Added Project and Report

Lessons/10-12/oneDTestFunctions.m

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+function TF = oneDTestFunctions()
+
+%function TF = oneDTestFunctions()
+%
+% Produces a cell array of function handlers, useful to test unconstrained
+% one-dimensional optimization algorithms.
+%
+% Each function in the array has the following interface:
+%
+%   [ v , varargout ] = f( x )
+%
+% Input:
+%
+% - x is either a scalar real denoting the input of f(), or [] (empty).
+%
+% Output:
+%
+% - v (real, scalar): if x == [] this is the best known lower bound on
+%   the global optimum of f() on the standard interval in which f() is
+%   supposed to be minimised (see next). If x ~= [] then v = f(x).
+%
+% - g (real, either scalar or a [ 1 x 2 ] matrix denoting an interval) is
+%   the first optional argument. This also depends on x. if x == [] then
+%   g is a [ 1 x 2 ] matrix denoting the standard interval in which f()
+%   is supposed to be minimised (into which v is the minimum). If x ~= []
+%   then g is a scalar containing the dervative g = f'(x) (or a
+%   subgradient of f() in x if f() is not differentiable at x).
+%
+% - H (real, scalar) is the second optional argument. This must only be
+%   specified if x ~= [], and it is the second derivative h = f''(x).
+%   If no such information is available, the function throws error.
+%
+% The current list of functions is the following:
+%
+%  1 Taylored nasty 10-th degree polynomial with a local minimum in 0
+%    (with value 0), a local minimum around -1, a saddle point around
+%    +1, a local maximum around +2 and another local minimum around +3.
+%    Thought to be plotted in [ -1.2 , 3.1 ], and minimised in
+%    [ -0.5 , 3 ] (the global minimim there being 0). Note that, by
+%    taking [ -1 , 3 ] instead, blind dycothomic search would hit jackpot
+%    in two iterations by sheer luck.
+%
+%{
+ =======================================
+ Author: Antonio Frangioni
+ Date: 08-11-18
+ Version 1.01
+ Copyright Antonio Frangioni
+ =======================================
+%}
+
+TF = cell( 1 , 1 );
+%TF{ 1 } = @(x) genericquad( [ 6 -2 ; -2 6 ] , [ 10 ; 5 ] , x );
+TF{ 1 } = @custom;
+
+% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
+
+% function [ v , varargout ] = genericquad( Q , q , x )
+%  % generic quadratic function f(x) = x' * Q * x / 2 + q' * x
+%
+% if isempty( x )  % informative call
+%    if min( eig( Q ) ) > 1e-14
+%       xStar = Q \ -q;
+%       v = 0.5 * xStar' * Q * xStar + q' * xStar;
+%    else
+%       v = - Inf;
+%    end
+%    if nargout > 1
+%       varargout{ 1 } = [ 0 ; 0 ];
+%    end
+% else
+%    if ~ isequal( size( x ) , [ 2 1 ] )
+%       error( 'genericquad: x is of wrong size' );
+%    end
+%    v = 0.5 * x' * Q * x + q' * x;  % f(x)
+%    if nargout > 1
+%       varargout{ 1 } = Q * x + q;  % \nabla f(x)
+%       if nargout > 2
+%          varargout{ 2 } = Q;       % \nabla^2 f(x)
+%       end
+%    end
+% end
+% end  % genericquad
+
+% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
+
+function [ v , varargout ] = custom( x )
+% custom 10-th degree polynomial
+% syms x
+% f = @(x) 91 * x^2 / 30 - 19 * x^3 / 6 - 54 * x^4 / 25 ...
+%   + 93 * x^5 / 23 - 23 * x^6 / 36 -  121 * x^7 / 93 ...
+%   + 72 * x^8 / 91 - 13 * x^9 / 74 + 9 * x^10 / 640
+%
+% diff( f , x )
+% 9*x^9)/64 - (117*x^8)/74 + (576*x^7)/91 - (847*x^6)/93
+% - (23*x^5)/6 + (465*x^4)/23 - (216*x^3)/25 - (19*x^2)/2 + (91*x)/15
+%
+% diff( f , x , 2 )
+% (81*x^8)/64 - (468*x^7)/37 + (576*x^6)/13 - (1694*x^5)/31
+% - (115*x^4)/6 + (1860*x^3)/23 - (648*x^2)/25 - 19*x + 91/15
+
+if isempty( x )  % informative call
+   v = 0;
+   if nargout > 1
+      varargout{ 1 } = [ -0.5 3 ];
+   end
+else
+   v = 91 * x^2 / 30 - 19 * x^3 / 6 - 54 * x^4 / 25 ...
+     + 93 * x^5 / 23 - 23 * x^6 / 36 -  121 * x^7 / 93 ...
+     + 72 * x^8 / 91 - 13 * x^9 / 74 + 9 * x^10 / 640;
+   if nargout > 1
+      g = 91 * x / 15 - 19 * x^2 / 2 - 216 * x^3 / 25 + 465 * x^4 / 23 ...
+	- 23 * x^5 / 6 - 847 * x^6 / 93 + 576 * x^7 / 91 ...
+	- 117 * x^8 / 74 + 9 * x^9 / 64;
+      varargout{ 1 } = g;  % f'(x)
+      if nargout > 2
+	 h = 91 / 15 - 19 * x - 648 * x^2 / 25 + 1860 * x^3 / 23 ...
+	   - 115 * x^4 / 6 - 1694 * x^5 / 31 + 576 * x^6 / 13 ...
+	   - 468 * x^7 / 37 + 81 * x^8 / 64;
+	 varargout{ 2 } = h;       % f''(x)
+      end
+   end
+end
+end  % custom
+
+% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
+
+end