Quadratically Perturbed Chance Constrained Programming with Fitted Distribution: t-Distribution vs. Gaussian

For chance-constrained programming (CCP) with non-Gaussian uncertainty, the optimization is generally intractable owing to the complicated probability density function (PDF). Using a simple fitted distribution with Kullback-Leibler (KL) divergence to represent the PDF mismatch is a systematic way to tackle CCP with non-Gaussian uncertainty. However, the essential difficulty of this methodology is to choose the fitted PDF, which should be close to the true PDF and make the resulting CCP problem solvable. In this paper, among the well-known PDFs, t-distribution rather than Gaussian is selected to be the fitted PDF. After the CCP with non-Gaussian uncertainty is transformed into CCP with a fitted PDF, the property of the regularized outage probability is analysed first. Then, the unimodal distributional properties are established for the quadratic form under t or Gaussian perturbation. Finally, based on the unimodal property and the regularized outage analysis, the analytical condition to make the safe approximation with fitted t-distribution have larger feasible set than the safe approximation with fitted Gaussian is obtained. With transceiver design under Logistic and Gaussian mixture channel uncertainties as an example, simulation results validate the less conservativeness property of the CCP with fitted t-distribution, compared to the CCP with fitted Gaussian and the classic moment method.