Hello

I'm trying to solve some problems on Nonlinear Optimization. I have the book "Nonlinear Optimization" - by Andrzej Ruszczyński, 2006.

It's problem 3.8 and sounds as follows.

The output variable y \in \mathbb{R} of a system depends on the input variable x \in \mathbb{R}. We have collecred observations \tilde{y_i}, \, i=1, \dots , N, for N different settings x_i of the input variable, with x_1 < x_2 < \dots < x_N.

(a) We know that the dependence of y on x should be nondecreasing. Formulate the problem of finding new values y_i such that y_1 \leq y_2 \leq \cdots \leq y_N and the sum of the squares of adjustments is minimized.

(b) We know that the dependence of y on x should be convex and nondecreasing. Formulate the problem of finding new values y_i such that this is satisfied and the sum of the squares of adjustments is minimized.

I'm a bit puzzled of what he means with "the dependence of y on x should be nondecreasing" and "the dependence of y on x should be convex and nondecreasing"

In (a) I have formulated the optimization problem as:
\min \left( \sum_{i=1}^N (\tilde{y}_i - y_i)^2 \right)
and have defined my y_i's as:
y_1 \leq y_2 \leq \cdots \leq y_N
and
y_1 \geq x_1, y_2 \geq x_2 , \cdots  y_N \geq x_N.
But as I wrote, I'm a bit lost on what he wants me to do ?

Can anyone enlighten me ?