} $ $ {\bf{O}}\left(n^{\frac{2}{3}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 16713 121 [17]
    0.98 606 5
    $ \frac{1}{2}(t^{2}-1)+\frac{t^{1-q}-1}{q-1} $ $ {\bf{O}}\left(qn^{\frac{q+1}{2q}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(q^{2}\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 12242 121 [19]
    0.98 651 5
    $ \psi_{m}(t) $ $ {\bf{O}}\left(mn^{\frac{2m+1}{4m}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(m^{2}\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 5754 121
    0.98 104 5
     | Show Table
    DownLoad: CSV

    Table 6.  Comparison for $ p = 10, n = 20 $

    Kernel functions Large update Small update $ \theta $ Inner It. Outer It. Reference
    $ \frac{1}{2}(t^{2}-1)-\log(t) $ $ {\bf{O}}\left(n \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 38278 128 [12]
    0.98 2613 5
    $ \frac{t^{2}-1-\log(t)}{2}+\frac{t^{1-q}-1}{2(q-1)} $ $ {\bf{O}}\left(qn^{\frac{q+1}{2q}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(q^{2}\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 51198 128 [7]
    0.98 2307 5
    $ \frac{1}{2}(t-\frac{1}{t})^{2} $ $ {\bf{O}}\left(n^{\frac{2}{3}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 25553 128 [17]
    0.98 857 5
    $ \frac{1}{2}(t^{2}-1)+\frac{t^{1-q}-1}{q-1} $ $ {\bf{O}}\left(qn^{\frac{q+1}{2q}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(q^{2}\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 21549 128 [19]
    0.98 897 5
    $ \psi_{m}(t) $ $ {\bf{O}}\left(mn^{\frac{2m+1}{4m}} \log\left( \frac{n}{\epsilon} \right)\right) $ $ {\bf{O}}\left(m^{2}\sqrt{n}\log\left( \frac{n}{\epsilon} \right) \right) $ 0.10 13054 128
    0.98 269 5
     | Show Table
    DownLoad: CSV
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