Euler Buckling Problem

Problem 4: Miser3 manual

Problem description
Problem setup
Solve the problem
Plot result
Footnote

Problem description

Over t in [0; 1 ], minimize

$$ J = -z_1 $$

subject to:

$\frac{dx_1}{dt} = x_2$

$\frac{dx_2}{dt} = \frac{-z_1*x_1}{x_3^2}$

$\int_0^1 x_3(t) \mathrm{d}t - 1 = 0$

$$ x_1(0) = 0 $$

$$ x_1(1) = 0 $$

$$ x_2(0) = 1 $$

$$ x_3(0) >= 0.5 $$

$$ x_3(t) >= 0.5 $$

% Copyright (c) 2007-2008 by Tomlab Optimization Inc.

Problem setup

toms t
toms z1
p = tomPhase('p', t, 0, 1, 40);
setPhase(p);

% States
tomStates x1 x2 x3 x4
% We don't need to introduce any control variables.

% Initial guess
x0 = {icollocate({
    x1 == 0;   x2 == 1
    x3 == 0.5; x4 == t/40})
    z1 == 10};

% Box constraints
cbox = {
    icollocate({-10 <= x1 <= 10
    -10 <= x2 <= 10; 0.5 <= x3 <= 10})
      0 <= z1 <= 500};

% Boundary constraints
cbnd = {initial({x3 >= 0.5; x1 == 0
    x2 == 1; x4 == 0})
    final({x1 == 0; x4 == 1})};

% ODEs and path constraints
ceq = {collocate({
    dot(x1) == x2
    dot(x2) == -z1*x1./x3.^2
    x3 >= 0.5 % Path constr.
     % Integral constr.
    })
    integrate(x3) == 1};

% Objective
objective = z1;

Solve the problem

options = struct;
options.name = 'Euler Buckling';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);

% Extract optimal states and controls from solution
t  = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);

Problem type appears to be: lpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Euler Buckling                 f_k       9.891564777222249900
                                       sum(|constr|)      0.000000000136989172
                              f(x_k) + sum(|constr|)      9.891564777359239000
                                              f(x_0)     10.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   40 ConJacEv   39 Iter   20 MinorIter  185
CPU time: 0.109375 sec. Elapsed time: 0.110000 sec.

Plot result

figure(1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-');
legend('x1','x2','x3');
title('Euler Buckling state variables');

Footnote

In the original [Miser3] problem formulation, it is requested to compute "u", equal to x3_t. u is not included in the optimization problem, thereby speeding up the solution process. x3_t can be obtained by simple numeric differentiation of x3.