Why some believe that continuous optimisation problems might be easier to solve than discrete optimisation problems?


Some believe that continuous optimisation problems might be easier to solve than their discrete counterparts, owing to the smoothness of the objective functions, which allows the use of calculus techniques. In general, calculus techniques mean to use objective and constraint information at a particular point x, to deduce information about the behaviour of the objective function at all surrounding points close to x. The deduced information is then used to guide the search direction. In contrast, combinatorial optimisation problems are to some extent harder to solve than continuous ones. This is because the behaviour of the objective and constraints may change significantly as we move from one feasible point to another, even if the two points are close according to some measure. Thus it is usually not possible to deduce information about the neighbouring points from the current one.