Simulated annealing

Author: Gwen Zhang (xz929), Yingjie Wang (yw2749), Junchi Xiao (jx422), Yichen Li (yl3938), Xiaoxiao Ge (xg353) (ChemE 6800 Fall 2024)

Stewards: Nathan Preuss, Wei-Han Chen, Tianqi Xiao, Guoqing Hu

Introduction

Simulated annealing (SA) is a probabilistic optimization algorithm inspired by the metallurgical annealing process, which reduces defects in a material by controlling the cooling rate to achieve a stable state. The core concept of SA is to allow algorithms to escape the constraints of local optima by occasionally accepting suboptimal solutions. This characteristic enables SA to find near-global optima in large and complex search spaces. During the convergence process, the probability of accepting a suboptimal solution diminishes over time.

SA is widely applied in diverse fields such as scheduling, machine learning, and engineering design, and is particularly effective for combinatorial optimization problems that are challenging for deterministic methods. First proposed in the 1980s by Kirkpatrick, Gelatt, and Vecchi, SA demonstrated its efficacy in solving various complex optimization problems through its analogy to thermodynamics. Today, it remains a powerful heuristic often combined with other optimization techniques to enhance performance in challenging problem spaces.

Algorithm Discussion

Formal Description of the Algorithm

SA is a probabilistic technique for finding approximate solutions to optimization problems, particularly those with large search spaces. Inspired by the annealing process in metallurgy, the algorithm explores the solution space by occasionally accepting worse solutions with a probability that diminishes over time. This reduces the risk of getting stuck in local optima and increases the likelihood of discovering the global optimum.

The basic steps of the SA algorithm are as follows:

1. Initialization: Begin with an initial solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_0} and an initial temperature $T_{0}$​.
2. Iterative Improvement: While the stopping criteria are not met:
   • Generate a new solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_{new}} ​in the neighborhood of the current solution s.
   • Calculate the difference in objective values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E = f(s_{new}) - f(s)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is the objective function.
   • If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E <0} (i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_{new}} is a better solution), accept Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_{new}} ​ as the current solution.
   • If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta E >0} , accept Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_{new}} ​ with a probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \exp\left(-\frac{\Delta E}{T}\right)} .
   • Update the temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T } according to a cooling schedule, typically Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=\alpha \cdot T } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha } is a constant between 0 and 1.
3. Termination: Stop the process when the temperature falls below a minimum threshold , or after a predefined number of iterations. The best solution encountered is returned as the approximate optimum.

Pseudocode for Simulated Annealing

def SimulatedAnnealing(f, s_0, T_0, alpha, T_min):
    s_current = s_0
    T = T_0
    while T > T_min:
        s_new = GenerateNeighbor(s_current)
        delta_E = f(s_new) - f(s_current)
        if delta_E < 0 or Random(0, 1) < math.exp(-delta_E / T):
            s_current = s_new
        T = alpha * T
    return s_current

Assumptions and Conditions

Objective Function Continuity:

The objective function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } is assumed to be defined and continuous in the search space.

Cooling Schedule:

The cooling schedule, typically geometric, determines how Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T } decreases over iterations. The cooling rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha } affects the convergence speed and solution quality.

Acceptance Probability:

Based on the Metropolis criterion, the probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \exp\left(-\frac{\Delta E}{T}\right)} allows the algorithm to accept worse solutions initially, encouraging broader exploration.

Termination Condition:

SA terminates when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T } falls below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{min} } ​ or when a predefined number of iterations is reached.