README.md

Metaheuristic Optimization Using Population-Based Simulated Annealing

There are two versions: one using the Eigen library for linear algebra (ver. 3.3.9), and the other using plain vanilla C++.

Features

• The code has been tested using g++ 8.1.0 (MinGW-W64).
• Variables can be real, integer, or mixed real/integer.
• Variables can be constrained to a specific interval or value setting the lower and the upper boundaries.
• Neighboroud search is performed using a normal distribution along randomly chosen dimensions.
• A nonlinear decreasing schedule is used for the temperature and for the standard deviation in the neighboroud search.
• Search space can be normalized to improve convergency.
• An arbitrary number of parameters can be passed (in a tuple) to the function to minimize.
• Solver parameters and results are passed using structures.
• Usage: test.exe example and test_Eigen.exe example.

Main Parameters

example Name of the example to run (Parabola, Alpine, Tripod, and Ackley.)

func Function to minimize. The position of all agents is passed to the function at the same time.

LB, UB Lower and upper boundaries of the search space.

nPop, epochs Number of agents (population) and number of iterations.

T0 Initial temperature.

alphaT Temperature reduction rate.

alphaS Standard deviation (neighboroud search) reduction rate.

nMove Number of neighbours of a state evaluated at each epoch.

prob Probability the dimension of a state is changed.

sigma0 Initial standard deviation used to search the neighboroud of a state.

IntVar List of indexes specifying which variable should be treated as integer. If all variables are real set IntVar=NULL. Indexes are specified in the range (0,nVar-1). It cannot be used when the search space is normalized.

normalize Specifies if the search space should be normalized (to improve convergency).

args Tuple containing any parameter that needs to be passed to the function to minimize. If no parameters are passed set args=NULL.

nVar Number of variables (dimensions of the search space).

nIntVar Number of integer variables.

X0 Global minimum point (used only to compare with the numerical solution).

seed Seeding value for the random number generator.

Examples

There are four examples: Parabola, Alpine, Tripod, and Ackley (see test.py for the specific equations and parameters). As illustration, a 3D plot of these functions is shown below.

• Parabola, Alpine, and Ackley can have an arbitrary number of dimensions, while Tripod has only two dimensions.

• Parabola, Tripod, and Ackley are examples where parameters (respectively, array X0, scalars kx and ky, and array X0) are passed using args.

• The global minimum for Parabola and Ackley is at X0; the global minimum for Alpine is at zero; the global minimum for Tripod is at [0,-ky] with local minimum at [-kx,+ky] and [+kx,+ky].

References

• Wikipedia, "Simulated Annealing".
• Kirkpatrick et al., 1983, "Optimization by Simulated Annealing", JSTOR.
• Jamil and Yang, 2013, "A Literature Survey of Benchmark Functions For Global Optimization Problems", arVix.
• Eigen template library for linear algebra.