Let $S \subset \mathbb{R}^n$ satisfy $0\leq x_1\leq\dots\leq x_n$ for all $\mathbf{x}\in S$, among other (possibly nonconvex) constraints, and suppose in addition that $\sum_{i=1}^n x_i \geq 1$ for all $\mathbf{x}\in S$. Suppose that the following process is guaranteed to find a point $\bar{\mathbf{x}}$ such that $\sum_{i=1}^n \bar{x}_i \leq 2$:
- Fix $\bar{x}_1$ to be the minimum possible value of $x_1$ in $S$, i.e. $\bar{x}_1 = \min_{\mathbf{x}\in S} x_1$.
- For each $i\geq 2$, fix $\bar{x}_i$ to be the minimum possible value of $x_i$ in $S$ with all previous $\bar{x}_1,\dots,\bar{x}_{i-1}$ fixed, i.e. $$\bar{x}_i = \min_{\mathbf{x}\in S:x_1=\bar{x}_1,\dots,x_{i-1}=\bar{x}_{i-1}} x_i$$.
We can say that $\bar{\mathbf{x}}$ is "near optimal" in terms of minimizing $\sum_{i=1}^n x_i$ over $S$ (since it is within a factor of $2$ of optimality).
My question is: is $\bar{\mathbf{x}}$ also "near optimal" for minimizing all other linear objectives $\mathbf{c}^\top\mathbf{x}$, when $c_1\geq \cdots \geq c_n$?
