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7 Nonlinear Programming
The TOMLAB bundle
testprob provides
three sets of test problems for nonlinear problems:
con_prob and
chs_prob.
7.1 An example of a nonlinear problem
The basic structure of a general nonlinear problem
is the following
|
|
|
f(x) |
| |
|
| s/t |
| xL |
≤ |
x |
≤ |
xU |
| bL |
≤ |
A x |
≤ |
bU |
| cL |
≤ |
c(x) |
≤ |
cU |
|
|
(8) |
where
x,
xL,
xU 
Rn,
f(
x)
R,
A Rm1 × n,
bL,
bU Rm1 and
cL,
c(
x),
cU Rm2.
An example of a problem of this class,
(that is also found in the TOMLAB quickguide) is nlpQG:
|
|
|
f(x)=α ( x2−x12 )2+ (1−x1 )2 |
| |
|
| s/t |
| −10 |
≤ |
x1 |
≤ |
2 |
| −10 |
≤ |
x2 |
≤ |
2 |
| α=100 |
|
|
(9) |
TOMLAB requires that general nonlinear problems are defined in
Matlab m-files. The function to be optimized must always be
supplied. It is recommended that the user supply as many
analytical functions as possible. There are six methods available
for numerical differentiation and also two for automatic.
The following files define the problem in TOMLAB.
File: tomlab/quickguide/rbbQG_f.m, rbbQG_g.m, rbbQG_H.m, rbbQG_c.m, rbbQG_dc.m, rbbQG_d2c.m
f: Function value
g: Gradient vector
H: Hessian matrix
c: Nonlinear constraint vector
dc: Nonlinear constraint gradient matrix
d2c: The second part of the Hessian to the Lagrangian
function for the nonlinear constraints.
The following file illustrates how to solve this NLP (CON) problem
in TOMLAB. Also view the m-files specified above for more
information.
File: tomlab/quickguide/nlpQG.m
Open the file for viewing, and execute nlpQG in Matlab.
% nlpQG is a small example problem for defining and solving
% nonlinear programming problems using the TOMLAB format.
Name = 'RBB Problem';
x_0 = [-1.2 1]'; % Starting values for the optimization.
x_L = [-10;-10]; % Lower bounds for x.
x_U = [2;2]; % Upper bounds for x.
fLowBnd = 0; % Lower bound on function.
c_L = -1000; % Lower bound on nonlinear constraints.
c_U = 0; % Upper bound on nonlinear constraints.
Prob = conAssign('rbbQG_f', 'rbbQG_g', 'rbbQG_H', [], x_L, x_U, Name, x_0,...
[], fLowBnd, [], [], [], 'rbbQG_c', 'rbbQG_dc', 'rbbQG_d2c', [], c_L, c_U);
Prob.Warning = 0; % Turning off warnings.
Result = tomRun('ucSolve', Prob, 1); % Ignores constraints.
% Result = tomRun('conopt', Prob, 1);
% Result = tomRun('snopt', Prob, 1);
7.2 con_prob
con_prob is a collection of 17 constrained nonlinear test problems
with 2 to 100 variables and up to 50 constrains.
In order to define the problem
n and solve it execute the following in Matlab:
Prob = probInit('con_prob',n);
Result = tomRun('',Prob);
7.3 chs_prob
chs_prob is a collection of 180 constrained nonlinear test problems from
the Hoch-Schittkowski set with 2 to 50 variables and about 10 constrains.
In order to define the problem
n and solve it execute the following in Matlab:
Prob = probInit('chs_prob',n);
Result = tomRun('',Prob);
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