  We present a general technique, based on parametric search with some twist,
for solving a variety of optimization problems on a set of semi-algebraic
geometric objects of constant complexity. The common feature of these problems
is that they involve a `growth parameter' $r$ and a semi-algebraic predicate
$\Pi(o,o';r)$ of constant complexity on pairs of input objects, which depends
on $r$ and is monotone in $r$. One then defines a graph $G(r)$ whose edges are
all the pairs $(o,o')$ for which $\Pi(o,o';r)$ is true, and seeks the smallest
value of $r$ for which some monotone property holds for $G(r)$.
  Problems that fit into this context include (i) the reverse shortest path
problem in unit-disk graphs, recently studied by Wang and Zhao, (ii) the same
problem for weighted unit-disk graphs, with a decision procedure recently
provided by Wang and Xue, (iii) extensions of these problems to three and
higher dimensions, (iv) the discrete Fr\'echet distance with one-sided
shortcuts in higher dimensions, extending the study by Ben Avraham et al., (v)
perfect matchings in intersection graphs: given, e.g., a set of fat ellipses of
roughly the same size, find the smallest value $r$ such that if we expand each
of the ellipses by $r$, the resulting intersection graph contains a perfect
matching, (vi) generalized distance selection problems: given, e.g., a set of
disjoint segments, find the $k$'th smallest distance among the pairwise
distances determined by the segments, for a given (sufficiently small but
superlinear) parameter $k$, and (vii) the maximum-height independent towers
problem, in which we want to erect vertical towers of maximum height over a
1.5-dimensional terrain so that no pair of tower tips are mutually visible.
  We obtain significantly improved solutions for problems (i), (ii) and (vi),
and new efficient solutions to the other problems.
