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feat: optimized route proof + scale space solver fix
Lean: - OptimizedRoute.lean: 2-opt route shorter than exactishRoute optimizedRoute cost: 345147 vs exactishRoute: 401666 (14.1% shorter) Proofs: optimizedRoute_length, optimizedRoute_shorter, costSavings_positive All via native_decide. lake build: 3571 jobs, 0 errors. Python: - scale_space_solver.py: replaced Gaussian cost smoothing with cluster-based multi-scale optimization. Single-linkage clustering at each sigma, reduced TSP on representatives, expand + 2-opt polish. Fixed voltage/scale mapping.
This commit is contained in:
Brandon Schneider
parent
e2f3a9e93b
commit
7884fd074b
2 changed files with 332 additions and 80 deletions
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@ -0,0 +1,42 @@
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import Semantics.RouteCost
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/-!
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# Optimized Route Proof
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2-opt local search over exactishRoute finds a strictly cheaper permutation
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of the same 39 nodes. The cost comparison is over `Nat` (Q16.16-scaled
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fixed-point arithmetic), so `native_decide` closes the inequality gate.
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Route discovered by 2-opt: cost 345147 < 401666 (exactishRoute).
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-/
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namespace Semantics.RouteCost
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/-- The 2-opt optimized route: same 39 nodes, different order. -/
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def optimizedRoute : List RouteNode :=
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[ nF01, nF02, nF03, nS1, nB2, nS2, nF04, nF05, nF06, nF07
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, nM2, nM3, nF12, nF11, nM5, nM1, nB3, nB1, nS4, nB4
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, nS3, nF08, nF09, nF10, nM4, nF37, nF34, nF24, nF17, nF16
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, nG0, nB5, nB8, nB7, nB6, nS5, nX3, nX2, nX1
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]
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/-- Both routes visit 39 nodes. -/
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theorem optimizedRoute_length :
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optimizedRoute.length = exactishRoute.length := by
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native_decide
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/-- The optimized route is strictly cheaper than the exactish route.
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Cost: 345147 < 401666 (Q16.16-scaled Nat). -/
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theorem optimizedRoute_shorter :
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routeCostSum optimizedRoute < routeCostSum exactishRoute := by
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native_decide
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/-- Cost savings from 2-opt optimization (Q16.16-scaled Nat). -/
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def costSavings : Nat :=
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routeCostSum exactishRoute - routeCostSum optimizedRoute
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theorem costSavings_positive : costSavings > 0 := by
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unfold costSavings
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exact Nat.sub_pos_of_lt optimizedRoute_shorter
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end Semantics.RouteCost
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@ -1,14 +1,18 @@
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"""
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Multi-scale optimization using scale space theory.
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Implements coarse-to-fine optimization via Gaussian smoothing at multiple
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scales, with Q16.16 fixed-point arithmetic for FPGA compatibility.
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Implements coarse-to-fine optimization via cluster-based route optimization
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at multiple scales, with Q16.16 fixed-point arithmetic for FPGA compatibility.
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Scale mapping:
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σ₃ (1.0): coarse LP relaxation → approximate solution
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σ₂ (0.75): tighter LP → better solution
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σ₁ (0.5): exact MIP → optimal solution
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σ₀ (0.25): formal verification target
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σ₃ (1.0): coarse — merge nearby nodes, solve small problem
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σ₂ (0.75): medium — tighter clustering
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σ₁ (0.5): fine — minimal clustering, warm-started
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σ₀ (0.25): formal verification target — full problem, 2-opt polish
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The Gaussian kernel is used for route-space smoothing, NOT cost matrix
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smoothing. At each scale σ, nodes whose pairwise cost is below σ·max_cost
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are clustered together. The reduced problem is solved, then expanded back.
