There are " n " dual constraints, each of which places a lower bound on a linear combination of " m " dual variables.
22.
Solving this problem is less computational hard because the amount of dual variables n is a lot of times much less than the amount of variables in the primal problem.
23.
For the one-dimensional case, the new patterns are introduced by solving an auxiliary optimization problem called the knapsack problem, using dual variable information from the linear program.
24.
The objective function of the subproblem is the reduced cost of the new variable with respect to the current dual variables, and the constraints require that the variable obey the naturally occurring constraints.
25.
Since each inequality can be replaced by an equality and a slack variable, this means each primal variable corresponds to a dual slack variable, and each dual variable corresponds to a primal slack variable.
26.
Then, for each constraint a generalized projection onto its feasible set is performed, updating both the constraint's dual variable and all primal variables for which there are non-zero coefficients in the constraint functions gradient.
27.
This is because the Fourier transformation takes differentiation into multiplication by the variable, and so a partial differential equation applied to the original function is transformed into multiplication by polynomial functions of the dual variables applied to the transformed function.
28.
For example, if C _ 2 and C _ 1 are joined by an edge whose label contains y, and so are C _ 1 and C _ 3, equality of y in all three dual variables is guaranteed.
29.
Without these constraints, one dual variable may take the value corresponding to the tuple x = 1, y = 2 while another dual variable takes the value corresponding to y = 3, z = 1, which assigns a different value to y.
30.
Without these constraints, one dual variable may take the value corresponding to the tuple x = 1, y = 2 while another dual variable takes the value corresponding to y = 3, z = 1, which assigns a different value to y.
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