<?php

$out_data = array (
    array("01_composita","Composita"),
    array("02_composita_p","Composita Plus"),
    array("03_contemporanea","Contemporanea"),
    array("04_epc","Epc"),
    array("05_hotel","Hotel"),
    array("06_plain_filomuro","Plait Filomuro"),
    array("07_pds","Pds"),
    array("08_soluzione","Soluzione"),
    array("09_teknoeuropa","Teknoeuropa"),
    array("10_telemako","Telemako")
);

$tmp = array_search(array("05_hotel","Hotel"), $out_data);

var_dump ($tmp); // выдает 4

?>

$needle = '05_hotel';

$result = false;
foreach ($out_data as $key => $val) {
    if ($val[0] == $needle) { $result = $key; break; }
}

var_dump($result);

<?php
$out_data = array (
    array("01_composita","Composita"),
    array("02_composita_p","Composita Plus"),
    array("03_contemporanea","Contemporanea"),
    array("04_epc","Epc"),
    array("05_hotel","Hotel"),
    array("06_plain_filomuro","Plait Filomuro"),
    array("07_pds","Pds"),
    array("08_soluzione","Soluzione"),
    array("09_teknoeuropa","Teknoeuropa"),
    array("10_telemako","Telemako"),
    array("08_soluzione","Telemako")
);

//Ищем
$needle = "08_soluzione";

//Собственно поиск
$result = array_filter($out_data, function($innerArray){
    global $needle;
    //return in_array($needle, $innerArray);    //Поиск по всему массиву
    return ($innerArray[0] == $needle); //Поиск по первому значению
});

//Результат
echo '<pre>'.print_r($result, true).'</pre>';
?>