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# Fit Using Inequality Constraint

Sometimes specifying boundaries using `min` and `max` are not sufficient, and more complicated (inequality) constraints are needed. In the example below the center of the Lorentzian peak is constrained to be between 0-5 away from the center of the Gaussian peak.

See also: https://lmfit.github.io/lmfit-py/constraints.html#using-inequality-constraints

    import matplotlib.pyplot as plt
    import numpy as np

    from lmfit import Minimizer, create_params, report_fit
    from lmfit.lineshapes import gaussian, lorentzian

    def residual(pars, x, data):
        model = (gaussian(x, pars['amp_g'], pars['cen_g'], pars['wid_g']) +
                 lorentzian(x, pars['amp_l'], pars['cen_l'], pars['wid_l']))
        return model - data

Generate the simulated data using a Gaussian and Lorentzian lineshape:

    np.random.seed(0)
    x = np.linspace(0, 20.0, 601)

    data = (gaussian(x, 21, 6.1, 1.2) + lorentzian(x, 10, 9.6, 1.3) +
            np.random.normal(scale=0.1, size=x.size))

Create the fitting parameters and set an inequality constraint for `cen_l`. First, we add a new fitting parameter `peak_split`, which can take values between 0 and 5. Afterwards, we constrain the value for `cen_l` using the expression to be `'peak_split+cen_g'`:

    pfit = create_params(amp_g=10, cen_g=5, wid_g=1, amp_l=10,
                         peak_split=dict(value=2.5, min=0, max=5),
                         cen_l=dict(expr='peak_split+cen_g'),
                         wid_l=dict(expr='wid_g'))

    mini = Minimizer(residual, pfit, fcn_args=(x, data))
    out = mini.leastsq()
    best_fit = data + out.residual

Performing a fit, here using the `leastsq` algorithm, gives the following fitting results:

    report_fit(out.params)

Out:

    [[Variables]]
        amp_g:       21.2722842 +/- 0.05138772 (0.24%) (init = 10)
        cen_g:       6.10496396 +/- 0.00334613 (0.05%) (init = 5)
        wid_g:       1.21434954 +/- 0.00327317 (0.27%) (init = 1)
        amp_l:       9.46504173 +/- 0.05445415 (0.58%) (init = 10)
        peak_split:  3.52163544 +/- 0.01004618 (0.29%) (init = 2.5)
        cen_l:       9.62659935 +/- 0.01066172 (0.11%) == 'peak_split+cen_g'
        wid_l:       1.21434954 +/- 0.00327317 (0.27%) == 'wid_g'
    [[Correlations]] (unreported correlations are < 0.100)
        C(amp_g, wid_g)      = +0.6199
        C(amp_g, peak_split) = +0.3796
        C(wid_g, peak_split) = +0.3445
        C(amp_g, amp_l)      = -0.2951
        C(cen_g, amp_l)      = -0.2761
        C(amp_g, cen_g)      = +0.1936
        C(wid_g, amp_l)      = -0.1651
        C(cen_g, wid_g)      = +0.1546

and figure:

    plt.plot(x, data, 'o')
    plt.plot(x, best_fit, '--', label='best fit')
    plt.legend()
    plt.show()

Total running time of the script: ( 0 minutes 0.211 seconds)

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