This paper is focused on analysis and reliable computations of limit loads in perfect plasticity. We recapitulate our recent results arising from a continuous setting of the so-called limit analysis problem. This problem is interpreted as a convex optimization subject to conic constraints. A related inf-sup condition on a convex cone is introduced and its importance for theoretical and numerical purposes is explained. Further, we introduce a penalization method for solving the kinematic limit analysis problem. The penalized problem may be solved by standard finite elements due to available convergence analysis using a simple local mesh adaptivity. This solution concept improves the simplest incremental method of limit analysis based on a load parametrization of an elastic-perfectly plastic problem.