1Blurb::
2Define coefficients of the linear inequality constraints
3Description::
4In the inequality case, the constraint matrix \f$A\f$ provides
5coefficients for the variables in the two-sided formulation:
6\f[a_l \leq Ax \leq a_u\f]
7
8Where the bounds are optionally specified by \c linear_inequality_lower_bounds,
9and \c linear_inequality_upper_bounds.
10The bounds, if not specified, will default to -infinity, and 0, respectively,
11resulting in one-sided inequalities of the form
12\f[Ax \leq 0.0\f].
13
14The linear_constraints topics page (linked above) outlines a few additional
15things to consider when using linear constraints.
16
17Topics::	linear_constraints
18Examples::
19
20In the first example, an optimization involving two variables, \c x1 and \c x2,
21is to be performed. These variables must satisfy two constraints:
22
23\f[ 1.5 \cdot x1 + 1.0 \cdot x2 \leq 5.0 \f]
24\f[ x1 \leq x2 \Longrightarrow x1 - x2 \leq 0.0 \f]
25
26The pair of constraints can be written in matrix form as:
27
28\f[\begin{bmatrix}
29  1.5 & 1.0 \\
30  1.0 & -1.0
31\end{bmatrix}
32
33\begin{bmatrix}
34  x1 \\
35  x2
36\end{bmatrix}
37\leq
38\begin{bmatrix}
39  5.0 \\
40  0.0
41\end{bmatrix}
42
43\f]
44
45The coefficient matrix and right hand side of this matrix inequality are expressed
46to Dakota in the variables section of the input file:
47
48\verbatim
49
50variables
51  continuous_design 2
52    descriptors 'x1' 'x2'
53
54  linear_inequality_constraint_matrix = 1.5   1.0
55                                        1.0  -1.0
56
57  linear_inequality_upper_bounds = 5.0
58                                   0.0
59
60\endverbatim
61<hr>
62The second example is more complex in two respects. First, some, but not all,
63of the constraints are "two sided", with both lower and upper bounds. Second,
64not all variables participate in all constraints.
65
66There are four variables, \c x1, \c x2, \c x3, and \c x4, and four constraints.
67
68\f[ -2.0 \leq 5.0 \cdot x1 + 2.0 \cdot x2 \leq 9.0 \f]
69\f[  0.0 \leq x1 + x3 \f]
70\f[ -8.0 \leq x2 + 6.0 \cdot x4 \leq 8.0 \f]
71\f[ x1 + x2 + x3 \leq 9.0 \f]
72
73Or, in matrix form,
74
75\f[
76\begin{bmatrix}
77  -2.0 \\
78  0.0 \\
79  -8.0 \\
80  -\infty
81\end{bmatrix}
82
83\leq
84
85\begin{bmatrix}
86  5.0 & 2.0 & 0.0 & 0.0 \\
87  1.0 & 0.0 & 1.0 & 0.0 \\
88  0.0 & 1.0 & 0.0 & 6.0 \\
89  1.0 & 1.0 & 1.0 & 0.0
90\end{bmatrix}
91
92\begin{bmatrix}
93  x1 \\
94  x2 \\
95  x3 \\
96  x4
97\end{bmatrix}
98\leq
99\begin{bmatrix}
100  9.0 \\
101  \infty \\
102  8.0 \\
103  9.0
104\end{bmatrix}
105\f]
106
107The Dakota specification for this matrix inequality is:
108
109\verbatim
110
111variables
112  continuous_design 4
113    descriptors 'x1' 'x2' 'x3' 'x4'
114
115  linear_inequality_constraint_matrix = 5.0  2.0  0.0  0.0
116                                        1.0  0.0  1.0  0.0
117                                        0.0  1.0  0.0  6.0
118                                        1.0  1.0  1.0  0.0
119
120  linear_inequality_lower_bounds =  -2.0
121                                     0.0
122                                    -8.0
123                                    -inf
124
125  linear_inequality_upper_bounds = 9.0
126                                   inf
127                                   8.0
128                                   9.0
129\endverbatim
130
131
132Theory::
133Faq::
134See_Also::
135