This directory contains a Matlab 4.2 implementation of Normal-Boundary
Intersection (NBI) as developed by Indraneel Das and John Dennis in 1996.
The command "help NBI" at the matlab prompt gives details on using this code.
You can also visit http://www.owlnet.rice.edu/~indra/NBIhomepage.html for help.

The NBI subproblem 
			max    t
		 s.t.      Phi*beta + t*n_hat = F(x)
				h(x) = 0
				g(x) <= 0
			    a <= x <=  b

	is solved for various settings of the barycentric coordinates beta.

	[ see Dept. of CAAM, Rice University TR 96-19 for details ]

User-defined files required are:

FUN.m:		function [fdummy, h, g] = FUN(x)
	
		Evaluates equality constraints h(x) = 0
		and inequality constraints g(x) <= 0 at a given x.
		fdummy is just a dummy output which doesn't enter calculations.

Fns.m:		function [F] = Fns(x)
		Returns the objective vector F(x) evaluated at x.

FUN1.m:		A file needed for finding the individual minima of the functions.

	Templates of the above are provided in the directory for a 5
	variable, 3 objective problem with 1 inequality and 2 equality
	constraints. The values of the relevant variables as they appear
	in the arguments of the function `NBI' in its Matlab help message 
	can be loaded using `load vars.mat' from the Matlab prompt. 
	
		The above files can also be generated internally by
	calling 'createshootfiles' from the matlab prompt and then entering
	the functions and constraints as strings at the prompt.

shadow_data.m 	function [shadowMin, shadow_x] = shadow_data()
		Returns the minimum of the ith objective in the ith component of shadowMin
		and the minimizer of the ith objective in the ith row of the matrix
		shadow_x. [This file is required if the user already knows the individual 
			minimizers and wants to bypass the step in which the shadow minimum
			is found.]

	A variety of options for 2-D and 3-D plots is provided to the user
after all the subproblems are solved. 
