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 $s_{0}$ and an initial temperature Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{0}} ​.
2. Iterative Improvement: While the stopping criteria are not met:
   • Generate a new solution $s_{new}$ ​in the neighborhood of the current solution s.
   • Calculate the difference in objective values $\Delta E=f(s_{new})-f(s)$, where $f$ is the objective function.
   • If $\Delta E<0$ (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.

Numerical Examples

The Ising model is a classic example from statistical physics and in order to reach its optimal state, we can use Simulated Annealing (SA) to reach a minimal energy state. In the SA approach, we start with a high "temperature" that allows random changes in the system’s configuration, which helps explore different arrangements of spins. As the temperature lowers, the algorithm gradually favors configurations with lower energy, eventually "freezing" into a low-energy state.

Problem Setup

Figure 1: Configuration for L

Consider a small 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 2\times 2} grid of spins, where each spin can be either 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 +1} or 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 -1} . The energy of the system is determined by the Hamiltonian: 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 E = -J \sum_{\langle i,j \rangle} S_i S_j} , 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 J} is a coupling constant (set to 1 for simplicity), 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_i} ​ and 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_j} are neighboring spins, and the summation is over nearest neighbors. The goal is to minimize 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 E} , achieved when all spins are aligned.

Algorithm Steps

1. Initialize the spins: start with a random configuration. For example: 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 L = \begin{bmatrix} +1 & -1 \\ -1 & +1 \end{bmatrix}}
2. Calculate Initial Energy: For this configuration, each pair of neighboring spins contributes 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 -JS_i S_j} to the energy. Calculate by summing these contributions: The pairs are 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 (+1, -1), (-1, +1), (-1, +1), (+1, -1)} , all of which contribute 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 +1} .  So, initial energy 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 E = 4} .
3. Neighboring Configurations: Select a random spin to flip. Suppose the spin in the top right corner is chosen, changing 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 +1} to 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 -1} : 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 \text{New L} = \begin{bmatrix} +1 & +1 \\ -1 & +1 \end{bmatrix}} Recalculate 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 E} for the new configuration: Now, three pairs are aligned 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 (+1,+1)} and contribute 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 -1} each, while one pair 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 (-1,+1)} contributes 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 +1} , so total energy 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 E = -2} .
4. Acceptance Decision: Calculate 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 = E_{\text{new}} - E_{\text{old}} = -2 - 4 = -6} . Since 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 the new configuration.