By Joel N. Franklin
This article makes an attempt to survey the middle matters in optimization and mathematical economics: linear and nonlinear programming, isolating airplane theorems, fixed-point theorems, and a few in their applications.
This textual content covers purely topics good: linear programming and fixed-point theorems. The sections on linear programming are situated round deriving equipment according to the simplex set of rules in addition to the various ordinary LP difficulties, resembling community flows and transportation challenge. I by no means had time to learn the part at the fixed-point theorems, yet i believe it will probably end up to be invaluable to investigate economists who paintings in microeconomic conception. This part provides 4 diversified proofs of Brouwer fixed-point theorem, an evidence of Kakutani's Fixed-Point Theorem, and concludes with an evidence of Nash's Theorem for n-person video games.
Unfortunately, an important math instruments in use by way of economists this present day, nonlinear programming and comparative statics, are slightly pointed out. this article has precisely one 15-page bankruptcy on nonlinear programming. This bankruptcy derives the Kuhn-Tucker stipulations yet says not anything in regards to the moment order stipulations or comparative statics results.
Most most likely, the unusual choice and insurance of subject matters (linear programming takes greater than 1/2 the textual content) easily displays the truth that the unique version got here out in 1980 and likewise that the writer is absolutely an utilized mathematician, now not an economist. this article is worthy a glance if you want to appreciate fixed-point theorems or how the simplex set of rules works and its functions. glance somewhere else for nonlinear programming or more moderen advancements in linear programming.
Read or Download Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems (Classics in Applied Mathematics, 37) PDF
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This article makes an attempt to survey the middle topics in optimization and mathematical economics: linear and nonlinear programming, keeping apart aircraft theorems, fixed-point theorems, and a few in their applications.
This textual content covers in simple terms matters good: linear programming and fixed-point theorems. The sections on linear programming are situated round deriving equipment in line with the simplex set of rules in addition to a few of the usual LP difficulties, corresponding to community flows and transportation challenge. I by no means had time to learn the part at the fixed-point theorems, yet i feel it could actually end up to be valuable to investigate economists who paintings in microeconomic concept. This part offers 4 assorted proofs of Brouwer fixed-point theorem, an explanation of Kakutani's Fixed-Point Theorem, and concludes with an evidence of Nash's Theorem for n-person video games.
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Extra info for Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems (Classics in Applied Mathematics, 37)
Example text
Are independent iff none of them is a linear combination of the others. EXAMPLE 1. Define the matrix We label the columns a1, a2, a3. The three columns are dependent, because they all lie in a 2-dimensional space. But every two of the columns are independent. For example, these two columns are independent: 28 1 Linear Programming EXAMPLE 2. For the matrix (3) let's compute all the basic solutions of First, there's a basic solution that depends on columns a1 and a2. We get this solution by solving We compute x1 = 3, x2 = 3.
In N dimensions, a convex polytope can be generated by any finite set of points x 1 , . . ,xp. The polytope consists of all the convex combinations where Every convex polytope is convex and closed. Lemma. Let C be a closed convex set that does not contain the origin x = 0. Then C contains a nearest point x°, with (The assumption that C is convex is superfluous but useful. ) PROOF. Let 6 be the greatest lower bound of |x| for all x in C: Let xl,x2,... be a sequence of points in C such that Then we can use convexity to prove that xk converges to the required nearest point x°.
Proof of (16): We know Ax = b, which says Form the equation (17) — A • (15): Since xa > 0, x^ > 0 , . . , we have for small |A|; then the components (19) give a new feasible solution. The new cost is If (16) were false, we could decrease the cost by letting A be some small positive or negative number. Then x would not be optimal. Contradiction; (16) is now proved. Let 2. start at zero and slowly increase. As long as the new components (19) remain ^ 0, they give a new optimal solution, since (16) implies new cost = old cost in (20).