  We study quasi-convex optimization problems, where only a subset of the
constraints can be sampled, and yet one would like a probabilistic guarantee on
the obtained solution with respect to the initial (unknown) optimization
problem. Even though our results are partly applicable to general quasi-convex
problems, in this work we introduce and study a particular subclass, which we
call "quasi-linear problems". We provide optimality conditions for these
problems. Thriving on this, we extend the approach of chance-constrained convex
optimization to quasi-linear optimization problems. Finally, we show that this
approach is useful for the stability analysis of black-box switched linear
systems, from a finite set of sampled trajectories. It allows us to compute
probabilistic upper bounds on the JSR of a large class of switched linear
systems.
