Decision-Review-System (DRS) optimizer

This program implements DRS optimizer to minimize sphere function.

1. Notation and problem statement

We solve the box-constrained minimization problemmin⁡x∈Ωf(x),Ω={x∈Rd:  ℓ≤x≤u},\min_{x\in\Omega} f(x),\qquad \Omega = \{x\in\mathbb{R}^d:\; \ell \le x \le u\},x∈Ωmin​f(x),Ω={x∈Rd:ℓ≤x≤u},

where f:Rd→Rf:\mathbb{R}^d\to\mathbb{R}f:Rd→R is the objective (for example Sphere f(x)=∑j=1dxj2f(x)=\sum_{j=1}^d x_j^2f(x)=∑j=1d​xj2​), ddd is dimension, and ℓ,u∈Rd\ell,u\in\mathbb{R}^dℓ,u∈Rd are lower/upper bounds.

Algorithmic parameters:

• NNN — population size (agents).
• TTT — max iterations.
• RRR — initial reviews per agent.
• kkk — number of probes per review (local tries).
• s0s_0s0​ — initial exploration step fraction (relative to bounds).
• ρ∈(0,1)\rho\in(0,1)ρ∈(0,1) — review step reduction factor.
• (Optional) τ\tauτ and ppp — replenishment period and fraction.

Population at iteration ttt:

• xi(t)∈Ω,  i=1,…,Nx_i^{(t)}\in\Omega,\; i=1,\dots,Nxi(t)​∈Ω,i=1,…,N (agent solutions).
• fi(t)=f(xi(t))f_i^{(t)} = f(x_i^{(t)})fi(t)​=f(xi(t)​).
• ri(t)∈{0,1,… }r_i^{(t)}\in\{0,1,\dots\}ri(t)​∈{0,1,…} reviews left for agent iii.

Global best:xt∗=arg⁡min⁡ifi(t),ft∗=min⁡ifi(t).x^*_t = \arg\min_{i} f_i^{(t)},\qquad f^*_t = \min_i f_i^{(t)}.xt∗​=argimin​fi(t)​,ft∗​=imin​fi(t)​.

2. Step-size schedule

A decreasing step fraction:s(t)=s0⋅(1−tT),t=0,1,…,T.s(t) = s_0\cdot\left(1 - \frac{t}{T}\right),\qquad t=0,1,\dots,T.s(t)=s0​⋅(1−Tt​),t=0,1,…,T.

Absolute step per coordinate uses the box range Δ=u−ℓ\Delta = u-\ellΔ=u−ℓ. A scalar/vector multiplication is elementwise.

3. Candidate proposal (on-field decision)

For agent iii at iteration ttt generate a candidate by Gaussian perturbation:x~i=xi(t)+s(t) Δ⊙ξ,ξ∼N(0,Id),\tilde{x}_i = x_i^{(t)} + s(t)\,\Delta\odot \xi,\qquad \xi\sim\mathcal{N}(0,I_d),x~i​=xi(t)​+s(t)Δ⊙ξ,ξ∼N(0,Id​),

then project to bounds:

x~i←proj⁡[ℓ,u](x~i)(elementwise clamp).\tilde{x}_i \leftarrow \operatorname{proj}_{[\ell,u]}(\tilde{x}_i) \quad(\text{elementwise clamp}).x~i​←proj[ℓ,u]​(x~i​)(elementwise clamp).

Evaluate f~i=f(x~i) \tilde{f}_i = f(\tilde{x}_i)f~​i​=f(x~i​).

Acceptance (standard Metropolis-free greedy):if f~i<fi(t)thenxi(t+1)=x~i,  fi(t+1)=f~i.\text{if }\tilde{f}_i < f_i^{(t)}\quad\text{then}\quad x_i^{(t+1)}=\tilde{x}_i,\; f_i^{(t+1)} = \tilde{f}_i.if f~​i​<fi(t)​thenxi(t+1)​=x~i​,fi(t+1)​=f~​i​.

If f~i\tilde{f}_if~​i​ improved global best, update xt∗,ft∗x_t^*, f_t^*xt∗​,ft∗​.

4. Review mechanism (intensification)

If f~i≥fi(t)\tilde{f}_i \ge f_i^{(t)}f~​i​≥fi(t)​ (proposal worse) and ri(t)>0r_i^{(t)}>0ri(t)​>0, agent uses one review:

• decrement review: ri(t)←ri(t)−1r_i^{(t)}\leftarrow r_i^{(t)}-1ri(t)​←ri(t)​−1.
• perform kkk focused probes with reduced step:srev(t)=ρ  s(t),yi,ℓ=xi(t)+srev(t) Δ⊙ζℓ,ζℓ∼N(0,Id),s_{\text{rev}}(t) = \rho\;s(t),\qquad y_{i,\ell} = x_i^{(t)} + s_{\text{rev}}(t)\,\Delta\odot \zeta_\ell,\quad \zeta_\ell\sim\mathcal{N}(0,I_d),srev​(t)=ρs(t),yi,ℓ​=xi(t)​+srev​(t)Δ⊙ζℓ​,ζℓ​∼N(0,Id​),project each yi,ℓy_{i,\ell}yi,ℓ​ into [ℓ,u][\ell,u][ℓ,u], compute f(yi,ℓ)f(y_{i,\ell})f(yi,ℓ​) for ℓ=1,…,k\ell=1,\dots,kℓ=1,…,k.
• let (yibest,fibest)=arg⁡min⁡ℓf(yi,ℓ)(y_i^\text{best}, f_i^\text{best})=\arg\min_{\ell} f(y_{i,\ell})(yibest​,fibest​)=argminℓ​f(yi,ℓ​).
• If fibest<fi(t)f_i^\text{best} < f_i^{(t)}fibest​<fi(t)​ accept xi(t+1)=yibestx_i^{(t+1)} = y_i^\text{best}xi(t+1)​=yibest​, else keep xi(t+1)=xi(t)x_i^{(t+1)}=x_i^{(t)}xi(t+1)​=xi(t)​.