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"""
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import math
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@ -140,17 +144,19 @@ def gaussian_kernel_2d_q16(sigma: float, size: int = 16) -> list[list[int]]:
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# Voltage ↔ scale mapping
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# ---------------------------------------------------------------------------
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# Voltage range: 0.6V → σ=1.0 (coarse), 1.2V → σ=0.0 (fine/identity)
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# Voltage range: 0.6V → σ=1.0 (coarse), 1.2V → σ=0.25 (fine)
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_VOLTAGE_MIN = 0.6
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_VOLTAGE_MAX = 1.2
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_SIGMA_AT_VMIN = 1.0
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_SIGMA_AT_VMAX = 0.01 # Not exactly 0 to avoid degenerate kernel
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_SIGMA_AT_VMAX = 0.25 # Matches finest scale in default sigmas
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def voltage_to_scale(voltage_mv: float) -> float:
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"""Map millivolt voltage to scale parameter σ.
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Range: 0.6V (600mV, σ=1.0) to 1.2V (1200mV, σ≈0.01).
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Range: 0.6V (600mV, σ=1.0) to 1.2V (1200mV, σ=0.25).
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Linear mapping: σ = 1.0 - (V - 0.6) / 0.6 * 0.75
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Args:
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voltage_mv: Voltage in millivolts.
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@ -161,7 +167,7 @@ def voltage_to_scale(voltage_mv: float) -> float:
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voltage_v = voltage_mv / 1000.0
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# Clamp to range
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voltage_v = max(_VOLTAGE_MIN, min(_VOLTAGE_MAX, voltage_v))
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# Linear interpolation: σ = 1.0 - (V - 0.6) / 0.6 * 0.99
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# Linear interpolation: σ = 1.0 - (V - 0.6) / 0.6 * 0.75
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t = (voltage_v - _VOLTAGE_MIN) / (_VOLTAGE_MAX - _VOLTAGE_MIN)
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sigma = _SIGMA_AT_VMIN + t * (_SIGMA_AT_VMAX - _SIGMA_AT_VMIN)
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return max(0.01, sigma)
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@ -184,58 +190,221 @@ def scale_to_voltage(sigma: float) -> float:
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# ---------------------------------------------------------------------------
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# Multi-scale solver
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# Cluster-based multi-scale solver
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# ---------------------------------------------------------------------------
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def _apply_smoothing_q16(matrix: list[list[float]],
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sigma: float) -> list[list[float]]:
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"""Apply Gaussian smoothing to a cost matrix using Q16.16 arithmetic.
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def _single_linkage_clusters(cost_matrix: list[list[float]],
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threshold: float) -> list[list[int]]:
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"""Cluster nodes using single-linkage clustering.
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Convolves each row and column with the Gaussian kernel.
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Merge nodes whose minimum pairwise cost is below the threshold.
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Args:
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cost_matrix: n×n cost matrix.
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threshold: Cost threshold for merging.
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Returns:
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List of clusters, where each cluster is a list of node indices.
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"""
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n = len(matrix)
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n = len(cost_matrix)
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if n == 0:
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return matrix
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return []
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kernel = gaussian_kernel_q16(sigma, size=min(n, 33))
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k_half = len(kernel) // 2
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# Union-find for single-linkage
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parent = list(range(n))
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# Convert matrix to Q16.16
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q16_matrix = [[q16_from_float(matrix[i][j]) for j in range(n)]
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for i in range(n)]
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def find(x: int) -> int:
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while parent[x] != x:
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parent[x] = parent[parent[x]]
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x = parent[x]
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return x
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# Smooth rows
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smoothed = [[0] * n for _ in range(n)]
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def union(x: int, y: int) -> None:
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rx, ry = find(x), find(y)
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if rx != ry:
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parent[rx] = ry
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# Merge pairs whose cost is below threshold
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for i in range(n):
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for j in range(n):
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total = 0
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for ki in range(len(kernel)):
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jj = j + ki - k_half
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if 0 <= jj < n:
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total += q16_multiply(q16_matrix[i][jj], kernel[ki])
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else:
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# Mirror boundary
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jj = max(0, min(n - 1, jj))
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total += q16_multiply(q16_matrix[i][jj], kernel[ki])
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smoothed[i][j] = q16_clamp(total)
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for j in range(i + 1, n):
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if cost_matrix[i][j] < threshold:
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union(i, j)
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# Smooth columns
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result = [[0] * n for _ in range(n)]
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# Group nodes by cluster root
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clusters: dict[int, list[int]] = {}
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for i in range(n):
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for j in range(n):
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total = 0
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for ki in range(len(kernel)):
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ii = i + ki - k_half
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if 0 <= ii < n:
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total += q16_multiply(smoothed[ii][j], kernel[ki])
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else:
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ii = max(0, min(n - 1, ii))
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total += q16_multiply(smoothed[ii][j], kernel[ki])
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result[i][j] = q16_clamp(total)
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root = find(i)
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if root not in clusters:
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clusters[root] = []
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clusters[root].append(i)
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# Convert back to float
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return [[q16_to_float(result[i][j]) for j in range(n)]
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for i in range(n)]
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return list(clusters.values())
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def _build_reduced_cost_matrix(cost_matrix: list[list[float]],
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clusters: list[list[int]]) -> list[list[float]]:
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"""Build a reduced cost matrix for cluster representatives.
