Abstract
In this chapter a new optimization method is presented, called the Heuristic Kalman Algorithm (HKA). This new algorithm is proposed as an alternative approach for solving continuous non-convex optimization problems. The principle of HKA is to consider explicitly the optimization problem as a measurement process intended to give an estimate of the optimum. A specific procedure, based on the Kalman estimator, was developed to improve the quality of the estimate obtained through the measurement process. A significant advantage of HKA against other metaheuristics lies mainly in the small number of parameters which have to be set by the user. In addition, it is shown that HKA converges almost surely to a near-optimal solution. The efficiency of HKA is evaluated in detail through several non-convex test problems, both in the unconstrained and constrained cases. The results are then compared to those obtained via other metaheuristics. These various numerical experiments show that the HKA has very interesting potentialities for solving non-convex optimization problems, especially with regard to the computation time and the success ratio.
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Notes
- 1.
The diagonal of the matrix \(P_{k}^{-}\) represents the variance vector of the Gaussian distribution, and the element-wise square root of the variance vector is defined as the standard deviation vector. This is the meaning of the shorthand notation \(S_{k}=(\mbox {vec}^{d}(P_{k}^{-}))^{1/2}\).
- 2.
ACOR: Ant colony optimization for continuous domains (see the Notes and References).
- 3.
CGA: Continuous Genetic Algorithm (see the Notes and References).
- 4.
ECTS: Enhanced Continuous Tabu Search (see the Notes and References).
- 5.
ESA: Enhanced Simulated Annealing (see the Notes and References).
- 6.
INTEROPT: Interactive Optimization (see the Notes and References).
- 7.
GP: Geometric Programming (see Notes and References).
- 8.
GAP: Genetic Algorithm and Penalty function (see Notes and References).
- 9.
CEGA: Co-Evolutionary Genetic Algorithm (see Notes and References).
- 10.
MGA: Multi-objective Genetic Algorithm (see Notes and References).
- 11.
CPSO: Co-evolutionary Particle Swarm Optimization (see Notes and References).
- 12.
ALPSO: Augmented Lagrangian Particle Swarm Optimization (see the Notes and References).
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Toscano, R. (2013). Heuristic Kalman Algorithm. In: Structured Controllers for Uncertain Systems. Advances in Industrial Control. Springer, London. https://doi.org/10.1007/978-1-4471-5188-3_5
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