Trajectory Methods in Global Optimization

We review the application of trajectory methods (not including homotopy methods) to global optimization problems. The main ideas and the most successful methods are described and directions of current and future research are indicated.

[1]  A. Mayne.  Parametric Optimization: Singularities, Pathfollowing and Jumps , 1990 .

[2]  E. G. Sturua,et al.  A trajectory algorithm based on the gradient method I. The search on the quasioptimal trajectories , 1991, J. Glob. Optim..

[3]  R. Palais.  Natural operations on differential forms , 1959 .

[4]  Immo Diener,et al.  Trajectory nets connecting all critical points of a smooth function , 1986, Math. Program..

[5]  H. B. Keller.  Global Homotopies and Newton Methods , 1978 .

[6]  Robert Schaback,et al.  An extended continuous Newton method , 1990 .

[7]  Francesco Zirilli,et al.  A global optimization algorithm using stochastic differential equations , 1988, TOMS.

[8]  J. Snyman.  A new and dynamic method for unconstrained minimization , 1982 .

[9]  I. Diener.  Newton Leaves and the Continuous Newton Method , 1993 .

[10]  V. Parisi,et al.  A New Method for Solving Nonlinear Simultaneous Equations , 1979 .

[11]  M. Hirsch,et al.  On Algorithms for Solving f(x)=0 , 1979 .

[12]  Alan J. Mayne,et al.  Towards Global Optimisation 2 , 1976 .

[13]  J. Snyman,et al.  A multi-start global minimization algorithm with dynamic search trajectories , 1987 .

[14]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[15]  F. H. Branin.  Widely convergent method for finding multiple solutions of simultaneous nonlinear equations , 1972 .

[16]  猪股 修二,et al.  On the Golf Method , 1961 .

[17]  R. D. Murphy,et al.  Iterative solution of nonlinear equations , 1994 .

[18]  A. Griewank.  Generalized descent for global optimization , 1981 .

[19]  F. Aluffi-Pentini,et al.  Global optimization and stochastic differential equations , 1985 .

[20]  P. Jonker,et al.  The continuous Newton-method for meromorphic functions , 1980 .

[21]  K. Schmitt,et al.  Global aspects of the continuous and discrete Newton method: A case study , 1988 .

[22]  Eugene L. Allgower,et al.  Predictor-Corrector and Simplicial Methods for Approximating Fixed Points and Zero Points of Nonlinear Mappings , 1982, ISMP.

[23]  S. Smale.  Convergent process of price adjust-ment and global newton methods , 1976 .

[24]  F. J. Gould,et al.  Relations Between Several Path Following Algorithms and Local and Global Newton Methods , 1980 .

[25]  A note on Branin's method for binding the critical points of smooth functions , 1987 .

[26]  Immo Diener.  On the global convergence of path-following methods to determine all solutions to a system of nonlinear equations , 1987, Math. Program..