Solving Systems of Non-Linear Equations by Broyden's Method with Projected Updates

We introduce a modification of Broyden's method for finding a zero of n nonlinear equations in n unknowns when analytic derivatives are not available. The method retains the local Q-superlinear convergence of Broyden's method and has the additional property that if any or all of the equations are linear, it locates a zero of these equations in n+1 or fewer iterations. Limited computational experience suggests that our modification often improves upon Eroyden's method.

Determining Feasibility of a Set of Nonlinear Inequality Constraints ; CU-CS-172-80

We present a new class of algorithms for determining whether there exists a point x ɛ Rn satisfying the nonlinear inequality constraints ci(x) ≤ 0, i=1, …, m, subject perhaps to satisfying linear inequality constraints lj¬(x) ≤0, j=1, …,k which are known to be feasible. Our algorithm consists of solving a sequence of linearly constrained optimization problems, using a sequence of objective functions ф(x,p) which are at least twice continuously differentiable, and which are generated by monotonically increasing the value of the non-negative parameter p. It is shown that in almost all cases, once p reaches or exceeds some finite value, that the solution to the linearily constrained optimization problem either is a feasible point, or establishes the infeasibility of the set of constraints. Computational results are presented in which the algorithm performs satisfactorily on feasible and on infeasible systems