Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization. In this paper, we extend theses concepts for nonlinear semidefinite programming. We define two sequential optimality conditions for nonlinear semidefinite programming. The first is a natural extension of the so-called Approximate-Karush–Kuhn–Tucker (AKKT), well known in nonlinear optimization. The second one, called Trace-AKKT, is more natural in the context of semidefinite programming as the computation of eigenvalues is avoided. We propose an augmented Lagrangian algorithm that generates these types of sequences and new constraint qualifications are proposed, weaker than previously considered ones, which are sufficient for the global convergence of the algorithm to a stationary point.

中文翻译：

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非线性半定规划的最优条件和全局收敛

顺序最优性条件在统一和扩展用于一般非线性优化的几类算法的全局收敛结果方面发挥了重要作用。在本文中，我们将这些概念扩展到非线性半定规划。我们为非线性半定规划定义了两个连续最优条件。第一个是所谓的 Approximate-Karush-Kuhn-Tucker (AKKT) 的自然扩展，在非线性优化中众所周知。第二个称为 Trace-AKKT，在半定规划的上下文中更自然，因为避免了特征值的计算。我们提出了一种增强拉格朗日算法来生成这些类型的序列，并提出了新的约束条件，比以前考虑的要弱，