  The theory of majorizing measures, extensively developed by Fernique,
Talagrand and many others, provides one of the most general frameworks for
controlling the behavior of stochastic processes. In particular, it can be
applied to derive quantitative bounds on the expected suprema and the degree of
continuity of sample paths for many processes.
  One of the crowning achievements of the theory is Talagrand's tight
alternative characterization of the suprema of Gaussian processes in terms of
majorizing measures. The proof of this theorem was difficult, and thus
considerable effort was put into the task of developing both shorter and easier
to understand proofs. A major reason for this difficulty was considered to be
theory of majorizing measures itself, which had the reputation of being opaque
and mysterious. As a consequence, most recent treatments of the theory
(including by Talagrand himself) have eschewed the use of majorizing measures
in favor of a purely combinatorial approach (the generic chaining) where
objects based on sequences of partitions provide roughly matching upper and
lower bounds on the desired expected supremum.
  In this paper, we return to majorizing measures as a primary object of study,
and give a viewpoint that we think is natural and clarifying from an
optimization perspective. As our main contribution, we give an algorithmic
proof of the majorizing measures theorem based on two parts: (1) We make the
simple (but apparently new) observation that finding the best majorizing
measure can be cast as a convex program. This also allows for efficiently
computing the measure using off-the-shelf methods from convex optimization. (2)
We obtain tree-based upper and lower bound certificates by rounding, in a
series of steps, the primal and dual solutions to this convex program.
  [...]