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The cost between two clusters is the minimum cost between any pair of
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nodes across the two clusters.
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Args:
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cost_matrix: Original n×n cost matrix.
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clusters: List of clusters (each a list of node indices).
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Returns:
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Reduced k×k cost matrix where k = number of clusters.
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"""
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k = len(clusters)
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reduced = [[0.0] * k for _ in range(k)]
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for ci in range(k):
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for cj in range(ci + 1, k):
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# Minimum cost across cluster boundaries
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min_cost = float('inf')
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for ni in clusters[ci]:
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for nj in clusters[cj]:
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if cost_matrix[ni][nj] < min_cost:
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min_cost = cost_matrix[ni][nj]
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reduced[ci][cj] = min_cost
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reduced[cj][ci] = min_cost
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return reduced
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def _expand_tour(cluster_tour: list[int],
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clusters: list[list[int]],
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cost_matrix: list[list[float]]) -> list[int]:
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"""Expand a cluster-level tour back to individual nodes.
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For each cluster in the tour, we need to enter and exit through specific
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nodes. We pick the entry/exit nodes that minimize the inter-cluster edges.
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Args:
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cluster_tour: Tour over cluster indices.
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clusters: List of clusters (each a list of node indices).
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cost_matrix: Original cost matrix.
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Returns:
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Tour over original node indices.
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"""
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if len(cluster_tour) <= 1:
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# Single cluster — order nodes greedily within cluster
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nodes = clusters[cluster_tour[0]]
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if len(nodes) <= 1:
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return nodes
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return _greedy_tour_subgraph(nodes, cost_matrix)[0]
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# For each consecutive pair of clusters, find the best entry/exit nodes
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k = len(cluster_tour)
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entry_node = [0] * k # Which node in cluster i we enter through
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exit_node = [0] * k # Which node in cluster i we exit through
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for i in range(k):
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ci = cluster_tour[i]
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cj = cluster_tour[(i + 1) % k]
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# Find the pair of nodes (one in ci, one in cj) with minimum cost
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best_cost = float('inf')
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best_exit = clusters[ci][0]
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best_entry = clusters[cj][0]
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for ni in clusters[ci]:
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for nj in clusters[cj]:
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c = cost_matrix[ni][nj]
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if c < best_cost:
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best_cost = c
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best_exit = ni
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best_entry = nj
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exit_node[i] = best_exit
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entry_node[(i + 1) % k] = best_entry
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# Build the full tour by visiting each cluster's nodes between entry/exit
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full_tour = []
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for i in range(k):
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ci = cluster_tour[i]
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cluster_nodes = clusters[ci]
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entry = entry_node[i]
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exit_nd = exit_node[i]
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if len(cluster_nodes) == 1:
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full_tour.append(cluster_nodes[0])
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else:
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# Build a path through the cluster from entry to exit
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# Use greedy nearest-neighbor within the cluster, starting at entry
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path = _build_cluster_path(cluster_nodes, entry, exit_nd, cost_matrix)
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full_tour.extend(path)
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return full_tour
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def _build_cluster_path(nodes: list[int], start: int, end: int,
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cost_matrix: list[list[float]]) -> list[int]:
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"""Build a path through cluster nodes from start to end.
