We discuss the asymptotic behavior of time-inhomogeneous Metropolis chains for solving constrained sampling and optimization problems. In addition to the usual inverse temperature schedule (βn)n∈[hollow N]*, the type of Markov processes under consideration is controlled by a divergent sequence (θn)n∈[hollow N]* of parameters acting as Lagrange multipliers. The associated transition probability matrices (Pβn,θn)n∈[hollow N]* are defined by Pβ,θ = q(x, y)exp(−β(Wθ(y) − Wθ(x))+) for all pairs (x, y) of distinct elements of a finite set Ω, where q is an irreducible and reversible Markov kernel and the energy function Wθ is of the form Wθ = U + θV for some functions U,V : Ω → [hollow R]. Our approach, which is based on a comparison of the distribution of the chain at time n with the invariant measure of Pβn,θn, requires the computation of an upper bound for the second largest eigenvalue in absolute value of Pβn,θn. We extend the geometric bounds derived by Ingrassia and we give new sufficient conditions on the control sequences for the algorithm to simulate a Gibbs distribution with energy U on the constrained set [Ω with tilde above] = {x ∈ Ω : V(x) = minz∈ΩV(z)} and to minimize U over [Ω with tilde above].