Applied and Computational Mathematics
Volume 6, Issue 4-1, July 2017, Pages: 72-82
Received: Apr. 8, 2017;
Accepted: Apr. 10, 2017;
Published: Jun. 9, 2017
Views 1774 Downloads 51
Loc Nguyen, Vietnam Institute of Mathematics, Hanoi, Vietnam
Global optimization is necessary in some cases when we want to achieve the best solution or we require a new solution which is better the old one. However global optimization is a hazard problem. Gradient descent method is a well-known technique to find out local optimizer whereas approximation solution approach aims to simplify how to solve the global optimization problem. In order to find out the global optimizer in the most practical way, I propose a so-called descending region (DR) algorithm which is combination of gradient descent method and approximation solution approach. The ideology of DR algorithm is that given a known local minimizer, the better minimizer is searched only in a so-called descending region under such local minimizer. Descending region is begun by a so-called descending point which is the main subject of DR algorithm. Descending point, in turn, is solution of intersection equation (A). Finally, I prove and provide a simpler linear equation system (B) which is derived from (A). So (B) is the most important result of this research because (A) is solved by solving (B) many enough times. In other words, DR algorithm is refined many times so as to produce such (B) for searching for the global optimizer. I propose a so-called simulated Newton – Raphson (SNR) algorithm which is a simulation of Newton – Raphson method to solve (B). The starting point is very important for SNR algorithm to converge. Therefore, I also propose a so-called RTP algorithm, which is refined and probabilistic process, in order to partition solution space and generate random testing points, which aims to estimate the starting point of SNR algorithm. In general, I combine three algorithms such as DR, SNR, and RTP to solve the hazard problem of global optimization. Although the approach is division and conquest methodology in which global optimization is split into local optimization, solving equation, and partitioning, the solution is synthesis in which DR is backbone to connect itself with SNR and RTP.
Global Optimization with Descending Region Algorithm, Applied and Computational Mathematics. Special Issue:Some Novel Algorithms for Global Optimization and Relevant Subjects.
Vol. 6, No. 4-1,
2017, pp. 72-82.
M. D. Le and Y. H. Le, “Lecture Notes on Optimization,” Vietnam Institute of Mathematics, Hanoi, 2014.
S. Boyd and L. Vandenberghe, Convex Optimization, New York, NY: Cambridge University Press, 2009, p. 716.
Y.-B. Jia, “Lagrange Multipliers,” 2013.
A. P. Ruszczyński, Nonlinear Optimization, Princeton, New Jersey: Princeton University Press, 2006, p. 463.
Wikipedia, “Karush–Kuhn–Tucker conditions,” Wikimedia Foundation, 4 August 2014. [Online]. Available: http://en.wikipedia.org/wiki/Karush–Kuhn–Tucker_conditions. [Accessed 16 November 2014].
P. D. Ta, “Numerical Analysis Lecture Notes,” Vietnam Institute of Mathematics, Hanoi, 2014.
T. Hoang, Convex Analysis and Global Optimization, Dordrecht: Kluwer, 1998, p. 350.
J. Kennedy and R. Eberhart, “Particle Swarm Optimization,” in Proceedings of IEEE International Conference on Neural Networks, 1995.
Wikipedia, “Particle swarm optimization,” Wikimedia Foundation, 7 March 2017. [Online]. Available: https://en.wikipedia.org/wiki/Particle_swarm_optimization. [Accessed 8 April 2017].
R. Poli, J. Kennedy and T. Blackwell, “Particle swarm optimization,” Swarm Intelligence, vol. 1, no. 1, pp. 33-57, June 2007.
Wikipedia, “Quasi-Newton method,” Wikimedia Foundation, 4 April 2017. [Online]. Available: https://en.wikipedia.org/wiki/Quasi-Newton_method. [Accessed 8 April 2017].
H. Jiao, Z. Wang and Y. Chen, “Global optimization algorithm for sum of generalized polynomial,” Applied Mathematical Modelling, vol. 37, no. 1-2, pp. 187-197, 18 February 2012.
T. Larsson and M. Patriksson, “Global optimality conditions for discrete and nonconvex optimization - With applications to Lagrangian heuristics and column generation,” Operations Research, vol. 54, no. 3, pp. 436-453, 21 April 2003.
P. Dawkins, “Gradient Vector, Tangent Planes and Normal Lines,” Lamar University, 2003. [Online]. Available: http://tutorial.math.lamar.edu/Classes/CalcIII/GradientVectorTangentPlane.aspx. [Accessed 2014].
V. H. H. Nguyen, Linear Algebra, Hanoi: Hanoi National University Publishing House, 1999, p. 291.
R. L. Burden and D. J. Faires, Numerical Analysis, 9th Edition ed., M. Julet, Ed., Brooks/Cole Cengage Learning, 2011, p. 872.
L. Nguyen, “Feasible length of Taylor polynomial on given interval and application to find the number of roots of equation,” International Journal of Mathematical Analysis and Applications, vol. 1, no. 5, pp. 80-83, 10 January 2015.
L. Nguyen, “Improving analytic function approximation by minimizing square error of Taylor polynomial,” International Journal of Mathematical Analysis and Applications, vol. 1, no. 4, pp. 63-67, 21 October 2014.
Wikipedia, “Sturm’s theorem,” Wikimedia Foundation, 2014. [Online]. Available: https://en.wikipedia.org/wiki/Sturm%27s_theorem. [Accessed 30 August 2014].