Abstract
The two-dimensional stochastic rootfinding problem (SRFP) is to findthe unique root of two equations with two unknowns, where thefunction value needs to be estimated. For such problems, numericalrootfinding algorithms might not be suitable because the algorithmmight be confused by random errors of function estimates. To solvetwo-dimensional SRFPs, we use retrospective approximation algorithms and propose the deterministic rootfinding method, two-dimensionalbisection search, for solving the sample-path equations,which is a major step of retrospective approximation (RA). The idea of two-dimensional bisection search arises from developing a bi-section like method for two-dimensional SRFPs. For the one-dimensional SRFP, solving g(x) = γ, the domain of x can be divided into two areas: x: g(x) - γ< 0 and x: g(x) > 0. For the two-dimensional SRFP, solving g1(x1, x2) = γ1, g2(x1, x2) = γ2, the domain of (x1, x2) can be divided into four areas with corresponding function values belong to: (+,+), (+, -), (-,+) and (-, -), where the first sign is the sign of g1(x1,x2) -γ1 and the second sign is referring to g2(x1,x2) –γ2.Two-dimensional bisection methods include two parts:(1) Finding one point in each of the four areas by Newton's method; the region defined by the four vertices is called a bounding area,hopefully bounding the true root. (2) Shrinking the bounding area until the stopping rule is satisfied. Simulation experiments are conducted to evaluate the algorithm performance. Two rootfinding problems, one deterministic and one stochastic, are solved. Empirical results show that the two-dimensional bisection method is less efficient than Newton's method when they are used in deterministic rootfinding. This result is similar to the fact that one-dimensional bisection methods converge slower than Newton's method. The RA with two-dimensional bisection method is, however, more efficient than the RA with Broyden's method when they are used in stochastic rootfinding.