Maximum theorem

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https://en.wikipedia.org/wiki/Maximum_theorem

The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959. [1] The theorem is primarily used in mathematical economics and optimal control.

Statement of theorem

Maximum Theorem. [2] [3] [4] [5] Let and be topological spaces, be a continuous function on the product , and be a compact-valued correspondence such that for all . Define the marginal function (or value function) by

and the set of maximizers by

.

If is continuous (i.e. both upper and lower hemicontinuous) at , then is continuous and is upper hemicontinuous with nonempty and compact values. As a consequence, the may be replaced by and the by .

Interpretation

The theorem is typically interpreted as providing conditions for a parametric optimization problem to have continuous solutions with regard to the parameter. In this case, is the parameter space, is the function to be maximized, and gives the constraint set that is maximized over. Then, is the maximized value of the function and is the set of points that maximize .

The result is that if the elements of an optimization problem are sufficiently continuous, then some, but not all, of that continuity is preserved in the solutions.

Proof

Throughout this proof we will use the term neighborhood to refer to an open set containing a particular point. We preface with a preliminary lemma, which is a general fact in the calculus of correspondences. Recall that a correspondence is closed if its graph is closed.

Lemma. [6] [7] [8] If are correspondences, is upper hemicontinuous and compact-valued, and is closed, then defined by is upper hemicontinuous.

Proof

Let , and suppose is an open set containing . If , then the result follows immediately. Otherwise, observe that for each we have , and since is closed there is a neighborhood of in which whenever . The collection of sets forms an open cover of the compact set , which allows us to extract a finite subcover . Then whenever , we have , and so . This completes the proof.

The continuity of in the maximum theorem is the result of combining two independent theorems together.

Theorem 1. [9] [10] [11] If is upper semicontinuous and is upper hemicontinuous, nonempty and compact-valued, then is upper semicontinuous.

Proof of Theorem 1

Fix , and let be arbitrary. For each , there exists a neighborhood of such that whenever , we have . The set of neighborhoods covers , which is compact, so suffice. Furthermore, since is upper hemicontinuous, there exists a neighborhood of such that whenever it follows that . Let . Then for all , we have for each , as for some . It follows that

which was desired.

Theorem 2. [12] [13] [14] If is lower semicontinuous and is lower hemicontinuous, then is lower semicontinuous.

Proof of Theorem 2

Fix , and let be arbitrary. By definition of , there exists such that . Now, since is lower semicontinuous, there exists a neighborhood of such that whenever we have . Observe that (in particular, ). Therefore, since is lower hemicontinuous, there exists a neighborhood such that whenever there exists . Let . Then whenever there exists , which implies

which was desired.

Under the hypotheses of the Maximum theorem, is continuous. It remains to verify that is an upper hemicontinuous correspondence with compact values. Let . To see that is nonempty, observe that the function by is continuous on the compact set . The Extreme Value theorem implies that is nonempty. In addition, since is continuous, it follows that a closed subset of the compact set , which implies is compact. Finally, let be defined by . Since is a continuous function, is a closed correspondence. Moreover, since , the preliminary Lemma implies that is upper hemicontinuous.

Variants and generalizations

A natural generalization from the above results gives sufficient local conditions for to be continuous and to be nonempty, compact-valued, and upper semi-continuous.

If in addition to the conditions above, is quasiconcave in for each and is convex-valued, then is also convex-valued. If is strictly quasiconcave in for each and is convex-valued, then is single-valued, and thus is a continuous function rather than a correspondence.

If is concave and has a convex graph, then is concave and is convex-valued. Similarly to above, if is strictly concave, then is a continuous function. [15]

It is also possible to generalize Berge's theorem to non-compact set-valued correspondences if the objective function is K-inf-compact. [16]

Examples

Consider a utility maximization problem where a consumer makes a choice from their budget set. Translating from the notation above to the standard consumer theory notation,

  • is the space of all bundles of commodities,
  • represents the price vector of the commodities and the consumer's wealth ,
  • is the consumer's utility function, and
  • is the consumer's budget set.

Then,

Proofs in general equilibrium theory often apply the Brouwer or Kakutani fixed-point theorems to the consumer's demand, which require compactness and continuity, and the maximum theorem provides the sufficient conditions to do so.

