Boundary conditions¶

Updated positions may leave the feasible box \([\mathbf{LB},\mathbf{UB}]\). Because many objectives are undefined outside their bounds — and optimal solutions often lie near the bounds — DEEM repairs infeasible components with a hybrid strategy that mixes damped reflection and periodic corrections. Mixing the two increases the diversity of repaired solutions and avoids the bias that a single rule would introduce.

For each dimension a uniform random number decides which rule is applied, and a second uniform random number \(r=U(0,1)\) scales the correction. The maximum allowed adjustment is the bound span \(\Delta^{max}=UB_{iD}-LB_{iD}\).

Damped reflection¶

If the component is out of bounds, it is moved back inside by a random fraction of the overshoot, capped at \(\Delta^{max}\):

\[ x^{\,i+1}_{j,iD} \leftarrow \begin{cases} LB_{iD} + r\,\Delta, & \Delta=\min\!\big(LB_{iD}-x^{\,i+1}_{j,iD},\ \Delta^{max}\big), & x^{\,i+1}_{j,iD} < LB_{iD},\\[6pt] UB_{iD} - r\,\Delta, & \Delta=\min\!\big(x^{\,i+1}_{j,iD}-UB_{iD},\ \Delta^{max}\big), & x^{\,i+1}_{j,iD} > UB_{iD}. \end{cases} \]

Periodic correction¶

The periodic rule wraps a component that exceeds one boundary to near the opposite boundary, preserving a continuous search-space topology. With \(\Delta x = \left(\frac{\Delta}{\Delta^{max}} - \big\lfloor \frac{\Delta}{\Delta^{max}}\big\rfloor\right)\Delta^{max}\):

\[ x^{\,i+1}_{j,iD} \leftarrow \begin{cases} UB_{iD} - r\,\Delta x, & \Delta = LB_{iD}-x^{\,i+1}_{j,iD}, & x^{\,i+1}_{j,iD} < LB_{iD},\\[6pt] LB_{iD} + r\,\Delta x, & \Delta = x^{\,i+1}_{j,iD}-UB_{iD}, & x^{\,i+1}_{j,iD} > UB_{iD}, \end{cases} \]

where \(\lfloor\cdot\rfloor\) is the floor function.

The hybrid rule¶

Algorithm 3 — Hybrid boundary-condition enforcement

function enforce_BC(x, LB, UB, method):
    for iD = 0 .. nD:
        u      <- U(0,1)
        r      <- U(0,1)
        Δmax   <- UB[iD] - LB[iD]
        if u >= 0.5:                      # damped reflection
            if   x[iD] < LB[iD]:  Δ = min(LB[iD]-x[iD], Δmax); x[iD] = LB[iD] + r·Δ
            elif x[iD] > UB[iD]:  Δ = min(x[iD]-UB[iD], Δmax); x[iD] = UB[iD] - r·Δ
        else:                             # periodic
            if   x[iD] < LB[iD]:  Δ = LB[iD]-x[iD]; Δx = frac(Δ/Δmax)·Δmax; x[iD] = UB[iD] - r·Δx
            elif x[iD] > UB[iD]:  Δ = x[iD]-UB[iD]; Δx = frac(Δ/Δmax)·Δmax; x[iD] = LB[iD] + r·Δx
    return x

Choosing the method in code¶

The boundary strategy is selected with the method_boundary argument. The implementation accepts the single rules and several combinations:

clip, random, damping, periodic, damping-periodic, damping-periodic-random, damping-periodic-clip.

The hybrid rule described above corresponds to damping-periodic (the per-dimension coin flip between damped reflection and periodic wrapping).