Abstract
We establish the convergence of a stochastic global optimization algorithm for general non-convex, smooth functions. The algorithm follows the trajectory of an appropriately defined stochastic differential equation (SDE). In order to achieve feasibility of the trajectory we introduce information from the Lagrange multipliers into the SDE. The analysis is performed in two steps. We first give a characterization of a probability measure (Π) that is defined on the set of global minima of the problem. We then study the transition density associated with the augmented diffusion process and show that its weak limit is given by Π.
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Aluffi-Pentini F., Parisi V., Zirilli F.: Global optimization and stochastic differential equations. J. Optim. Theory Appl. 47(1), 1–16 (1985)
Aluffi-Pentini F., Parisi V., Zirilli F.: A global optimization algorithm using stochastic differential equations. ACM Trans. Math. Softw. 14(4), 345–365 (1989)
Bender C.M., Orszag S.A.: Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory. Springer-Verlag, Berlin (1999)
Chiang T.S., Hwang C.R., Sheu S.J.: Diffusion for global optimization in R n. SIAM J. Control Optim. 25(3), 737–753 (1987)
Gelfand S.B., Mitter S.K.: Recursive stochastic algorithms for global optimization in R d. SIAM J. Control Optim. 29(5), 999–1018 (1991)
Geman S., Hwang C.R.: Diffusions for global optimization. SIAM J. Control Optim. 24(5), 1031–1043 (1986)
Gidas, B.: The Langevin equation as a global minimization algorithm. In: Disordered Systems and Biological Organization (Les Houches, 1985), Vol. 20 of NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., pp. 321–326. Springer, Berlin (1986)
Gidas, B.: Simulations and global optimization. In: Random Media (Minneapolis, Minn., 1985), Vol. 7 of IMA Vol. Math. Appl., pp. 129–145. Springer, New York (1987)
Gidas, B.: Metropolis-type Monte Carlo simulation algorithms and simulated annealing. In: Topics in Contemporary Probability and Its Applications, Probability Stochastics Series, pp. 159–232. CRC, Boca Raton, FL (1995)
Hwang C.R.: Laplace’s method revisited: weak convergence of probability measures. Ann. Probab. 8(6), 1177–1182 (1980)
Kushner H.J.: Asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects: global minimization via Monte Carlo. SIAM J. Appl. Math. 47(1), 169–185 (1987)
Li-Zhi L., Liqun Q., Hon W.T.: A gradient-based continuous method for large-scale optimization problems. J. Glob. Optim. 31(2), 271 (2005)
Luenberger D.G.: The gradient projection method along geodesics. Manag. Sci. 18, 620–631 (1972)
Luenberger D.G.: Linear and Nonlinear Programming, 2nd edn. Kluwer Academic Publishers, Boston (2003)
Maringer, D., Parpas, P.: Global optimization of higher order moments in portfolio selection. J. Glob. Optim. doi:10.1007/s10898-007-9224-3
Oksendal, B.: Stochastic Differential Equations, an Introduction with Applications, 6th edn. Springer, New York
Parpas, P., Rustem, B., Pistikopoulos, E.N.: Global optimization of robust chance constrained problems. J. Glob. Optim. doi:10.1007/s10898-007-9244-z
Parpas P., Rustem B., Pistikopoulos E.N.: Linearly constrained global optimization and stochastic differential equations. J. Glob. Optim. 36(2), 191–217 (2006)
Recchioni M.C., Scoccia A.: A stochastic algorithm for constrained global optimization. J. Glob. Optim. 16(3), 257–270 (2000)
Zirilli, F.: The use of ordinary differential equations in the solution of nonlinear systems of equations. In: Nonlinear Optimization, 1981 (Cambridge, 1981), NATO Conference Series II: Systems Science, pp. 39–46. Academic Press, London (1982)
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Parpas, P., Rustem, B. Convergence analysis of a global optimization algorithm using stochastic differential equations. J Glob Optim 45, 95–110 (2009). https://doi.org/10.1007/s10898-008-9397-4
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DOI: https://doi.org/10.1007/s10898-008-9397-4