# Lifts and Projections of Sets defined by Polynomial Inequalities

My research is on using design optimisation studies using ridge functions of quantities of interest. Ridge functions allow us to create surrogate models in lower dimensional subspaces using fewer function evaluations than would be required in the full $$n$$-dimensional space (defined to lie in the $$n$$-dimensional hypercube i.e. $$\mathbf{x} \in [-1,1]^n$$).

So for each quantity of interest, we have a ridge function $$f(\mathbf{x})\approx g(\mathbf{u})$$, where $$\mathbf{u} = U^T\mathbf{x}$$ ($$U \in \mathbb{R}^{n \times d}$$ and $$d) are the active variables in which the function varies the most. We also have the inactive variables $$\mathbf{v} = V^T\mathbf{x}$$ with which the quantity of interest does not vary a great deal, where $$V=\mathrm{null}(U^T)$$. Note, concatenating these matrices gives the orthogonal matrix $$W=[U,V]$$ and given the active and inactive variables, we can return to $$n$$-dimensional space using $$\mathbf{x}=U\mathbf{u} + V\mathbf{v}$$.

Given a ridge function $$f_1 \approx g_1(\mathbf{u}_1)$$, I can define a feasible set, say $$\Omega_1$$, in terms of its active variables by a set of $$m$$ inequalities i.e. $$\Omega_1 = \{ \mathbf{u}_1 | h_i(\mathbf{u}_1)\leq b_i, \quad i = 1,\dots,m \}$$. However, these active and inactive variables are determined by their effect on their respective quantities of interest, and so reside in different subspaces.

I am interested in what I can say about this set in the subspace defined by a different ridge function $$f_2 \approx g_2(\mathbf{u}_2)$$. That is, what is there to be said about the set $$\Omega_2 = \{ \mathbf{u}_2 | \mathbf{u}_1 \in \Omega_1 \}$$?

In the case where $$g_1(\mathbf{u}_1)$$ is a linear ridge function, the set $$\Omega_1$$ is an $$n$$-dimensional polytope (including the bounds from $$\mathbf{x}$$ being contained in the unit hypercube). This means we can use a number of available polytope projection algorithms to find the set $$\Omega_2$$ (relatively) easily.

However, I am unsure of where to even begin (or indeed if such a representation exists) in the case where $$g_1(\mathbf{u}_1)$$ is nonlinear (e.g. a quadratic). If anyone has any suggestions as to pertinent research or reading that I may look into to begin investigating this question, I would really appreciate it.

To give a visual aide, I used a very naive and inefficient sampling technique to investigate for relatively low dimensions to get some images of the sets I would expect. Feasible set defined in space $$\mathbf{u}_1$$ projected onto the space $$\mathbf{u}_2$$ The grey sets being the feasible sets and the black dashed line being the boundary of the active variables such that the condition that $$\mathbf{x}$$ being contained in the unit hypercube is met.

Thank you for taking the time to read my post.