Mirrored Pairs of Optimization Problems

Abstract

Often, a solution to one problem of constrained optimization is also a solution to another, closely related, problem. The binocular vision afforded by these mirrored pairs of optimization problems frequently adds clarity and insight to our understanding of the original problem, as it so markedly has the case of consumer's demand. There it is standard to remark that a solution to the problem of maximizing utility subject to given prices and wealth is, in cases (Varian, 1978, p. 92), also a solution to the problem of minimizing expenditures subject to achieving the appropriate level of utility, and vice versa, a remark that leads directly to important identities that tie together the expenditure function, the indirect utility function, the Marshallian demand function, and the Hicksian demand function (Varian, 1978, p. 92). The question then arises of delineating precisely what is meant by these non-perverse cases, which a solution to an original problem is also a solution to its reflection. The conditions for the consumer's problem offered Varian's textbook (1978, p. 112) are really quite sharp, while even those Debreu's classic treatise (1959, pp. 67-71) are still unnecessarily restrictive. Moreover, these discussions do not bring out the symmetries between the conditions under which a solution to an original maximum problem also solves its reflected minimum problem, and those under which a solution to a minimum problem also solves its reflected maximum problem. This paper presents an abstract formulation of such problems, a formulation that leads naturally to a simple, symmetric and complete answer to the question posed above. Because the sole concern is with the interrelations of solutions to different problems, the theory needs to deal only with a few primitive objects, less numerous and less subtle than would be required for deeper inquiries into such matters as the existence, uniqueness or stability of solutions. Thus, the results obtained here do not lie at all deep the mathematical sense, the simplicity of the proofs alone precluding that. Nevertheless, the high degree of abstraction gives the formal scheme considerable generality and power, capable of application a wide variety of concrete specializations, and so eliminating the need to prove similar theorems ad hoc as one encounters each separate application.