Appreciate
The foundational appreciation/depreciation penalty sequence of Park and Kim (2015, Eq. 4): the penalty is multiplied by the appreciation factor appreciationFactor (a, greater than 1) when the solution looks infeasible (standardized measure S greater than 0) and by the depreciation factor depreciationFactor (d, between 0 and 1) when it looks feasible (S at most 0). This makes the sequence diverge for infeasible solutions and converge to zero for interior feasible solutions, satisfying Park and Kim's Condition 1 for those cases. For boundary (active) solutions it converges only in distribution (their Theorem 3), with positive probability of not vanishing — the precise gap that the convergence-optimized PS1/PS2 close. This class is that foundational sequence, not PS1 or PS2.
Park and Kim's fuller PS1 (their Figure 3: a visit-count-dependent depreciation factor and an M0 * k^rho appreciation cap) and PS2 (their Section 3.3: infeasibility-probability-adaptive factors) are faithful refinements that additionally require global search state — chiefly the adaptive cap M0 (their Eq. 5), which is computed from the sample-best feasible objective across visited solutions. That global state is the deferred SearchStateSnapshot seam; those sequences drop into PenaltySequence once it is populated. This sequence needs only per-solution memory.
Parameters
a: the growth factor applied when S is greater than 0; must be greater than 1
d: the decay factor applied when S is at most 0; must be in (0, 1)
the starting penalty value; must be positive and finite