Nondifferentiable optimization algorithm for designing control systems having singular value inequalities (Q788989)

A decomposition nondifferentiable algorithm is proposed for the optimization problem f(x)\(\to \min\) subject to the constraints \(g^ j(x)\leq 0\), \(j=1,...,m_ g\); \(\max_{t\in T}h^ k(x,t)\leq 0\), \(k=1,...,m_ h\) and \(p_ j(\omega)\leq s^ i_ j(x,\omega)\leq q_ j(\omega),\quad i=1,...,m,\quad j=1,...,L,\quad \omega \in \Omega.\) Here f(.), \(g^ j(.)\), \(h^ k(.,.)\) are differentiable and \(p_ j(.)\), q(.) are continuous functions. The numbers \(s^ i_ j(x,\omega)\) are the singular values of an \(m\times m\) complex valued transfer matrix \(G_ j(x,\omega)\) and \(\Omega\) is a frequency interval. The algorithm itself is demonstrated on the simpler case f(x)\(\to \min\), \(p(\omega)\leq s^ i(x,\omega)\leq q(\omega),\quad i=1,...,m,\quad \omega \in \Omega,\) where \(s^ i(x,\omega)\) are the singular values of a single matrix. The decomposition technique reduces the last problem to a sequence of problems in which \(\omega\) varies over a finite subset of \(\Omega\).