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In this paper, we propose new Metropolis-Hastings and simulated annealingalgorithms on finite state space via modifying the energy landscape. The coreidea of landscape modification rests on introducing a parameter $c$, in whichthe landscape is modified once the algorithm is above this threshold parameterto encourage exploration, while the original landscape is utilized when thealgorithm is below the threshold for exploitation purpose. We illustrate thepower and benefits of landscape modification by investigating its effect on theclassical Curie-Weiss model with Glauber dynamics and external magnetic fieldin the subcritical regime. This leads to a landscape-modified mean-fieldequation, and with appropriate choice of $c$ the free energy landscape can betransformed from a double-well into a single-well, while the location of theglobal minimum is preserved on the modified landscape. Consequently, runningalgorithms on the modified landscape can improve the convergence to theground-state in the Curie-Weiss model. In the setting of simulated annealing,we demonstrate that landscape modification can yield improved or evensubexponential mean tunneling time between global minima in the low-temperatureregime by appropriate choice of $c$, and give convergence guarantee using animproved logarithmic cooling schedule with reduced critical height. We alsodiscuss connections between landscape modification and other accelerationtechniques such as Catoni's energy transformation algorithm, preconditioning,importance sampling and quantum annealing. The technique developed in thispaper is not only limited to simulated annealing and is broadly applicable toany difference-based discrete optimization algorithm by a change of landscape.

Improved Metropolis-Hastings algorithms via landscape modifcation with applications to simulated annealing and the Curie-Weiss model