论文标题
在非线性半限定和二阶锥体编程的弱二阶最优条件下
On the weak second-order optimality condition for nonlinear semidefinite and second-order cone programming
论文作者
论文摘要
非线性圆锥编程问题的二阶必需最佳条件通常是在非高度和严格的互补性下建立的。在本文中，假设罗宾逊的约束资格和弱恒定的级别型属性，则为两类非线性圆锥问题（即半决赛和二阶编程编程）建立了这种类型的条件，这些条件严格比非非等级弱弱。我们的方法是通过基于惩罚的策略来完成的，该战略旨在为一阶和二阶算法提供强大的全球收敛结果。由于我们没有假设严格的互补性，因此临界锥不会减少到子空间，因此，我们所获得的二阶条件是根据临界锥的直率空间来定义的。在非线性编程的情况下，这种条件还原为算法实践中广泛用作二阶平稳性测量的标准二阶条件。
Second-order necessary optimality conditions for nonlinear conic programming problems that depend on a single Lagrange multiplier are usually built under nondegeneracy and strict complementarity. In this paper we establish a condition of such type for two classes of nonlinear conic problems, namely semidefinite and second-order cone programming, assuming Robinson's constraint qualification and a weak constant rank-type property which are, together, strictly weaker than nondegeneracy. Our approach is done via a penalty-based strategy, which is aimed at providing strong global convergence results for first- and second-order algorithms. Since we are not assuming strict complementarity, the critical cone does not reduce to a subspace, thus, the second-order condition we arrive at is defined in terms of the lineality space of the critical cone. In the case of nonlinear programming, this condition reduces to the standard second-order condition widely used as second-order stationarity measure in the algorithmic practice.
