## Minimization

**-**

## Minimization

# Minimization

In his influential 1784 essay “Idea for a Universal History with a Cosmopolitan Intent,” Immanuel Kant (1724–1804) defends his third thesis in the following terms:

Nature does nothing unnecessary and is not prodigal in the use of means to her ends. Nature seems here to have taken delight in the greatest frugality and to have calculated her animal endowments so closely—so precisely to the most pressing needs of a primitive existence—that she seems to have willed that if man should ever work himself up … he alone would have the entire credit for it and would have only himself to thank. (Kant [1784] 1983, p. 31.)

This Leibnizian key to how Nature plans and attains its ends and objectives—approaches the principal-agent problem, if not a game, that it is confronted with—had already been identified by Leonhard Euler (1707–1783) at least forty years earlier:

Namely, because the shape of the whole universe is most perfect and, in fact, designed by the wisest creator, nothing in all of the world will occur in which no maximum or minimum rule is somehow shining forth [nihil omnino in mundo contingint, in quo non maximi minimive ratio quapiam eluciat] (Euler [1744] 1952, p. 411).

It is in Paul Samuelson’s 1947 *Foundations of Economic Analysis* that Lionel Robbins’s (1898–1984) redefinition of economics as the “science which studies human behavior as a relationship between ends and scarce means which have alternative uses” (Robbins 1932, p. 16) is taken and squarely rooted into the “general properties of minimum systems” (the latter being one of the two section headings of Samuelson’s 1964 foreward to the reprint of his book). The trajectory of the notion of *minimization*, and its mirror-image *maximization*, in the evolution of economic science and of the social sciences, insofar as they submit to the analytical lead of the former, is so well charted that the paired terms do not merit an entry in *The New Palgrave* (Eatwell et al. 1987; also see Samuelson 1971). Thus V. M. Tikhomirov begins *Stories about Maxima and Minima* with the blasé statement that in “daily life it is constantly necessary to choose the best possible (optimal) solution [and] a tremendous number of such problems arise in economics and in technology” (Tikhomirov [1986] 1990, p. ix). Pareto-optimal and core allocations, to take only two solution concepts pertaining to the collective level, can be seen, respectively, as the outcomes at the individual level of expenditure-minimization and of preference maximization problems at some suitably chosen set of prices. (For the precise definitions and assumptions underlying these results, see Lange 1942; Arrow 1951; and Debreu 1959 for the former; and Edgeworth 1881; Debreu and Scarf 1963; and Aumann 1964 for the latter.)

In conclusion, two points need flagging: an unfortunate tendency to single out calculus as the only key to problems involving minimization and maximization (Koopmans 1957; Dorfman et al. 1958; Debreu 1959; and Niven 1981 are vigorous antidotes) and, more controversially perhaps, a tendency (as in Euler, this entry and also possibly in Kant) to single out the principle of minimization (and maximization), and the consequent submission to quantification that it implies, as the only explanatory key to social science phenomena. A fuller elaboration of these postmodern demurrals is best left to longer entries.

**SEE ALSO** *Kant, Immanuel; Mathematical Economics; Mathematics in the Social Sciences; Maximization; Methods, Quantitative; Models and Modeling; Optimizing Behavior; Quantification; Rationality; Samuelson, Paul A*.

## BIBLIOGRAPHY

Arrow, Kenneth J. 1951. An Extension of the Basic Theorems of Classical Welfare Economics. In *Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability*, ed. Jerzy Neyman, 507–532. Berkeley: University of California Press.

Aumann, Robert J. 1964. Markets with a Continuum of Traders.*Econometrica* 32: 39–50.

Debreu, Gerard. 1959. *Theory of Value: An Axiomatic Analysis of Economic Equilibrium*. New York: Wiley.

Debreu, Gerard, and Herbert Scarf. 1963. A Limit Theorem on the Core of an Economy. *International Economic Review* 4:235–246.

