Summary: We consider two problems which can be posed as spectral radius minimization problems. Firstly, we consider the fastest average agreement problem on multi-agent networks adopting a linear information exchange protocol. Mathematically, this problem can be cast as finding an optimal \(W\in\mathbb R^{n\times n}\) such that \(x(k+1)=Wx(k)\), \(W{\mathbf 1}={\mathbf 1}\), \({\mathbf 1}^TW={\mathbf 1}^T\) and \(W\in{\mathcal S}({\mathcal E})\). Here, \(x(k)\in\mathbb R^n\) is the value possessed by the agents at the \(k\)th time step, \({\mathbf 1}\in\mathbb R^n\) is an all-one vector and \({\mathcal S}({\mathcal E})\) is the set of real matrices in \(\mathbb R^{n\times n}\) with zeros at the same positions specified by a network graph \({\mathcal G}({\mathcal V},{\mathcal E})\), where \({\mathcal V}\) is the set of agents and \({\mathcal E}\) is the set of communication links between agents. The optimal \(W\) is such that the spectral radius \(\rho(W-{\mathbf{11}}^T/n)\) is minimized. To this end, we consider two numerical solution schemes: one using the \(q\)th-order Spectral Norm (2-norm) Minimization (\(q\)-SNM) and the other Gradient Sampling (GS), inspired by the methods proposed in [J. V. Burke, A. S. Lewis and M. L. Overton, Linear Algebra Appl. 351–352, 117–145 (2002; Zbl 1005.65041); L. Xiao and S. Boyd, Syst. Control Lett. 53, No. 1, 65–78 (2004; Zbl 1157.90347)]. In this context, we theoretically show that when \({\mathcal E}\) is symmetric, i.e., no information flow from the \(i\)th to the \(j\)th agent implies no information flow from the \(j\)th to the \(i\)th agent, the solution \(W_s^{(1)}\) from the 1-SNM method can be chosen to be symmetric and \(W_s^{(1)}\) is a local minimum of the function \(\rho(W-{\mathbf{11}}^T/n)\). Numerically, we show that the \(q\)-SNM method performs much better than the GS method when \({\mathcal E}\) is not symmetric. Secondly, we consider the famous static output feedback stabilization problem, which is considered to be a hard problem (some think NP-hard): for a given linear system \((A,B,C)\), find a stabilizing control gain \(K\) such that all the real parts of the eigenvalues of \(A+BKC\) are strictly negative. In spite of its computational complexity, we show numerically that \(q\)-SNM successfully yields stabilizing controllers for several benchmark problems with little effort.