$
$ #
$ echo "FILE_NAME_20180413" |
  perl -lne 'BEGIN {use Time::Piece}
             ($x = $_) =~ s/^.*_//;
             $t = Time::Piece->strptime($x, "%Y%m%d");
             $weekday = $t->wday;
             $delta = 3 - $weekday;
             if ($weekday < 3) { $delta -= 7 };
             $t += 24*60*60*$delta;
             print $t->strftime("%Y%m%d");
            '
20180410
  
$
$

$
$ # Generate some test data
$ perl -le 'BEGIN {use Time::Piece}
            sub get_prev_tuesday {
                ($x = shift) =~ s/^.*_//;
                $t = Time::Piece->strptime($x, "%Y%m%d");
                $weekday = $t->wday;
                $delta = 3 - $weekday;
                if ($weekday < 3) { $delta -= 7 };
                $t += 24*60*60*$delta;
                return $t->strftime("%Y%m%d");
            }
            # Generate a list of filenames for testing
            $start_date = Time::Piece->strptime("20180401", "%Y%m%d");
            foreach $offset (0..29) {
                $date = $start_date + 24*60*60*$offset;
                $file_name = "FILE_NAME_".$date->strftime("%Y%m%d");
                $val = get_prev_tuesday($file_name);
                print $file_name, " => ", $val;
            }
           '
FILE_NAME_20180401 => 20180327
FILE_NAME_20180402 => 20180327
FILE_NAME_20180403 => 20180403
FILE_NAME_20180404 => 20180403
FILE_NAME_20180405 => 20180403
FILE_NAME_20180406 => 20180403
FILE_NAME_20180407 => 20180403
FILE_NAME_20180408 => 20180403
FILE_NAME_20180409 => 20180403
FILE_NAME_20180410 => 20180410
FILE_NAME_20180411 => 20180410
FILE_NAME_20180412 => 20180410
FILE_NAME_20180413 => 20180410
FILE_NAME_20180414 => 20180410
FILE_NAME_20180415 => 20180410
FILE_NAME_20180416 => 20180410
FILE_NAME_20180417 => 20180417
FILE_NAME_20180418 => 20180417
FILE_NAME_20180419 => 20180417
FILE_NAME_20180420 => 20180417
FILE_NAME_20180421 => 20180417
FILE_NAME_20180422 => 20180417
FILE_NAME_20180423 => 20180417
FILE_NAME_20180424 => 20180424
FILE_NAME_20180425 => 20180424
FILE_NAME_20180426 => 20180424
FILE_NAME_20180427 => 20180424
FILE_NAME_20180428 => 20180424
FILE_NAME_20180429 => 20180424
FILE_NAME_20180430 => 20180424
  
$
$