function?[hv]?=?approximate_hypervolume_ms（F?ub?samples）
%?[hv]?=?approximate_hypervolume_ms（F?ub?samples）
%
%?Computes?the?hypervolume?（or?Lebesgue?measure）?of?the?M?x?l
%?matrix?F?of?l?vectors?of?M?objective?function?values?by?means
%?of?a?Monte-Carlo?approximation?method.
%
%?IMPORTANT:
%???Considers?Minimization?of?the?objective?function?values!
%
%?Input:
%?-?F??????????????-?An?M?x?l?matrix?where?each?of?the?l?columns?
%????????????????????represents?a?vector?of?M?objective?function?values
%????????????????????（note?that?this?function?assumes?that?all?
%????????????????????solutions?of?F?are?non-dominated）.
%?-?ub?????????????-?Optional:?Upper?bound?reference?point?（default:
%????????????????????the?boundary?point?containing?the?maximum?of?F
%????????????????????for?each?objective）.
%?-?samples????????-?Optional:?The?number?of?samples?used?for?the?Monte-
%????????????????????Carlo?approximation?（default:?100000）.
%
%?Output:
%?-?hv?????????????-?The?hypervolume?（or?lebesgue?measure）?of?F.
%
%?Author:?Johannes?W.?Kruisselbrink
%?Last?modified:?March?17?2011

	if?（nargin?	if?（nargin?
	[M?l]?=?size（F）;
	samples?=?100000;
	lb?=?min（F‘）‘;

	F_samples?=?repmat（lb1samples）?+?rand（Msamples）?.*?repmat（（ub?-?lb）1samples）;
	is_dominated_count?=?0;
	for?i?=?1:samples
		for?j?=?1:l
			if?（dominates（F（:j）?F_samples（:i）））
				is_dominated_count?=?is_dominated_count?+?1;
				break;
			end
		end
	end

	hv?=?prod（ub?-?lb）?*?（is_dominated_count?/?samples）;

end

?屬性????????????大小?????日期????時(shí)間???名稱
-----------?---------??----------?-----??----
?????文件????????1592??2011-03-17?17:30??approximate_hypervolume_ms.m
?????文件????????1219??2011-03-17?17:30??compute_crowding_distances.m
?????文件?????????912??2011-03-17?17:31??compute_dominating_solutions.m
?????文件????????3115??2011-03-17?17:30??compute_hypervolume_contributions.m
?????文件????????1260??2011-03-17?17:30??compute_reference_point.m
?????文件?????????999??2011-03-25?19:23??crowding_distance_selection.m
?????文件?????????714??2011-03-17?17:30??dominates.m
?????文件????????1672??2011-03-17?17:30??generate_hv_testset.m
?????文件????????4042??2011-12-08?19:36??lebesgue_measure.m
?????文件????????1351??2011-03-17?17:30??non_dominated_front.m
?????文件?????????858??2011-03-17?17:30??non_dominated_sorting.m
?????文件????????1079??2011-12-08?16:24??test_lebesgue_measure.m
?????文件?????????694??2011-03-17?17:30??weakly_dominates.m
?????文件????????1308??2014-02-12?13:44??license.txt