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examples.m
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examples.m
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classdef examples < handle
properties
ConstraintMethod = ''
algorithm = ''
end
methods
function this = examples(algorithm,ConstraintMethod)
if nargin < 1 || isempty(algorithm)
this.algorithm = 'DFP';
else
this.algorithm = algorithm;
end
if nargin < 2 || isempty(ConstraintMethod)
this.ConstraintMethod = 'AL';
else
this.ConstraintMethod = ConstraintMethod;
end
end
function [x,fval,exitflag,output,ex] = getExample(this,No)
switch No
case 1
% exmple 1
% Minimize f(x1,x2) = -2x1 -x2
%
% S.t. g1(x1,x2) = x1^2 + x2^2 <= 25 (nonlinear inequality constraint)
% g2(x1,x2) = x1^2 - x2^2 <= 7 (nonlinear inequality constraint)
% 0 <= x1 <= 10; 0 <= x2 <= 10 (box constraints)
%
% Starting point: (x1,x2) = (1,1) (a feasible point)
x0 = [1;1];
lb = [0;0];
ub = [7;10];
f =@(x) examples.f1(x);
g =@(x) examples.g1(x);
ex = fmindescent(f,x0,[],[],[],[],lb,ub,g,'display','iter','Algorithm',this.algorithm);
[x,fval,exitflag,output] = ex.solve;
ex.postprocess;
case 2
% exmple 2
% Minimize f(x1,x2) = x1^4 - 2*x1*x1*x2 + x1*x1 + x1*x2*x2 - 2*x1 + 4
%
% S.t. h(x1,x2) = x1^2 + x2^2 = 2 (nonlinear equality constraint)
% g(x1,x2) = 1/4*x1^2 +3/4*x2^2 <= 1 (nonlinear inequality constraint)
% 0 <= x1 <= 4; 0 <= x2 <= 4 (box constraints)
%
% Starting point: (x1,x2) = (sqrt(2),0) (a feasible point)
x0 = [sqrt(2);0];
lb = [0;0];
ub = [4;4];
f =@(x) examples.f2(x);
g =@(x) examples.g2(x);
ex = fmindescent(f,x0,[],[],[],[],lb,ub,g,'display','iter','Algorithm',this.algorithm);
[x,fval,exitflag,output] = ex.solve;
ex.postprocess;
case 3
% Example 12.3 from Arora (2012) Example 7.3, pp. 285-287 (Arora (2004) pp. 418-420):
%
% Minimize f(x1,x2) = (x1-10)^3 + (x2-20)^3
%
% S.t. g1(x1,x2) = -(x1-5)^2 -(x2-5)^2 <= -100 (nonlinear inequality constraint)
% g2(x1,x2) = (x1-6)^2 + (x2-5)^2 <= 82.81 (nonlinear inequality constraint)
% 13 <= x1 <= 100; 0 <= x2 <= 100 (box constraints)
%
% Starting point: (x1,x2) = (20,5) (a infeasible point)
x0 = [13;5];
lb = [13;0];
ub = [100;100];
f =@(x) examples.f3(x);
g =@(x) examples.g3(x);
ex = fmindescent(f,x0,[],[],[],[],lb,ub,g,'display','iter','Algorithm',this.algorithm);
[x,fval,exitflag,output] = ex.solve;
ex.postprocess;
case 4
% Minimize max (f1(x1,x2) = 7 - 2*x1 + 2*x2; f2(x1,x2) = 5 - x1 + 3*x2)
%
% S.t. g1(x1,x2) = x1^2 + x2^2 <= 25 (nonlinear inequality constraint)
% g2(x1,x2) = x1^2 - x2^2 <= 7 (nonlinear inequality constraint)
% 0 <= x1 <= 10; 0 <= x2 <= 10 (box constraints)
%
% Starting point: (x1,x2) = (1,1) (a feasible point)
x0 = [1;1;1000];
lb = [0;10;0];
ub = [0;10;1e6];
f =@(x) examples.f4(x);
g =@(x) examples.g4(x);
ex = fmindescent(f,x0,[],[],[],[],lb,ub,g,'display','iter','Algorithm',this.algorithm);
[x,fval,exitflag,output] = ex.solve;
ex.postprocess;
case 5
% Minimize f(x1,x2) = x1^2+2*x2^2
% S.t. g1(x1,x2) = x1 + x2 >= 1
x0 = [1;1];
lb = [-10;-10];
