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# xpress.pwl

Purpose

Returns a piecewise linear function over a variable.

Synopsis
`xpress.pwl(dict)`
Argument

dict | Python dictionary containing, as keys, two-elements tuples, and, as values, linear expressions in a variable. If the piecewise linear function has only constant values (i.e. it is a piecewise constant function), the input variable can be specified with the key-value pair None: x.

Example
The following example shows various usages of xpress.pwl to model nonlinear functions as piecewise-linear functions :

    p = xp.problem()

    x = p.addVariable()  # Nonnegative variable
    y = p.addVariable(lb=-xp.infinity)  # dependent variable, unrestricted
    t = p.addVariable()
    w = p.addVariable()

    # Sets a piecewise linear objective: a ramp function
    p.setObjective (xp.pwl({(-xp.infinity, -1):  -2,
                            (-1, 1):             2*x,
                            (1, xp.infinity):    2}))

    p.addConstraint (t == xp.pwl({(1,2): 4*x, (2,4): 2, (4,5): -1}))

    # Piecewise CONSTANT function: add a key-value pair None: x to specify
    # input variable
    p.addConstraint (t == xp.pwl({(1,2): 4, (2,4): 2, (4,5): -1, None: x}))

    p.addConstraint (xp.pwl({(-1,0): x, (0,1): 2*x, (1,10): 2}) <=
                     xp.pwl({(0,10): 2*z, (10,20): z+2, (20,xp.infinity): 4}))

Further information
1. A piecewise linear function must use only one variable in all of the dictionary's values;
2. All values in the dictionary must be either constants or linear functions;
3. The intervals, specified as two-element tuples in the dictionary's keys, must be pairwise disjoint, i.e., they must not overlap.
4. Discontinuities in the function are allowed, i.e., one can declare a function as follows: xp.pwl({(1, 2): 2*x + 4, (2,3): x - 1}), which is obviously discontinuous at 2. The value of the function if the optimal solution has x=2 will be then either 8 or 1.
Related topics
problem.addPwlCons, problem.delPwlCons, problem.getPwlCons.
Parent Topic Xpress operators

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