I have an optimization problem which I think might be fairly difficult, so I want to ask if it's practically possible.
To start out, I have a linear programming problem, for which I obtain the optimal objective value. The problem has multiple optimal solutions. Next, I want to investigate possible solutions subject to the objective value being at least a given fraction of the optimal value (so, just another linear constraint). Specifically, I want to minimize the euclidean distance between two vectors, one of them given and the other being a vector of ratios of variables in the problem. Are there any algorithms available that can solve this problem?
The problem has typically at least 100 variables, and the matrix A in Ax = 0 is sparse. Here's an attempt at formulating the problem:
\begin{gathered} \min \sum\limits_i {{{\left( {\frac{{{x_{a(i)}}}}{{{x_{b(i)}}}}} \right)}^2} - 2{y_i}\left( {\frac{{{x_{a(i)}}}}{{{x_{b(i)}}}}} \right)} \\ s.t: \\ Ax = 0 \\ Z = {c^T}x \\ {\alpha _i} \leqslant {x_i} \leqslant {\beta _i} \\ Z \geqslant {Z^ * } \\ 0 \leqslant {Z^ *} \leqslant {Z_{\max}} \end{gathered}
