Jordan Algebras are an important tool for dealing with semidefinite programming and optimization over symmetric cones in general. In this paper, a judicious use of Jordan Algebras in the context of sequential optimality conditions is done in order to generalize the global convergence theory of an Augmented Lagrangian method for nonlinear semidefinite programming. An approximate complementarity measure in this context is typically defined in terms of the eigenvalues of the constraint matrix and the eigenvalues of an approximate Lagrange multiplier. By exploiting the Jordan Algebra structure of the problem, we show that a simpler complementarity measure, defined in terms of the Jordan product, is stronger than the one defined in terms of eigenvalues. Thus, besides avoiding a tricky analysis of eigenvalues, a stronger necessary optimality condition is presented. We then prove the global convergence of an Augmented Lagrangian algorithm to this improved necessary optimality condition. The results are also extended to an interior point method. The optimality conditions we present are sequential ones, and no constraint qualification is employed; in particular, a global convergence result is available even when Lagrange multipliers are unbounded.

通常，约旦代数是处理对称圆锥上的半定规划和优化的重要工具。在本文中，在顺序最优性条件下明智地使用了约旦代数，以推广用于非线性半定规划的增强拉格朗日方法的全局收敛理论。在这种情况下，通常根据约束矩阵的特征值和近似拉格朗日乘数的特征值来定义近似互补性度量。通过利用问题的约旦代数结构，我们表明，根据约旦乘积定义的一种更简单的互补性度量要强于根据特征值定义的一种互补性度量。因此，除了避免对特征值进行棘手的分析外，提出了一个更强的必要最优性条件。然后，我们证明了扩展拉格朗日算法对该改进的必要最优性条件的全局收敛性。结果也扩展到内点法。我们提出的最优条件是顺序条件，没有采用约束条件。特别是，即使拉格朗日乘数不受限制，也可以获得全局收敛结果。