Many convex optimisation problems can be formulated as the minimisation of a convex function of a probability measure over a given set. Typical examples include determining the ellipsoid of minimum volume, or the smallest ball, containing a set of points. When it is known in advance that the optimal measure will be supported by a small number of points, it is advantageous to eliminate unnecessary points (candidates) in order to simplify the problem. Safe screening rules aim to eliminate such points: a rule defines a test to be applied to the candidates in order to eliminate those that are useless; a rule is safe when no point supporting an optimal measure is eliminated. The aim is to apply the screening rule during optimisation, regardless of the optimisation algorithm used. Usually, the efficiency of elimination increases when approaching the optimum, so the rule should be applied several (many) times and be as simple as possible. In addition to the construction of ellipsoids and balls of minimal volume, I will present the construction of screening rules for different criteria in optimal design of experiments, some of them with a link to (quadratic) Lasso. Parts of this work are based on collaborations with Radoslav Harman (Comenius University, Bratislava) and Guillaume Sagnol (TU Berlin).
