The dragon exceeds the proper limits; there will be occasion for repentance.
The Book of Changes (Yi Qing) (c. 1200 BC)
In modeling an optimization problem, the easiest and most common mistake is to leave something out. This chapter shows how to reduce such omissions by systematically checking the model before trying to compute with it. Such a check can detect formulation errors, prevent wasteful computations, and avoid wrong answers. As a perhaps unexpected bonus, such a preliminary study may lead to a simpler and more clearly understandable model with fewer variables and constraints than the original one.
The methods of this chapter, informally referred to as boundedness analysis, should be regarded as a model reduction and verification process to be carried out routinely before attempting any numerical optimization procedure. At the same time, one should be cautious about the limitations of boundedness arguments because they are based on necessary conditions, namely mathematical truths that hold assuming an optimal solution exists. Such existence, derived from sufficient conditions, is not always easy to prove. The complete optimality theory in Chapters 4 and 5 provides important additional tools to those presented in this chapter.
One of the common themes in this book is to advocate “model reduction” whenever possible; that is, rigorously cutting down the number of constraint combinations and variables that could lead to the optimum – before too much numerical computation is done. This reduction has two motivations. The first is to seek a particular solution, that is, a numerical answer for a given set of parameter values. The second is to construct a specific parametric optimization procedure, which for any set of parameter values would directly generate the globally optimal solution with a minimum of iteration or searching.
The motivation to reduce models to construct optimized parametric design procedures comes from three applications. The first is to generate, without unnecessary iterative computation, the optimal design directly from a set of input parameter values. The second is to reoptimize specific equipment configurations in the face of changing parameter values. Designers call this “resizing.”