  In this paper, we further develop the approach, originating in [GJN], to
"computation-friendly" hypothesis testing via Convex Programming. Most of the
existing results on hypothesis testing aim to quantify in a closed analytic
form separation between sets of distributions allowing for reliable decision in
precisely stated observation models. In contrast to this descriptive (and
highly instructive) traditional framework, the approach we promote here can be
qualified as operational -- the testing routines and their risks are yielded by
an efficient computation. All we know in advance is that, under favorable
circumstances, specified in [GJN], the risk of such test, whether high or low,
is provably near-optimal under the circumstances. As a compensation for the
lack of "explanatory power," this approach is applicable to a much wider family
of observation schemes and hypotheses to be tested than those where "closed
form descriptive analysis" is possible.
  In the present paper our primary emphasis is on computation: we make a step
further in extending the principal tool developed in the cited paper -- testing
routines based on affine detectors -- to a large variety of testing problems.
The price of this development is the loss of blanket near-optimality of the
proposed procedures (though it is still preserved in the observation schemes
studied in [GJN], which now become particular cases of the general setting
considered here).
  [GJN]: Goldenshluger, A., Juditsky, A., Nemirovski, A. "Hypothesis testing by
convex optimization," Electronic Journal of Statistics 9(2), 2015
