Bi7740: Scientific computing
Optimization: a brief summary - Exercise
Vlad Popovici
popovici@iba.muni.cz
Institute of Biostatistics and Analyses
Masaryk University, Brno
Vlad Bi7740: Scientific computing
The beer can problem
variables: x1 the diameter, x2 the height of the can
Problem: let x = [x1, x2]T
minimize f(x) =
πx2
1
2
+ πx1x2 with respect to x1, x2
subject to
−2x1 + x2 ≤ 0
ceq(x) = 333 −
πx2
1
4
x2 = 0
Vlad Bi7740: Scientific computing
The differentials:
the gradient of the objective function:
f(x) =
∂f
∂x1
,
∂f
∂x2
T
= [π(x1 + x2), πx1]T
Vlad Bi7740: Scientific computing
The differentials:
the gradient of the objective function:
f(x) =
∂f
∂x1
,
∂f
∂x2
T
= [π(x1 + x2), πx1]T
the Jacobian of the nonlinear constraint function (which is a
scalar function, so it’s a gradient finally):
Jceq =
∂ceq
∂x1
,
∂ceq
∂x2
= −
π
2
x1x2, −
π
4
x2
1
Vlad Bi7740: Scientific computing
Matlab implementation
Constraint function implementation (save it as
beer_constraint.m file):
function [c ceq c_j ceq_j] = beer_constraint(x, vol)
% Evaluate the constraint for the beer can.
%
% x(1) − diameter
% x(2) − height
c = []; % no nonlinear inequalities
ceq = vol − 0.25*pi*x(1)^2*x(2);
% Jacobian:
if nargout > 2
c_j = [];
ceq_j = [−0.5*pi*x(1)*x(2), −0.25*pi*x(1)^2];
end
return
Vlad Bi7740: Scientific computing
Objective function implementation (save it as beer_area.m file):
function [f nabla_f] = beer_area(x)
% Compute the area of a beer can.
% x(1) − diameter
% x(2) − height
f = pi*x(1)*x(2) + 0.5*pi*x(1)^2;
% gradient:
if nargout > 1
nabla_f = [pi*x(1)+pi*x(2); pi*x(1)]; % column ...
vector!
end
return
Vlad Bi7740: Scientific computing
Matlab session: try also optimtool!
vol = 333; % cm^3
% to bound the parameter:
h_constraint = @(x) (beer_constraint(x, vol));
A = [−2 1]; b = 0;
lb = [4; 5];
ub = [8; 15];
x0 = [6; 10];
% set the options:
options = optimoptions('fmincon');
options = optimoptions(options,'Display', 'off');
options = optimoptions(options,'Algorithm', ...
'active−set');
x = fmincon(@beer_area, x0, A, b, [], [], lb, ub, ...
h_constraint, options);
Vlad Bi7740: Scientific computing