Fenchel Factors

The methodology of this book is to solve the primal and the dual simultaneously. The primal less the dual is nonnegative, and one can factor this difference into a sum of Fenchel factors. Each is greater than or equal to zero, so when there is no duality gap, each factor must be zero. Together these conditions are necessary and sufficient for a simultaneous solution. To reverse this procedure furnishes an effective and simple way to calculate the dual. Start with Fenchel inequalities. Add them together and rearrange to obtain an inequality with the primal objective function on the left and the dual objective function on the right. This chapter presents two examples of this reversal methodology. The reversed sequence conforms with how economists treat this topic. Each firm maximizes profit by producing where the output price equals marginal cost. Starting our analysis with the Fenchel factors means starting with profit maximization by firms. Deriving the primal and the dual from the Fenchel factors discovers how profit maximization implies industry-cost minimization. In applications, reversal is a simple and direct way to derive the dual. Valuable for both derivation and exposition, it facilitates the analysis of complicated primal/dual pairs. The reversal methodology is the technique mostly used throughout this book.