关于PAGMO/PYGMO中Corana模拟退火算法参数的技术问询

Hey there! Let's break down PyGMO's simulated annealing parameters, their real-world purpose, how they align with Corana et al.'s classic SA paper, and tailor this to your 4D homography registration problem.

PyGMO Parameter Deep Dive #

Let's go through each parameter with your specific problem in mind (4 variables with defined ranges, initial cost ~50, optimal cost ~0):

• Ts (initial temperature): You’ve got the right idea—this sets the starting threshold for accepting worse solutions via the Metropolis criterion. For your case, since your initial bad solution has a cost of 50, set Ts to match the scale of early cost fluctuations: aim for 20–50. This lets the algorithm explore broadly at first instead of getting stuck in the local minimum of your initial poor guess.
• Tf (final temperature): This is the "cooled down" temperature where the algorithm stops accepting worse solutions. Since your optimal cost is near 0, set Tf to a tiny value like 1e-3 or 1e-4—low enough that only solutions with negligible cost improvements get accepted, ensuring convergence to the optimal region.
• n_T_adj (number of temperature adjustments): This controls how many times the temperature is lowered during the annealing process. Each adjustment triggers a cooling step (using a fixed reduction coefficient, more on that later). Think of this as the number of "cooling stages" your algorithm goes through.
• n_range_adj (number of range adjustments per temperature): For each constant temperature stage, this is how many times the algorithm will tweak the mutation range (step size) of your decision variables. Before each adjustment, it runs a set number of mutations to calculate acceptance rates, then expands or shrinks the range based on whether acceptance is too high (needs more exploration) or too low (needs more exploitation).
• bin_size (number of mutations for acceptance rate calculation): This is the sample size used to compute acceptance rates before adjusting the mutation range. More iterations mean a more reliable acceptance rate, preventing overreactive step size changes.
• start_range (initial mutation range): This defines the initial step size for mutating each decision variable. For your problem, since variables have different ranges, ideally you’d set per-dimension values (if PyGMO supports array inputs here):
  • x1 ([-10,10]): 2 (10% of the total range)
  • x2 ([10,30]): 2 (10% of the total range)
  • x3 ([-45,0]): 4.5 (10% of the total range)
  • x4 ([0.5,2]): 0.15 (10% of the total range)
    If you can only use a single float, pick a middle ground like 2 and let the algorithm refine ranges per variable over time.

Mapping to Corana et al.'s Original SA Parameters #

Corana’s 1987 paper laid out the core SA framework, and here’s how PyGMO’s parameters map directly:

• Corana’s N_s: Equivalent to PyGMO’s bin_size. N_s was the number of evaluations per step-size adjustment cycle—PyGMO uses bin_size mutations to compute acceptance rates for the same purpose.
• Corana’s N_T: Equivalent to PyGMO’s n_range_adj. N_T was the number of step-size adjustments per temperature level, which is exactly what n_range_adj controls in PyGMO.
• Corana’s r_T: This is the temperature reduction coefficient (e.g., 0.85) that lowers the temperature at each n_T_adj step. PyGMO doesn’t expose this as a direct input—it uses a default value aligned with Corana’s recommendations (usually ~0.85). You can calculate how Ts decays to Tf using Tf = Ts * (r_T)^n_T_adj if you want to fine-tune n_T_adj.
• Corana’s initial step size: Maps directly to PyGMO’s start_range, the initial mutation amplitude for each variable.

Initial Parameter Recommendations for Your Problem #

Based on your 4D homography setup, here’s a starting point for tuning:

• Ts: 30 (balances early exploration and focus)
• Tf: 1e-3 (ensures late-stage convergence)
• n_T_adj: 30 (with r_T=0.85, this takes Ts=30 down to ~1e-3 perfectly)
• n_range_adj: 7 (enough adjustments per temperature to refine step sizes)
• bin_size: 150 (reliable acceptance rate stats without wasting computation)
• start_range: Use per-dimension values if possible, else 2 as a compromise

内容的提问来源于stack exchange，提问作者AstroBen