Max-Plus Algebra and Mathematical Fear in Dynamic Optimization

Max-plus algebra, cost measures, and mathematical fear have proved useful tools in dynamic optimization. Indeed, the first two have even become a central tool in some fields of investigation such as discrete event systems. We first recall the fundamentals of max-plus algebra with simple examples of max-plus linear models, and simple consequences of that remark. We then introduce cost measures, the natural equivalent of probability measures in the max-plus algebra, and their fundamental properties, including the definition of the mathematical fear (the equivalent of the mathematical expectation), induced measures and conditioning. Finally, we concentrate on those aspects that are put in use in dynamical optimization and state a separation theorem which was first derived using these tools.