Global optimization for robust control synthesis based on the matrix product eigenvalue problem (Q2755397)

The authors of this paper propose a new framework for solving robost control synthesis problems, namely the matrix product eigenvalue problem (MPEP), consisting of minimizing the maximum eigenvalue of the product of two block-diagonal positive-definite symmetric matrices under convex constraints. If \({\mathcal B}{\mathcal D}\) is a set of block-diagonal matrices and \({\mathcal Z}_c\) is a closed bounded convex subset of \({\mathcal B}{\mathcal D}\times{\mathcal B}{\mathcal D}\), then the MPEP is formulated as follows: NEWLINE\[NEWLINE\text{minimize }\lambda^{1/2}_{\max}(\Sigma,\Lambda),\quad\text{subject to }(\Sigma,\Lambda)\in{\mathcal Z}_c,NEWLINE\]NEWLINE where \(\lambda^{1/2}_{\max}(\cdot)\) denotes the square root of the maximum eigenvalue and \({\mathcal Z}_c\) satisfies the following monotonicity property NEWLINE\[NEWLINE(\widehat\Sigma,\widehat\Lambda)\in{\mathcal Z}\Rightarrow \{(\Sigma,\Lambda)\in {\mathcal B}{\mathcal D}\times{\mathcal B}{\mathcal D}\mid\Sigma\geq\widehat\Sigma, \Lambda\geq\widehat\Lambda\}\subset {\mathcal Z}_c.NEWLINE\]NEWLINE The authors do not intend to find the exact optimal value for MPEP, but instead give an algorithm to find a suboptimal value within any given tolerance \(\varepsilon\). For any given \(\varepsilon\in (0,1)\), an \(\varepsilon\)-global optimization algorithm is achieved by solving \(N\) linear matrix inequality problems satisfying \(N\leq N_{\max}\), where \(N_{\max}= O((1/\varepsilon)^{q/2})\) and \(q\) denotes a number of scalar parameters in \(\Sigma\in{\mathcal B}{\mathcal D}\). For a fixed number \(q\) the computational complexity is polynomial in the inverse of the tolerance \(1/\varepsilon\).NEWLINENEWLINENEWLINEA comparison is made with elementwise bounding for the bilinear matrix inequality optimization problem and matrix-based bounding for the MPEP.