Contrast Limited Histogram Equalisation moves the input image histogram gently towards one which has a more uniform distribution. Viewed as a tone mapping operation, CLHE generates a tone curve with bounded max and min slopes. It is this boundedness which ensures that the processed images have more detail but few artefacts. Outside of limiting contrast, recent improvements to histogram equalisation include constraining the tone curve to make good whites and blacks and constraining the tone curve to be smooth. This paper makes three contributions. First, we show that the CLHE formalism is not least-squares optimal but optimality can be achieved by reformulating the problem in a quadratic programming framework. Second, we incorporate the additional constraints of tone curve smoothness and good whites and blacks in our quadratic programming CLHE framework. Third, experiments demonstrate the utility of our method.