See also

Notes

  1. ^ Ok, Efe (2007). Real Analysis with Economics Applications. Princeton University Press. p.  306. ISBN  978-0-691-11768-3.
  2. ^ The original reference is the Maximum Theorem in Chapter 6, Section 3 Claude Berge (1963). Topological Spaces. Oliver and Boyd. p. 116. Famously, or perhaps infamously, Berge only considers Hausdorff topological spaces and only allows those compact sets which are themselves Hausdorff spaces. He also requires that upper hemicontinuous correspondences be compact-valued. These properties have been clarified and disaggregated in later literature.
  3. ^ Compare with Theorem 17.31 in Charalambos D. Aliprantis; Kim C. Border (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer. pp.  570. This is given for arbitrary topological spaces. They also consider the possibility that may only be defined on the graph of .
  4. ^ Compare with Theorem 3.5 in Shouchuan Hu; Nikolas S. Papageorgiou (1997). Handbook of Multivalued Analysis. 1: Theory. Springer-Science + Business Media, B. V. p. 84. They consider the case that and are Hausdorff spaces.
  5. ^ Theorem 3.6 in Beavis, Brian; Dobbs, Ian (1990). Optimization and Stability Theory for Economic Analysis. New York: Cambridge University Press. pp. 83–84. ISBN  0-521-33605-8.
  6. ^ Compare with Theorem 7 in Chapter 6, Section 1 of Claude Berge (1963). Topological Spaces. Oliver and Boyd. p. 112. Berge assumes that the underlying spaces are Hausdorff and employs this property for (but not for ) in his proof.
  7. ^ Compare with Proposition 2.46 in Shouchuan Hu; Nikolas S. Papageorgiou (1997). Handbook of Multivalued Analysis. 1: Theory. Springer-Science + Business Media, B. V. p. 53. They assume implicitly that and are Hausdorff spaces, but their proof is general.
  8. ^ Compare with Corollary 17.18 in Charalambos D. Aliprantis; Kim C. Border (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer. pp.  564. This is given for arbitrary topological spaces, but the proof relies on the machinery of topological nets.
  9. ^ Compare with Theorem 2 in Chapter 6, Section 3 of Claude Berge (1963). Topological Spaces. Oliver and Boyd. p. 116. Berge's argument is essentially the one presented here, but he again uses auxiliary results proven with the assumptions that the underlying spaces are Hausdorff.
  10. ^ Compare with Proposition 3.1 in Shouchuan Hu; Nikolas S. Papageorgiou (1997). Handbook of Multivalued Analysis. 1: Theory. Springer-Science + Business Media, B. V. p. 82. They work exclusively with Hausdorff spaces, and their proof again relies on topological nets. Their result also allows for to take on the values .
  11. ^ Compare with Lemma 17.30 in Charalambos D. Aliprantis; Kim C. Border (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer. pp.  569. They consider arbitrary topological spaces, and use an argument based on topological nets.
  12. ^ Compare with Theorem 1 in Chapter 6, Section 3 of Claude Berge (1963). Topological Spaces. Oliver and Boyd. p. 115. The argument presented here is essentially his.
  13. ^ Compare with Proposition 3.3 in Shouchuan Hu; Nikolas S. Papageorgiou (1997). Handbook of Multivalued Analysis. 1: Theory. Springer-Science + Business Media, B. V. p. 83. They work exclusively with Hausdorff spaces, and their proof again relies on topological nets. Their result also allows for to take on the values .
  14. ^ Compare with Lemma 17.29 in Charalambos D. Aliprantis; Kim C. Border (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer. pp.  569. They consider arbitrary topological spaces and use an argument involving topological nets.
  15. ^ Sundaram, Rangarajan K. (1996). A First Course in Optimization Theory. Cambridge University Press. p.  239. ISBN  0-521-49770-1.
  16. ^ Theorem 1.2 in Feinberg, Eugene A.; Kasyanov, Pavlo O.; Zadoianchuk, Nina V. (January 2013). "Berge's theorem for noncompact image sets". Journal of Mathematical Analysis and Applications. 397 (1): 255–259. arXiv: 1203.1340. doi: 10.1016/j.jmaa.2012.07.051. S2CID  8603060.

References

  • Claude Berge (1963). Topological Spaces. Oliver and Boyd. pp. 115–117.
  • Charalambos D. Aliprantis; Kim C. Border (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer. pp.  569-571.
  • Shouchuan Hu; Nikolas S. Papageorgiou (1997). Handbook of Multivalued Analysis. 1: Theory. Springer-Science + Business Media, B. V. pp. 82–89.