In this paper, we are concerned with efficient algorithms for solving the least squares semidefinite programming which contains many equalities and inequalities constraints. Our proposed method is built upon its dual formulation and is a type of active-set approach. In particular, by exploiting the nonnegative constraints in the dual form, our method first uses the information from the Barzlai–Borwein step to estimate the active/inactive sets, and within an adaptive framework, it then accelerates the convergence by switching the L-BFGS iteration and the semi-smooth Newton iteration dynamically. We show the global convergence under mild conditions, and furthermore, the local quadratic convergence under the additional nondegeneracy condition. Various types of synthetic as well as real-world examples are tested, and preliminary but promising numerical experiments are reported.

在本文中，我们关注用于求解包含许多等式和不等式约束的最小二乘半定规划的有效算法。我们提出的方法建立在其双重表述之上，是一种主动设定方法。特别地，通过利用对偶形式的非负约束，我们的方法首先使用来自Barzlai–Borwein步骤的信息来估计有效/无效集，然后在自适应框架内，通过切换L-BFGS来加速收敛。动态迭代和半平滑牛顿迭代。我们显示了在温和条件下的全局收敛性，此外，还显示了在附加非退化条件下的局部二次收敛性。测试了各种类型的合成示例和实际示例，