An approximation method for computing the expected value of max-affine expressions

Expected values of max-affine expressions appear in optimization problems for various stochastic systems, such as in model predictive control (MPC) for stochastic max-plus-linear systems, in identification of stochastic max-plus-linear systems, and in control of stochastic monotonic piecewise affine systems. Solving these optimization problems involves the computation of the expected value of the maximum of affine expressions, which will then appear in the objective function or in the constraints. The computation of this expected value can be highly complex and expensive, which also results in a high computation time to solve the optimization problem. Therefore, the focus of this paper is on decreasing the computational complexity of the calculation of these expected values. To this end, we use an approximation method based on the moments of a random variable. We illustrate in an example that this method results in a much lower computation time and a much lower computational complexity than the existing computational methods while still guaranteeing a performance that is comparable to the performance of those methods.