So reviews allow an agent to perform a short, intensified local search to “overturn” a worse on-field decision.

5. No-review case (exploration fallback)

If f~i≥fi(t)\tilde{f}_i\ge f_i^{(t)}f~​i​≥fi(t)​ and ri(t)=0r_i^{(t)}=0ri(t)​=0, perform a small jitter:x^i=xi(t)+γ s(t) Δ⊙η,η∼U([−1,1]d),\hat{x}_i = x_i^{(t)} + \gamma\, s(t)\,\Delta\odot \eta,\quad \eta\sim\mathcal{U}([-1,1]^d),x^i​=xi(t)​+γs(t)Δ⊙η,η∼U([−1,1]d),

with tiny γ\gammaγ (e.g. 0.05). Accept if improves.

6. Replenishment (optional)

Every τ\tauτ iterations, restore reviews to a random subset of agents: choose m=⌈pN⌉m=\lceil pN\rceilm=⌈pN⌉ agents and set rj←Rr_j \leftarrow Rrj​←R for those. This encourages intermittent deep search. In practice τ≈T/5\tau\approx T/5τ≈T/5, p∈(0,0.3)p\in(0,0.3)p∈(0,0.3).

7. Projection operator

proj⁡[ℓ,u](z)=min⁡(max⁡(z,ℓ),u)(elementwise).\operatorname{proj}_{[\ell,u]}(z) = \min(\max(z,\ell),u) \quad \text{(elementwise)}.proj[ℓ,u]​(z)=min(max(z,ℓ),u)(elementwise).

8. Termination

Stop after t=Tt=Tt=T or earlier if ft∗f^*_tft∗​ below desired tolerance ϵ\epsilonϵ.

9. Algorithm summary (pseudocode)

For completeness:

1. Initialize xi(0)∼ℓ+Δ⊙Uniform(0,1)dx_i^{(0)}\sim \ell + \Delta\odot\text{Uniform}(0,1)^dxi(0)​∼ℓ+Δ⊙Uniform(0,1)d, fi(0)f_i^{(0)}fi(0)​, and ri(0)=Rr_i^{(0)}=Rri(0)​=R.
2. For t=0t=0t=0 to T−1T-1T−1:

• compute s(t)s(t)s(t).
• for each agent iii:
• propose x~i\tilde{x}_ix~i​ and f~i\tilde{f}_if~​i​.
• if f~i<fi(t)\tilde{f}_i < f_i^{(t)}f~​i​<fi(t)​ accept.
• else if ri(t)>0r_i^{(t)}>0ri(t)​>0 use review: run kkk probes with reduced step, accept best probe if improves.
• else attempt small jitter and accept if improves.
• optionally replenish some reviews every τ\tauτ iterations.
• update global best.

1. Return xT∗x^*_TxT∗​, fT∗f^*_TfT∗​.

10. Complexity

Let CfC_fCf​ be cost per evaluation of fff. Per iteration:

• baseline proposals: NNN evaluations.
• reviews: each used review adds up to kkk extra evaluations. If average reviews used per iter is rˉ\bar{r}rˉ, cost per iter ≈(N+rˉk)Cf\approx (N + \bar{r}k)C_f≈(N+rˉk)Cf​.Total cost O(T (N+rˉk) Cf)\mathcal{O}\big(T\,(N + \bar{r}k)\,C_f\big)O(T(N+rˉk)Cf​).

11. Behavior & design rationale

• Greedy acceptance enforces monotone improvement for each agent.
• Reviews introduce occasional intensified local search (exploitation) but are limited by RRR, preserving exploration.
• Step schedule s(t)s(t)s(t) reduces step sizes over time for convergence-like behavior.
• Replenishment injects episodic intensification to escape stagnation.
• Projection keeps iterates feasible.

12. Parameter recommendations (practical)

• NNN: 20–100 (40 is a good default for moderate dimensions).
• TTT: depends on budget — 200–1000 typical.
• RRR: 1–4 reviews per agent (2 is standard).

The DRS optimization algorithm provides a balance of exploration (standard proposals) and exploitation (review-based local searches) inspired by the cricket DRS process.

It ensures monotonic improvement under greedy acceptance, bounded search space handling, and probabilistic intensification through limited review opportunities.

• kkk: 3–8 probes per review (4 used in code).
• s0s_0s0​: 0.2–0.6 (0.4 moderate).
• ρ\rhoρ: 0.3–0.7 (0.5 reduces step notably during reviews).
• γ\gammaγ (jitter scale): 0.01–0.1.
• τ\tauτ: T/5T/5T/5, ppp: 0.1–0.25 if using replenishment.

13. Specialization to Sphere

If f(x)=∑j=1dxj2f(x)=\sum_{j=1}^d x_j^2f(x)=∑j=1d​xj2​, the method is derivative-free and will converge because:

• decreasing step s(t)→0s(t)\to 0s(t)→0,
• occasional local probes allow moving toward origin,
• greedy acceptance ensures nonincreasing per-agent objective.

14. Variants and extensions

• Replace Gaussian proposal with Lévy or differential moves.
• Use acceptance with Metropolis probability to allow uphill moves.
• Maintain an external archive of best samples for crossover.
• Adaptive reviews: increase RRR for agents achieving large improvements.