Discovery and Proof via Duality

Convex optimization via duality furnishes a unified development of economic theory, and this book applies this mathematical technique to a wide variety of problems in economic and econometric theory. Given a convex function f, an optimization problem defines its conjugate f*, another convex function: $$f^{\ast }\left ( \boldsymbol {x}^{\ast }\right ) :=\sup _{\boldsymbol {x}}\left [ \left \langle \boldsymbol {x}^{\ast },\boldsymbol {x}\right \rangle -f\left ( \boldsymbol {x}\right ) \right ] \!.$$ Under general conditions, “conjugate duality” holds: the conjugate of the conjugate is the original function. The conjugate is the building block of convex optimization by Fenchel and Rockafellar. Given a convex optimization problem (the “primal”), one can associate with it a “dual” optimization problem. Perturbing the primal makes the problem slightly different, and this perturbation determines the dual. One calculates the dual from the primal via the conjugate. If the primal is a minimization, then the dual is a maximization. Under general conditions, there is “no duality gap”: the minimum of the primal is the maximum of the dual. “Proof by optimization” via duality furnishes a unified development of economic theory. To discover and prove an economic theorem, set up a primal that embodies some economic condition. Calculate a dual and interpret it. What is the economic meaning of a solution to the dual? Our standard method is to solve the primal and the dual simultaneously—an algebraic solution—by setting the primal equal to the dual. To figure out the economic meaning of this simultaneous solution both discovers and proves an economic theorem.