一种序列半定规划方法及其在无源降阶建模中的应用
A sequential semidefinite programming method and an application in passive reduced-order modeling
AI总结：
提出一种序列半定规划（SSP）方法，通过对称化处理互补条件并转化为线性半定子问题，用于求解含非线性半定约束的非线性规划，并证明其局部二次收敛性，最后应用于无源降阶建模中的非线性半定规划问题。
摘要：
我们考虑具有非线性半定约束的非线性规划的求解。由于需要有效利用半正定矩阵锥，求解此类非线性半定规划比标准非线性规划更为复杂。特别地，需要选择合适的对称化过程来线性化互补条件。对称化过程的选择可以很自然地转移到某些线性半定子问题中，从而简化为一个经过充分研究的问题。由此产生的序列半定规划（SSP）方法是标准非线性规划中著名的SQP方法的推广。我们给出了非线性半定规划的灵敏度结果，并基于该结果给出了SSP方法局部二次收敛的自包含证明。我们还描述了在无源降阶建模中出现的一类非线性半定规划，并报告了将SSP方法应用于该类问题的数值实验结果。
英文摘要：
We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. In particular, a suitable symmetrization procedure needs to be chosen for the linearization of the complementarity condition. The choice of the symmetrization procedure can be shifted in a very natural way to certain linear semidefinite subproblems, and can thus be reduced to a well-studied problem. The resulting sequential semidefinite programming (SSP) method is a generalization of the well-known SQP method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class.
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