Monte Carlo method-based optimization techniques sample the objective function by randomly "hopping" from the current solution vector to another with a difference in the function value of ${\displaystyle \Delta E}$ . The acceptance probability of such a trial jump is in most cases chosen to be ${\displaystyle \min \left(1;\exp \left(-\beta \cdot \Delta E\right)\right)}$  (Metropolis criterion) with an appropriate parameter ${\displaystyle \beta }$ .

The general idea of STUN is to circumvent the slow dynamics of ill-shaped energy functions that one encounters for example in spin glasses by tunneling through such barriers.

This goal is achieved by Monte Carlo sampling of a transformed function that lacks this slow dynamics. In the "standard-form" the transformation reads ${\displaystyle f_{STUN}:=1-\exp \left(-\gamma \cdot \left(E(x)-E_{o}\right)\right)}$  where ${\displaystyle E_{o}}$  is the lowest function value found so far. This transformation preserves the loci of the minima.

${\displaystyle f_{STUN}}$  is then used in place of ${\displaystyle E}$  in the original algorithm giving a new acceptance probability of ${\displaystyle \min \left(1;\exp \left(-\beta \cdot \Delta f_{STUN}\right)\right)}$ 

The effect of such a transformation is shown in the graph.

A variation on always tunneling is to do so only when trapped at a local minimum. ${\displaystyle \gamma }$  is then adjusted to tunnel out of the minimum and pursue a more globally optimum solution. Detrended fluctuation analysis is the recommended way of determining if trapped at a local minimum.

• Simulated annealing
• Parallel tempering
• Genetic algorithm
• Differential evolution

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