Large-Scale Minimization of the Pseudospectral Abscissa

TL;DR

This work tackles the large-scale problem of minimizing the $\epsilon$-pseudospectral abscissa $\alpha_{\epsilon}(A(x))$ for an analytic, affine matrix-valued function $A(x)$. A one-sided subspace framework reduces $A(x)$ to a small, evolving subspace $A^V(x)$, solving a sequence of reduced nonconvex minimax problems and enriching the subspace with right singular vectors of $A(x)-zI$, thereby achieving Hermite interpolation between full and reduced problems. The authors prove global convergence and, for a single parameter, superlinear convergence of the reduced problems to the global minimizer, with extensions to multi-parameter cases and a real-pseudospectral extension discussed. Numerical results on synthetic tests and COMP$\ell_{e}ib$ benchmarks demonstrate the method’s effectiveness for matrices with thousands of degrees of freedom, including large-scale H-infinity and stability-related problems, and provide a publicly available MATLAB implementation. The work highlights robust stability and transient-growth insights gained by optimizing the pseudospectral abscissa, offering a scalable approach for parameter-dependent controller design and reliability analysis in large systems.

Abstract

This work concerns the minimization of the pseudospectral abscissa of a matrix-valued function dependent on parameters analytically. The problem is motivated by robust stability and transient behavior considerations for a linear control system that has optimization parameters. We describe a subspace procedure to cope with the setting when the matrix-valued function is of large size. The proposed subspace procedure solves a sequence of reduced problems obtained by restricting the matrix-valued function to small subspaces, whose dimensions increase gradually. It possesses desirable features such as a superlinear convergence exhibited by the decay in the errors of the minimizers of the reduced problems. In mathematical terms, the problem we consider is a large-scale nonconvex minimax eigenvalue optimization problem such that the eigenvalue function appears in the constraint of the inner maximization problem. Devising and analyzing a subspace framework for the minimax eigenvalue optimization problem at hand with the eigenvalue function in the constraint require special treatment that makes use of a Lagrangian and dual variables. There are notable advantages in minimizing the pseudospectral abscissa over maximizing the distance to instability or minimizing the $\mathcal{H}_\infty$ norm; the optimized pseudospectral abscissa provides quantitative information about the worst-case transient growth, and the initial guesses for the parameter values to optimize the pseudospectral abscissa can be arbitrary, unlike the case to optimize the distance to instability and $\mathcal{H}_\infty$ norm that would normally require initial guesses yielding asymptotically stable systems.

Figures (5)

• Figure 1: The progress of the subspace framework to minimize $\alpha_\epsilon(A(x))$ over $x \in [-0.3, 0.2]$ for the example $A(x)$ in Mengi2018 of order $n = 400$, and with $\epsilon$ equal to the computed value of the maximum of ${\mathcal{D}}(A(x))$ over $x \in [-0.3,0.2]$. The black circles mark the interpolation points, while the crosses mark the global minimizers of $\alpha_\epsilon^{{\mathcal{V}}_1}(x)$ and $\alpha_\epsilon^{{\mathcal{V}}_2}(x)$.
• Figure 2: The plots of $\alpha_{\epsilon}(x)$ as a function of $x$ for the NN18 example in the COMP$l_e ib$ collection. The right-hand plot is a zoomed version of the left-hand plot focusing on $x \in[-1, 0.02]$.
• Figure 3: The plots of $\alpha_{\epsilon}(x)$ for $\epsilon = 0.2$ as a function of $x$ for the HF1 example from the COMP$l_e ib$ collection. In each plot, the circle marks the computed global minimizer. (Left) Contour diagram of $\alpha_{\epsilon}(x)$. (Right) 3-dimensional plot of the map $x \mapsto \alpha_{\epsilon}( x )$.
• Figure 4: The plots of $\alpha_{\epsilon}(\widetilde{x}^{(j)})$ for $\epsilon = 0.2$ for the HF2D2 example for randomly chosen five thousand points $\widetilde{x}^{(j)} \in [-1,1]^6$ for $j = 1, \dots , 5000$. Every dot in the plot corresponds to $(j, \alpha_{\epsilon}(\widetilde{x}^{(j)}))$ for some $j$. The horizontal line at the bottom is $y = \alpha_{\epsilon}(x_\ast) = -0.4124020$, where $x_\ast$ is the computed global minimizer.
• Figure 5: The plots of the boundaries of the rightmost components of $\Lambda_{\epsilon}(0)$ and $\Lambda_{\epsilon}(x_\ast)$ for the HF2D2 example. The crosses mark the rightmost two eigenvalues of $A$, while the vertical line represents the imaginary axis.

Theorems & Definitions (19)

• Lemma 2.2: Monotonicity
• Lemma 2.3
• Theorem 2.4
• Definition 2.5
• Theorem 2.6: Derivatives of the Pseudospectral Abscissa Function
• Definition 2.7
• Theorem 2.8: Derivatives of the Reduced Pseudospectral Abscissa Function
• Lemma 2.9: Hermite Interpolation
• Proof 1
• Theorem 3.1