THE GLOBAL OPTIMIZATION OF VARIATIONAL-PROBLEMS WITH DISCONTINUOUS SOLUTIONS

We consider variational problems in which the slope of the admissible curves is not necessarily bounded, so that they admit discontinuous so lutions. A problem is first reformulated as one consisting of the mini mization of an integral in a space of functions satisfying a set of in tegral equalities; this is then transfered to a nonstandard framework, in which Loeb measures rake the place of the functions and a near-min imizer can always be found. This is mapped back to the standard world by means of the standard part map; its image is a minimizer, so that t he optimization is global. The minimizer is shown to be the solution o f an infinite dimensional linear program and by well-proven approximat ion procedures a finite dimensional linear program is found by means o f which nearly-optimal curves can be constructed for the original prob lem. A numerical example is given.