Probability-one homotopies in computational science
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 9th International Congress on computational and applied mathematics
A globally convergent interior point algorithm for non-convex nonlinear programming
Journal of Computational and Applied Mathematics
Solving generalized Nash equilibrium problem with equality and inequality constraints
Optimization Methods & Software
A continuation method for Nash equilibria in structured games
Journal of Artificial Intelligence Research
A continuation method for Nash equilibria in structured games
IJCAI'03 Proceedings of the 18th international joint conference on Artificial intelligence
Intent-leveraged optimization of analog circuits via homotopy
Proceedings of the Conference on Design, Automation and Test in Europe
An aggregate deformation homotopy method for min-max-min problems with max-min constraints
Computational Optimization and Applications
Continuation methods for mixing heterogeneous sources
UAI'02 Proceedings of the Eighteenth conference on Uncertainty in artificial intelligence
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For many years globally convergent probability-one homotopy methods have been remarkably successful on difficult realistic engineering optimization problems, most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective. Convergence theory has been derived for a few particular problems, and considerable fixed point theory exists, but generally convergence theory for the homotopy maps used in practice for nonlinear constrained optimization has been lacking. This paper derives some probability-one homotopy convergence theorems for unconstrained and inequality constrained optimization, for linear and nonlinear inequality constraints, and with and without convexity. Some insight is provided into why the homotopies used in engineering practice are so successful, and why this success is more than dumb luck. By presenting the theory as variations on a prototype probability-one homotopy convergence theorem, the essence of such convergence theory is elucidated.