  We investigate the use of optimization to compute bounds for extremal
performance measures. This approach takes a non-parametric viewpoint that aims
to alleviate the issue of model misspecification possibly encountered by
conventional methods in extreme event analysis. We make two contributions
towards solving these formulations, paying especial attention to the arising
tail issues. First, we provide a technique in parallel to Choquet's theory, via
a combination of integration by parts and change of measures, to transform
shape constrained problems (e.g., monotonicity of derivatives) into families of
moment problems. Second, we show how a moment problem cast over infinite
support can be reformulated into a problem over compact support with an
additional slack variable. In the context of optimization over tail
distributions, the latter helps resolve the issue of non-convergence of
solutions when using algorithms such as generalized linear programming. We
further demonstrate the applicability of this result to problems with
infinite-value constraints, which can arise in modeling heavy tails.
