4.10 Appendix

4.10.1 Proof of Proposition 4.1 in Section 4.4

We follow the technique used in [203] by defining the function

Equation 4.184

Notice that

Equation 4.185

Equation 4.186

Equation 4.187

Equation 4.188

where (4.188) follows from the assumption that y '( x ) m . In (4.188), denotes that the matrix ( A - B ) is positive semidefinite. It then follows from (4.185), (4.187), and (4.188) that d ( t ) 0, for any t . Now on setting

Equation 4.189

we obtain

Equation 4.190

Assume that the penalty function r ( x ) is convex and bounded from below; then the cost function C ( q ) is convex and has a unique minimum C ( q *). Therefore, q * is the unique solution to (4.15) such that z ( q *) = . Since the sequence C ( q ^l ) is decreasing and bounded from below, it converges. Therefore, from (4.190) we have

Equation 4.191

Since for any realization of r , the probability that z ( q ^l ) falls in the null space of the matrix ( SR ^-1 S ^T ) is zero, then (4.191) implies that z ( q ^l ) with probability 1. Since z ( q ) is a continuous function of q and has a unique minimum point q *, we then have q ^l q * with probability 1, as l .

4.10.2 Proof of Proposition 4.2 in Section 4.5

Denote . Then (4.75) can be written in matrix form as

Equation 4.192

Denote . Then from (4.73) and (4.74) we obtain

Equation 4.193

Using (4.192) and (4.193), we obtain

Equation 4.194

Equation 4.195

Equation 4.196

where in (4.194) denotes the Moore “Penrose generalized matrix inverse [189]; in (4.195) we have used the fact that , which can easily be verified by using the definition of the Moore “Penrose generalized matrix inverse [189]; in (4.196) we have used the facts that ; and (4.196) is the matrix form of (4.76). Finally, we notice that

Equation 4.197

It follows from (4.197) that the k th diagonal element of the diagonal matrix A ^-2 satisfies

Equation 4.198