By Vivina Barutello, Susanna Terracinni
We advise a positive evidence for the Ambrosetti-Rabinowitz Mountain cross Theorem offering an set of rules, in accordance with a bisection approach, for its implementation. The potency of our set of rules, rather compatible for difficulties in excessive dimensions, is composed within the low variety of stream traces to be computed for its convergence; hence it improves the single at present used and proposed via Y.S. Choi and P.J. McKenna"
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Additional info for A bisection algorithm for the numerical Mountain Pass
This situation has problems involving both discrete and continuous decisions represented by the subsystems and switching times. A transformed technique is introduced for solving this mixed discrete and continuous optimal control problem. The basic idea behind this technique is transforming the mixed continuous-discrete optimal control problem into an optimal parameter selection problem [84, 1991], which only deals with continuous decision variables. Since the transformed problem still involves the switching times located within subdivisions, which make the numerical solution of such an optimal control problem difficult.
7 represent all the computing results of this problem. 8 2. Piecewise-linear Transformation The idea of piecewise-linear transformation of the time variable was first introduced by Teo [40, 1991], but the time intervals were mapped into instead of where is the number of time intervals. In Lee, Teo, Rehbock and Jennings [51, 1997], the time transformation is described by where is a piece-wise constant transformation. In this book, a similar idea is used, but the implementation is a little simpler, not requiring another differential equation.
The vector um must satisfy the upper and lower bounds 0 and 1, thus and a constraint must be satisfied. The vector represents the values of the state function takes at each switching time. 1) Step 1. Initialization. Set the maximum number of function evaluations, par, which is the system parameter of the MATLAB “constr” function, and also another system parameter par(13) = 1 (1 represents the number of the equation constraint in the minimization problem), and a vector of the parameters which are used in the whole subroutines, par = [number of the state components, number of control components, nn= number of total subintervals], arbitrary starting lengths of switching time intervals OPTIMAL CONTROL MODELS IN FINANCE 28 vector uu = upper bound of um, thus of the state function xinit.
A bisection algorithm for the numerical Mountain Pass by Vivina Barutello, Susanna Terracinni