Dorfman, Robert, Paul A. Samuelson, and Robert M. Solow. 1958. *Linear Programming and Economic Analysis*. New York: McGraw-Hill.

Eatwell, John, Murray Milgate, and Peter K. Newman, eds. 1987. *The New Palgrave: A Dictionary of Economics*. London: Macmillan.

Edgeworth, Francis Y. 1881. *Mathematical Psychics*. London:Kegan-Paul.

Euler, Leonhard. [1744] 1952. *Methodus inveniendi curvas lineas maximi minimive proprietate gaudentes: Sive solution problematis viso isoperimetrici latissimo sense accepti*. In *Opera Omnia*, series 1, vol. 24. Leipzig, Germany: Tuebner.

Kant, Immanuel. [1784] 1983. Idea for a Universal History with a Cosmopolitan Intent. In *Perpetual Peace and Other Essays on Politics, History, and Morals*. Trans. Ted Humphrey, 29–40. Indianapolis, IN: Hackett.

Koopmans, Tjalling C. 1957. *Three Essays on the State of Economic Science*. New York: McGraw-Hill.

Lange, Oskar. 1942. The Foundations of Welfare Economics. *Econometrica* 10: 215–228.

Mordukhovich, Boris S. 2006. *Variational Analysis and Generalized Differentiation*. Berlin: Springer.

Niven, Ivan. 1981. *Maxima and Minima without Calculus*. Washington, DC: Mathematical Association of America.

Robbins, Lionel. 1932. *An Essay on the Nature and Significance of Economic Science*. London: Macmillan.

Samuelson, Paul A. 1947. *Foundations of Economic Analysis*. Cambridge, MA: Harvard University Press.

Samuelson, Paul A. 1971. Maximum Principles in Analytical Economics. In *Les Prix Nobel en 1970,* 273–288. Stockholm: The Nobel Foundation.

Tikhomirov, V. M. [1986] 1990. *Stories about Maxima and Minima*. Trans. Abe Shenitzer. Providence, RI: American Mathematical Society.

*M. Ali Khan*

## minimization

**minimization** **1.** The process of manipulating a logical expression and thereby transforming it into a simpler but equivalent expression with the same truth table. In practice this commonly means reducing the number of logic gates, gate inputs, or logic levels in a combinational circuit that realizes the logical expression. Minimization methods include use of Karnaugh maps and algebraic manipulation (often computer-aided).

**2.** The process of converting a finite-state machine to an equivalent minimal machine.

**3.** In the study of effective computability, the process of defining a new function by searching for values of a given function using the *minimization operator* or μ-*operator*. The functions involved are usually over the natural numbers. Let *g* be a function of *n*+1 variables. Then, for any given values of *x*_{1},…,*x _{n}*, the expression μ

*y*.

*g*(

*x*

_{1},…,

*x*,

_{n}*y*)

is evaluated by searching for the smallest value of

*y*for which

*g*(

*x*

_{1},…,

*x*,

_{n}*y*) = 0

This can be done by letting

*y*run through all natural numbers, in increasing order, until a suitable

*y*is found, whereupon that value of

*y*is returned as the value of the μ-expression. If no suitable

*y*exists the μ-expression is undefined. Also it may happen that before a suitable

*y*is found a value of

*y*is encountered for which

*g*(

*x*

_{1},…,

*x*,

_{n}*y*)

is itself undefined; in this case again the μ-expression is undefined.

This construct is used to define a function

*f*of

*n*variables from the function

*g*of

*n*+1 variables:

*f*(

*x*

_{1},…,

*x*) = μ

_{n}*y*.

*g*(

*x*

_{1},…,

*x*,

_{n}*y*)

Because of the possibility of the μ-expression being undefined,

*f*is a partial function. The process of searching for values, and the use of minimization, are essential factors that allow the formalism of recursive functions to define all the computable functions.

**4.**See optimization.