Last Updated：2022/12/24
One of the main reasons for the interest in convex envelopes is the fact that the set of global minimum points of f on S is contained in the set of global minimum points of conv_S(f) on S and the two minimum values coincide (see, e.g., [11, 17]). Hence, if the convex envelope were efficiently computable or available in closed form, one could replace the nonconvex problem of minimizing f on S with the convex problem of minimizing f on conv_S(f).
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One
of
the
main
reasons
for
the
interest
in
convex
envelopes
is
the
fact
that
the
set
of
global
minimum
points
of
f
on
S
is
contained
in
the
set
of
global
minimum
points
of
conv_S(f)
on
S
and
the
two
minimum
values
coincide
(see,
e.g.,
[11,
17]).
Hence,
if
the
convex
envelope
were
efficiently
computable
or
available
in
closed
form,
one
could
replace
the
nonconvex
problem
of
minimizing
f
on
S
with
the
convex
problem
of
minimizing
f
on
conv_S(f).
