Approximation with inequality constraints

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Suppose $\mathbf x = [x_1\; x_2\; \cdots\; x_n]$ is a discrete approximation of a function at $n$ points. I want to get another approximation of this function at $n/2$ even points, say $\mathbf y = [x'_2\; x'_4\; \cdots\; x'_{n/2}]$ based on the approximation $\mathbf x$. And I want it to satisfy inequality conditions $A\mathbf y + \mathbf b \geq \mathbf 0$. It looks to me like a constrained optimization problem, but I haven't taken a course on that. And I also don't know how to set the target function if that is really an optimization problem. Any suggestions?