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1 @c Copyright (C) 1996 John W. Eaton |
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2 @c This is part of the Octave manual. |
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3 @c For copying conditions, see the file gpl.texi. |
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4 |
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5 @node Optimization, Quadrature, Differential Equations, Top |
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6 @chapter Optimization |
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7 |
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8 @menu |
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9 * Quadratic Programming:: |
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10 * Nonlinear Programming:: |
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11 * Linear Least Squares:: |
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12 @end menu |
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13 |
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14 @c @cindex linear programming |
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15 @cindex quadratic programming |
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16 @cindex nonlinear programming |
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17 @cindex optimization |
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18 @cindex LP |
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19 @cindex QP |
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20 @cindex NLP |
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21 |
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22 @node Quadratic Programming, Nonlinear Programming, Optimization, Optimization |
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23 @section Quadratic Programming |
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24 |
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25 @ftable @code |
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26 @item qpsol |
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27 |
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28 @example |
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29 [x, obj, info, lambda] |
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30 = qpsol (x, H, c, lb, ub, lb, A, ub) |
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31 @end example |
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32 Solve quadratic programs using Gill and Murray's QPSOL. Because QPSOL |
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33 is not freely redistributable, this function is only available if you |
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34 have obtained your own copy of QPSOL. @xref{Installation}. |
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35 @end ftable |
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36 |
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37 @findex qpsol_options |
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38 Tolerances and other options for @code{qpsol} may be specified using the |
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39 function @code{qpsol_options}. |
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40 |
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41 @node Nonlinear Programming, Linear Least Squares, Quadratic Programming, Optimization |
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42 @section Nonlinear Programming |
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43 |
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44 @ignore |
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45 @ftable @code |
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46 @item fsqp |
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47 |
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48 @example |
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49 fsqp () |
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50 @end example |
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51 |
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52 @findex fsqp_options |
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53 Tolerances and other options for @code{fsqp} may be specified using the |
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54 function @code{fsqp_options}. |
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55 |
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56 Sorry, this hasn't been implemented yet. |
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57 @end ignore |
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58 |
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59 @ftable @code |
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60 @item npsol |
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61 |
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62 @example |
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63 [x, obj, info, lambda] |
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64 = npsol (x, 'phi', lb, ub, lb, A, ub, lb, 'g', ub) |
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65 @end example |
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66 Solve nonlinear programs using Gill and Murray's NPSOL. Because NPSOL |
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67 is not freely redistributable, this function is only available if you |
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68 have obtained your own copy of NPSOL. @xref{Installation}. |
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69 |
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70 The second argument is a string containing the name of the objective |
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71 function to call. The objective function must be of the form |
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72 |
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73 @example |
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74 y = phi (x) |
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75 @end example |
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76 |
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77 @noindent |
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78 where x is a vector and y is a scalar. |
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79 @end ftable |
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80 |
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81 @findex npsol_options |
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82 Tolerances and other options for @code{npsol} may be specified using the |
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83 function @code{npsol_options}. |
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84 |
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85 @node Linear Least Squares, , Nonlinear Programming, Optimization |
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86 @section Linear Least Squares |
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87 |
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88 @ftable @code |
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89 @item gls (@var{Y}, @var{X}, @var{O}) |
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90 Generalized least squares (GLS) estimation for the multivariate model |
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91 |
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92 @example |
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93 Y = X * B + E, mean(E) = 0, cov(vec(E)) = (s^2)*O |
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94 @end example |
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95 |
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96 @noindent |
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97 with |
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98 |
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99 @example |
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100 Y an T x p matrix |
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101 X an T x k matrix |
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102 B an k x p matrix |
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103 E an T x p matrix |
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104 O an Tp x Tp matrix |
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105 @end example |
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106 |
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107 @noindent |
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108 Each row of Y and X is an observation and each column a variable. |
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109 |
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110 Returns BETA, v, and, R, where BETA is the GLS estimator for B, v is the |
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111 GLS estimator for s^2, and R = Y - X*BETA is the matrix of GLS residuals. |
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112 |
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113 @item ols (@var{Y}, @var{X}) |
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114 Ordinary Least Squares (OLS) estimation for the multivariate model |
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115 |
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116 @example |
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117 Y = X*B + E, mean (E) = 0, cov (vec (E)) = kron (S, I) |
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118 @end example |
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119 |
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120 @noindent |
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121 with |
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122 |
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123 @example |
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124 Y an T x p matrix |
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125 X an T x k matrix |
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126 B an k x p matrix |
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127 E an T x p matrix |
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128 @end example |
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129 |
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130 @noindent |
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131 Each row of Y and X is an observation and each column a variable. |
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132 |
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133 Returns BETA, SIGMA, and R, where BETA is the OLS estimator for B, i.e. |
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134 |
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135 @example |
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136 BETA = pinv(X)*Y, |
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137 @end example |
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138 |
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139 @noindent |
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140 where pinv(X) denotes the pseudoinverse of X, SIGMA is the OLS estimator |
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141 for the matrix S, i.e. |
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142 |
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143 @example |
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144 SIGMA = (Y - X*BETA)'*(Y - X*BETA) / (T - rank(X)) |
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145 @end example |
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146 |
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147 and R = Y - X*BETA is the matrix of OLS residuals. |
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148 @end ftable |
