Problem Decomposition and Multi-Objective Optimization

1 Sub-Problems and Objectives Divide and conquer techniques in problem solving are familiar and intuitive; first find the solution to sub-problems and then re-use these to find solutions to the whole problem. For example, we may decompose the problem of designing a vehicle into designing the engine and designing the body. It is acknowledged that most real-world problems (vehicles included) do not decompose neatly into separable sub-problems. For example, the optimal properties of a drive system have dependencies with the passenger capacity. Nonetheless, it is very often possible to simplify a problem greatly by identifying sub-problems that exhibit some degree of independence. Multi-Objective Optimization, MOO, is similarly familiar and intuitive; there are several features of a system that we wish to optimize simultaneously and we wish to examine the alternatives that optimize each of the features independently, and/or offer a compromise of multiple objectives simultaneously. For example, we wish to minimize both the materials cost and construction time for our vehicle. It is acknowledged that sometimes multiple objectives can be satisfied simultaneously. For example, perhaps there is a simple design that is both cheap and fast to manufacture. This is the basis of Pareto dominance; a solution that is preferred with respect to all objectives. Nonetheless, it is often useful to acknowledge that objectives are constrained and to accept a set of solutions that optimize different objectives, rather than a single compromise. Both these forms of optimization recognize some form of component structure in the problems they address. In problem decomposition we think of a problem with multiple sub-problems, in MOO we think of a problem with multiple objectives. We propose that the difference between the two approaches is primarily one of emphasis. The main difference between a sub-problem and an objective is that the former expects some degree of independence from other sub-problems, whereas the latter expects some degree of constraint with other objectives. Yet, problem decomposition accepts that compromise may be necessary, and MOO accepts the possibility of a solution that may be good in respect to all objectives. In reality, both approaches acknowledge that components of a problem exhibit both independence and constraint. Ideally, in both these approaches we would like to take a solution that is good with respect to one objective or sub-problem, and put it together with a solution that is good with respect to another objective or sub-problem, and somehow combine them to …