Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems
Let \(S^n\) denote the Euclidean space of all \((n \times n)\)-symmetric matrices provided with the Frobenius norm. Let \(\lambda_1(A) \geq \lambda_2(A) \geq \dots \geq \lambda_n(A)\) be the eigenvalues of \(A\) arranged in the descending order and \[ \sigma_k(A) = \sum_{i = 1}^{k} \lambda_i(A) . \] Let \(A: \mathbb R^m \to S^n\) be a continuously differentiable mapping, i.e., \(A(x)\) is an \((n \times n)\)-symmetric matrix for any \(x \in \mathbb R^m\). The authors study the following optimization problem: \[ \min_{x \in \mathbb R^m} (f(x) + \sigma_k(A(x))),\tag{1} \] where \(f: \mathbb R^m \to \mathbb R\) is a second-order continuously differentiable function and the mapping \(A: \mathbb R^n \to S^n\) is second-order continuously differentiable. The authors derive second-order conditions, which ensure the existence of augmented Lagrange multipliers for the optimization problem (1).
eigenvalue composite optimization problems
scientific article; zbMATH DE number 6484157
scientific article