Abstract
Motivated by instability analysis of unstable (excited state) solutions in computational physics/chemistry, in this paper, the minimax method for solving an optimal control problem with partially uncontrollable variables is embedded into a more general min-equilibrium problem. Results in saddle critical point analysis and computation are modified to provide more information on the minimized objective values and their corresponding riskiness for one to choose in decision making. A numerical algorithm to compute such minimized objective values and their corresponding riskiness is devised. Some convergence results of the algorithm are also established.
Highlights
In the study of self-guided light waves in nonlinear optics [1,2,6], solutions that are not ground states are the excited states and called solitons, are of great interests
One wishes to locally minimize an objective functional J : X × Y → R where X represents the space of controllable variable x and Y represents the space of variable y uncontrollable upto small perturbations
It is clear that if u∗ = (x∗, y∗) is a minimax solution, u∗ is a min-equilibrium solution and in this case, a small perturbation in the y-variable will only decrease the value of J, which is certainly welcome
Summary
In the study of self-guided light waves in nonlinear optics [1,2,6], solutions that are not ground states are the excited states and called solitons, are of great interests. One wishes to locally minimize an objective functional J : X × Y → R where X represents the space of controllable variable x and Y represents the space of variable y uncontrollable upto small perturbations For this class of problems, to get a reliable solution, the most conservative strategy widely used in the optimal control literature is to seek a minimax solution of the form u∗ = (x∗, y∗) = arg min max J(x, y). It is clear that if u∗ = (x∗, y∗) is a minimax solution, u∗ is a min-equilibrium solution and in this case, a small perturbation in the y-variable will only decrease the value of J, which is certainly welcome.
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