min is concave. So to do what you want requires introduction of binary variables to “linearize” the non-convex use of min. Look at the last bullet of https://www.fico.com/fico-xpress-optimization/docs/latest/mipform/dhtml/chap2s1.html?scroll=ssecminval to see how. I leave you to work out the details for your program.
Also, please read
Why isn't CVX accepting my model? READ THIS FIRST!