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OPTIMIZATION OVER AN OPEN SET

In this chapter we study in detail the first example of Chapter 1. We first establish some notation which will be in force throughout these Notes. Then we study our example. This will generalize to a canonical problem, the properties of whose solution are stated as a theorem. Some additional properties are mentioned in the last section.

All vectors are column vectors, with two consistent exceptions mentioned in 2.1.3 and 2.1.5 below and some other minor and convenient exceptions in the text. Prime denotes transpose so that if x Є Rⁿ then x ' is the row vector x' = (x ₁,..., x _n), and x = (x ₁,..., x _n) '. Vectors are normally denoted by lower case letters, the ith component of a vector x Є Rⁿ is denoted x_i , and differentvectors denoted by the same symbol are distinguished by superscripts as in x ^j and x ^k. 0 denoted both the zero vector and the real number zero, but no confusion will result.

Thus if x = (x ₁,..., x _n) ' and y = (y ₁,..., y _n) ' then x'y = x ₁ y ₁ +... + x_ny_n as in ordinary matrix multiplication. If x Є Rⁿ we define| x | = √ x'x.

If x = (x ₁,..., x _n) ' and y = (y ₁,..., y _n) ' then x ≥ y means x _i ≥ y _i , i = 1,..., n. In particular if x Є Rⁿ , then x ≥ 0, if x _i ≥ 0, i = 1,..., n.

Matrices are normally denoted by capital letters. If A is an m x n matrix, then A^j denotes the jth column of A, and A_i denotes the ith row of A. Note that A_i is a row vector. A^j_i denotes the entry of A in the ith row and jth column; this entry is sometimes also denoted by the lower case letter a_ij, and then we also write A = { a_ij }. I denotes the identity matrix; its size will be clear from the context. If confusion is likely, we write I_n, to denote the n x n identity matrix.