My approach would be to reduce this to a 1-variable minimization over x(11) and solve that with fminsearch,
The objective function, which I call innerProblem here, simply does a combinatoric search for the minimum of the conic program over the integer variables subject to a fixed known input value for the eleventh variable x11. Because your integer variables only range from 1 to 3, there aren't that many combinations to search through. All the combinations can be easily pre-computed as columns of the 10x59049 matrix Combinations. Also, the operations needed to evaluate all the combinations are highly vectorizable - no for-loops involved.