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Uses nearest-neighbor heuristic constrained to the cluster.
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"""
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if len(nodes) <= 2:
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# Just return all nodes, start first
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result = [start]
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for n in nodes:
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if n != start:
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result.append(n)
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return result
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visited = {start}
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path = [start]
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current = start
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# Visit all nodes except the end node
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remaining = set(nodes) - {start, end}
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while remaining:
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best_next: int = nodes[0] # will be overwritten
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best_cost = float('inf')
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for n in remaining:
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c = cost_matrix[current][n]
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if c < best_cost:
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best_cost = c
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best_next = n
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path.append(best_next)
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visited.add(best_next)
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remaining.remove(best_next)
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current = best_next
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# End at the exit node
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if end != start:
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path.append(end)
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return path
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def _greedy_tour_subgraph(nodes: list[int],
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cost_matrix: list[list[float]]) -> tuple[list[int], float]:
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"""Greedy nearest-neighbor tour on a subset of nodes."""
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if len(nodes) <= 1:
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return nodes, 0.0
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visited = {nodes[0]}
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tour = [nodes[0]]
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current = nodes[0]
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total_cost = 0.0
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for _ in range(len(nodes) - 1):
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best_j = -1
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best_c = float('inf')
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for j in nodes:
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if j not in visited and cost_matrix[current][j] < best_c:
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best_c = cost_matrix[current][j]
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best_j = j
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tour.append(best_j)
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visited.add(best_j)
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total_cost += best_c
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current = best_j
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total_cost += cost_matrix[current][tour[0]]
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return tour, total_cost
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def _greedy_tour(cost_matrix: list[list[float]]) -> tuple[list[int], float]:
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@ -295,17 +464,18 @@ def _2opt_improve(tour: list[int],
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def solve_multiscale(cost_matrix: list[list[float]],
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sigmas: Optional[list[float]] = None) -> dict:
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"""Solve routing problem at multiple scales.
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"""Solve routing problem at multiple scales using cluster-based optimization.
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Coarse-to-fine strategy:
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σ₃ (1.0): coarse LP relaxation → approximate solution
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σ₂ (0.75): tighter LP → better solution
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σ₁ (0.5): exact MIP → optimal solution
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σ₀ (0.25): formal verification target
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Coarse-to-fine strategy with clustering:
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σ=1.0: coarse — merge nearby nodes, solve small problem
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σ=0.75: medium — tighter clustering
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σ=0.5: fine — minimal clustering, warm-started
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σ=0.25: formal verification target — full problem, 2-opt polish
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At each scale, the cost matrix is Gaussian-smoothed, then solved
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with progressively tighter methods. Solutions from coarser scales
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seed finer scales.
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At each scale σ, nodes whose pairwise cost is below σ·max_cost are
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clustered together. The reduced problem is solved on cluster representatives,
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then expanded back to individual nodes. Solutions from coarser scales
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warm-start finer scales.
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Args:
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cost_matrix: n×n cost matrix.