ub = [10;10];
f =@(x) examples.f5(x);
g =@(x) examples.g5(x);
ex = fmindescent(f,x0,[],[],[],[],lb,ub,g,'display','iter','Algorithm',this.algorithm);
[x,fval,exitflag,output] = ex.solve;
ex.postprocess;
case 6
% Minimize f(x1,x2) = x1^2+2*x2^2
% S.t. g1(x1,x2) = x1 + x2 >= 1
x0 = [-1;3];
lb = [-10;-10];
ub = [10;10];
f =@(x) examples.f6(x);
g =[];
ex = fmindescent(f,x0,[],[],[],[],lb,ub,g,'display','iter','Algorithm',this.algorithm);
[x,fval,exitflag,output] = ex.solve;
ex.postprocess;
end % switch
end % function
end % methods
methods(Static = true, Hidden = false)
function [f,df] = f1(x)
f = [];
df = [];
switch nargout
case 1
f = -2*x(1) - x(2);
case 2
df(1,1) = -2;
df(2,1) = 1;
end
end
function [g,geq,dg,dgeq] = g1(x)
g = [];
geq = [];
dg = [];
dgeq = [];
switch nargout
case {1,2}
g(1,1)= x(1)^2 + x(2)^2 - 25;
g(2,1) = x(1)^2 - x(2)^2 - 7;
case {3,4}
dg(1,1) = 2*x(1);
dg(2,1) = 2*x(2);
dg(1,2) = 2*x(1);
dg(2,2) = -2*x(2);
end
end
function [f,df] = f2(x)
f = [];
df = [];
switch nargout
case 1
f = x(1)^4 - 2*x(1)^2*x(2) + x(1)^2 + x(1)*x(2)^2 - 2*x(1) + 4;
case 2
df(1,1) = 4*x(1)^3 -4*x(1)*x(2) + 2*x(1) + x(2)^2 -2;
df(2,1) = -2*x(1)^2 + 2*x(1);
end
end
function [g,geq,dg,dgeq] = g2(x)
g = [];
geq = [];
dg = [];
dgeq = [];
switch nargout
case {1,2}
g(1)= 1/4*x(1)^2 + 3/4*x(2)^2-1;
geq(1) = x(1)^2 + x(2)^2 - 2;
case {3,4}
dg(1,1) = 2/4*x(1);
dg(2,1) = 6/4*x(2);
dgeq(1,1) = 2*x(1);
dgeq(2,1) = 2*x(2);
end
end
function [f,df] = f3(x)
f = [];
df = [];
switch nargout
case 1
f = (x(1)-10)^3 + (x(2)-20)^3;
case 2
df(1,1) =3*(x(1)-10)^2;
df(2,1) = 3*(x(2)-20)^2;
end
end
function [g,geq,dg,dgeq] = g3(x)
g = [];
geq = [];
dg = [];
dgeq = [];
switch nargout
case {1,2}
g(1,1)= -(x(1)-5)^2/100 - (x(2)-5)^2/100 + 1;
g(2,1) = (x(1)-6)^2/82.81 + (x(2)-5)^2/82.81 - 1;
case {3,4}
dg(1,1) = -2*(x(1)-5)/100;
dg(2,1) = -2*(x(2)-5)/100;
dg(1,2) = 2*(x(1)-6)/82.81;
dg(2,2) = -2*(x(2)-5)/82.81;
end
end
function [f,df] = f4(x)
f = [];
df = [];
switch nargout
case 1
f = x(3);
case 2
df(1,1) = 0;
df(2,1) = 0;
df(3,1) = 1;
end
end
function [g,geq,dg,dgeq] = g4(x)
g = [];
geq = [];
dg = [];
dgeq = [];
switch nargout
case {1,2}
g(1,1) = 7 - 2*x(1) + 2*x(2) - x(3);
g(2,1) = 5 - x(1) + 3*x(2) - x(3);
g(3,1) = (x(1)^2 + x(2)^2) - 25;
g(4,1) = (x(1)^2 - x(2)^2) - 7;
case {3,4}
dg(1,1) = -2;
dg(2,1) = 2;
dg(3,1) = -1;
dg(1,2) = -1;
dg(2,2) = 3;
dg(3,2) = -1;
dg(1,3) = 2*(x(1));
dg(2,3) = 2*(x(2));
dg(3,3) = 0;
dg(1,4) = 2*(x(1));
dg(2,4) = -2*(x(2));
dg(3,4) = 0;
end
end
function [f,df] = f5(x)
f = [];
df = [];
switch nargout
case 1
f = x(1)^2+2*x(2)^2;
case 2
df(1,1) = 2*x(1);
df(2,1) = 4*x(2);
end
end
function [g,geq,dg,dgeq] = g5(x)
g = [];
geq = [];
dg = [];
dgeq = [];
switch nargout
case {1,2}
g(1,1)= -x(1)-x(2)+1;
case {3,4}
dg(1,1) = -1;
dg(2,1) = -1;
end
end
function [f,df] = f6(x)
f = [];
df = [];
switch nargout
case 1
f = (1-x(1))^2+100*(x(2)-x(1)^2)^2;
case 2
df(1,1) = 400*x(1)^3 - 400*x(1)*x(2)+2*x(1)-2;
df(2,1) = 200*(x(2)-x(1)^2);
end
end
end % Methods
end % class