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