I want to solve the following problem, the cvx gives me the solution, but this solution doesn’t satisfy the constraint ( w_tm’Vw_tm <= (1+tol)^2* var_mini)？

When optimality is claimed, the solvers called by CVX are only “guaranteed” to provide solutions which satisfy constraints to within a feasibility tolerance, which might typically be approximately 1e-8. If the constraint is violated by 1e-9, that is considered acceptable; but if that occurred in the constraint you claim is violated, it would produce logical 0 in your constraint satisfaction assessment. You should subtract one side of the inequality constraint from the other and look at the numerical value.

If you need a constraint to be strictly satisfied, even though the solver might violate it by a small amount, you need to include a buffer amount. Pick a small_positive _number, such as 1e-6; change LHS <= RHS
into LHS + small_positive_number <= RHS.
Then if CVX/solver claim the problem is solved to optimality, it is actually the case that LHS <= RHS.

General advice for the future:
Don’t use quiet until you are sure everything is working correctly. That way, you will see the solver and CVX output.

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