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By Barry Evans

In 30 essays--filled with anecdotes and illustrations--Evans takes such ordinary techniques as gravity, water, and breath and turns them into delightfully documented adventures. specific interviews with Stephen Jay Gould, Linus Pauling, and different inventive and articulate scientists upload an additional measurement. photographs. Line drawings. Puzzles.

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16 Chapter Two Linear Inequalities and Theorems of the Alternative 1. Introduction It was mentioned in Chap. 1 that the presence of inequality constraints in a minimization problem constitutes the distinguishing feature between the minimization problems of the classical calculus and those of nonlinear programming. Although our main interest lies in nonlinear problems, and hence in nonlinear inequalities, linearization (that is, approximating nonlinear constraints by linear ones) will be frequently resorted to.

The type of theorem that will concern us in this chapter will involve two systems of linear inequalities and/or equalities, say systems I and II. A typical theorem of the alternative asserts that either system I has a solution, or that system II has a solution, but never both. The most famous theorem of this type is perhaps Farkas' theorem [Farkas 02, Tucker 56, Gale 60]. i Fig. 1 Geometric interpretation of Farkas' theorem: II' has solution, I' has no solution. has a solution but never both.

For 0 ^ X g 1 Hence r + A is convex. Theorem The product pY of a convex set T in Rn and the real number p is a convex set. 2 Nonlinear Programming PROOF Let zl,z* £ »T, then z1 = »xl, z2 = ^x\ where xl,x2 £ T. 0 ^ X^ 1 For Corollary // F and A are two convex sets in R", then F — A is a convex set. 2. Separation theorems for convex sets It is intuitively plausible that if we had two disjoint convex sets in Rn, then we could construct a plane such that one set would lie on one side of the plane and the other set on the other side.

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