  The Quantum Alternating Operator Ansatz (QAOA+) is one of the Variational
Quantum Algorithm (VQA) specifically developed to tackle combinatorial
optimization problems by exploring the feasible space in search of a target
solution. For constrained optimization problems with unconstrained variables,
which we call Unconstrained-Variables Problems (UVPs), the mixed operators in
the QAOA+ circuit are applied to the constrained variables, while the
single-qubit rotating gates $R_X$ operate on the unconstrained variables. The
expressibility of this circuit is limited by the shortage of two-qubit gates
and the parameter sharing in the $R_X$, which consequently impacts the
performance of QAOA+ for solving UVPs. Therefore, it is crucial to develop a
suitable ansatz for UVPs. In this paper, we propose the Variational Quantum
Algorithm-Preserving Feasible Space (VQA-PFS) ansatz, exemplified by the
Uncapacitated Facility Location Problem (UFLP), that applies mixed operators on
constrained variables while employing Hardware-Efficient Ansatz (HEA) on
unconstrained variables. The numerical results demonstrate that VQA-PFS
significantly enhances the success probability and exhibits faster convergence
compared to QAOA+, Quantum Approximation Optimization Algorithm (QAOA), and
HEA. Furthermore, VQA-PFS reduces the circuit depth dramatically in comparison
to QAOA+ and QAOA. Our algorithm is general and instructive in tackling UVPs.
