Sources of nonlinear acoustic emissions, particularly those associated with cavitation activity, play a key role in the safety and efficacy of current and emerging therapeutic ultrasound applications, such as oncological drug delivery, blood-brain barrier opening, and histotripsy. Passive acoustic mapping (PAM) is the first technique to enable real-time and non-invasive imaging of cavitation activity during therapeutic ultrasound exposure, through the recording and passive beamforming of broadband acoustic emissions using an array of ultrasound detectors. Initial limitations in PAM spatial resolution led to the adoption of optimal data-adaptive beamforming algorithms, such as the robust capon beamformer (RCB), that provide improved interference suppression and calibration error mitigation compared to non-adaptive beamformers. However, such approaches are restricted by the assumption that the recorded signals have a Gaussian distribution. To overcome this limitation and further improve the source resolvability of PAM, we propose a new beamforming approach termed robust beamforming by linear programming (RLPB). Along with the variance, this optimization-based method uses higher-order-statistics of the recorded signals, making no prior assumption on the statistical distribution of the acoustic signals. The RLPB is found via numerical simulations to improve resolvability over time exposure acoustics and RCB. In vitro experimentation yielded improved resolvability with respect to the source-to-array distance on the order of 22% axially and 13% transversely relative to RCB, whilst successfully accounting for array calibration errors. The improved resolution and decreased dependence on accurate calibration of RLPB is expected to facilitate the clinical translation of PAM for diagnostic, including super-resolution, and therapeutic ultrasound applications.