In this paper, we present a primal-dual interior point algorithm for semidefinite optimization problems based on a new class of kernel functions. These functions constitute a combination of the classic kernel function and a barrier term. We derive the complexity bounds for large and small-update methods respectively. We show that the best result of iteration bounds for large and small-update methods can be achieved, namely $$O\left({q\sqrt n {{\left({\log \sqrt n} \right)}^{{{q + 1} \over q}}}\log {n \over \varepsilon}} \right)$$ O ( q n ( log n ) q + 1 q log n ε ) for large-update methods and $$O\left({{q^{{3 \over 2}}}{{\left({\log \sqrt n} \right)}^{{{q + 1} \over q}}}\sqrt n \log {n \over \varepsilon}} \right)$$ O ( q 3 2 ( log n ) q + 1 q n log n ε ) for small-update methods. We test the efficiency and the validity of our algorithm by running some computational tests, then we compare our numerical results with results obtained by algorithms based on different kernel functions.

中文翻译：

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用于半定优化的有效参数化对数核函数

在本文中，我们基于一类新的核函数提出了一种用于半定优化问题的原始对偶内点算法。这些函数构成了经典核函数和屏障项的组合。我们分别推导出大和小更新方法的复杂度界限。我们表明可以实现大和小更新方法的迭代边界的最佳结果，即 $$O\left({q\sqrt n {{\left({\log \sqrt n} \right)}^{ {{q + 1} \over q}}}\log {n \over \varepsilon}} \right)$$ O ( qn ( log n ) q + 1 q log n ε ) 用于大更新方法和 $$ O\left({{q^{{3 \over 2}}}{{\left({\log \sqrt n} \right)}^{{{q + 1} \over q}}}\sqrt n \log {n \over \varepsilon}} \right)$$ O ( q 3 2 ( log n ) q + 1 qn log n ε ) 用于小更新方法。