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@ -322,41 +492,81 @@ def solve_multiscale(cost_matrix: list[list[float]],
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if n == 0:
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return {'solutions': {}, 'converged': True, 'best_sigma': 0.0}
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# Find the maximum cost for threshold computation
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max_cost = 0.0
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for i in range(n):
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for j in range(n):
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if cost_matrix[i][j] > max_cost:
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max_cost = cost_matrix[i][j]
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if max_cost == 0.0:
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# All costs are zero — any tour is optimal
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tour = list(range(n))
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return {
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'solutions': {sigmas[0]: {'tour': tour, 'cost': 0.0,
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'n_clusters': 1, 'sigma': sigmas[0]}},
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'converged': True,
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'best_sigma': sigmas[0],
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'best_cost': 0.0,
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}
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solutions = {}
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best_cost = float('inf')
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best_sigma = sigmas[0]
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prev_tour = None
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for sigma in sigmas:
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# Smooth the cost matrix at this scale
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smoothed = _apply_smoothing_q16(cost_matrix, sigma)
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# Compute clustering threshold: merge nodes with cost < sigma * max_cost
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threshold = sigma * max_cost
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# Solve on smoothed costs
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if prev_tour is not None:
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# Warm-start: use previous solution as seed
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# Compute cost on smoothed matrix
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seed_cost = sum(smoothed[prev_tour[i]][prev_tour[(i + 1) % n]]
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for i in range(n))
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# Run 2-opt on smoothed matrix starting from previous tour
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tour, smoothed_cost = _2opt_improve(prev_tour[:], smoothed)
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# Cluster nodes using single-linkage
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clusters = _single_linkage_clusters(cost_matrix, threshold)
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n_clusters = len(clusters)
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if n_clusters == 1:
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# All nodes in one cluster — solve the full problem
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if prev_tour is not None:
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tour, cost = _2opt_improve(prev_tour[:], cost_matrix)
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else:
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tour, cost = _greedy_tour(cost_matrix)
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tour, cost = _2opt_improve(tour, cost_matrix)
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else:
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# Cold start: greedy + 2-opt
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tour, smoothed_cost = _greedy_tour(smoothed)
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tour, smoothed_cost = _2opt_improve(tour, smoothed)
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# Build reduced cost matrix for cluster representatives
|
||||
reduced_matrix = _build_reduced_cost_matrix(cost_matrix, clusters)
|
||||
|
||||
# Evaluate on original cost matrix
|
||||
real_cost = sum(cost_matrix[tour[i]][tour[(i + 1) % n]]
|
||||
for i in range(n))
|
||||
# Solve the reduced problem
|
||||
if n_clusters <= 2:
|
||||
# Trivial: just order the clusters
|
||||
cluster_tour = list(range(n_clusters))
|
||||
else:
|
||||
cluster_tour, _ = _greedy_tour(reduced_matrix)
|
||||
if n_clusters >= 4:
|
||||
cluster_tour, _ = _2opt_improve(cluster_tour, reduced_matrix)
|
||||
|
||||
# Expand cluster tour back to individual nodes
|
||||
tour = _expand_tour(cluster_tour, clusters, cost_matrix)
|
||||
|
||||
# Polish with 2-opt on the full problem
|
||||
if prev_tour is not None:
|
||||
# Warm-start: try both the expanded tour and the previous tour
|
||||
tour_a, cost_a = _2opt_improve(tour, cost_matrix)
|
||||
tour_b, cost_b = _2opt_improve(prev_tour[:], cost_matrix)
|
||||
if cost_a <= cost_b:
|
||||
tour, cost = tour_a, cost_a
|
||||
else:
|
||||
tour, cost = tour_b, cost_b
|
||||
else:
|
||||
tour, cost = _2opt_improve(tour, cost_matrix)
|
||||
|
||||
solutions[sigma] = {
|
||||
'tour': tour,
|
||||
'cost': real_cost,
|
||||
'smoothed_cost': smoothed_cost,
|
||||
'cost': cost,
|
||||
'n_clusters': n_clusters,
|
||||
'sigma': sigma,
|
||||
}
|
||||
|
||||
if real_cost < best_cost:
|
||||
best_cost = real_cost
|
||||
if cost < best_cost:
|
||||
best_cost = cost
|
||||
best_sigma = sigma
|
||||
|
||||
prev_tour = tour
|
||||
|
|
@ -415,7 +625,7 @@ if __name__ == '__main__':
|
|||
|
||||
for sigma, data in sorted(result['solutions'].items(), reverse=True):
|
||||
print(f" σ={sigma:.2f}: tour cost={data['cost']:.2f}, "
|
||||
f"smoothed={data['smoothed_cost']:.2f}")
|
||||
f"clusters={data['n_clusters']}")
|
||||
|
||||
# Demo voltage mapping
|
||||
print("\nVoltage ↔ Scale mapping:")
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue