Quasi-newton Modifications to Snopt

subject to c(x) ≥ 0, where f : R → R and c : R → R are twice continuously differentiable functions. Let g(x) denote the gradient of f(x), and J(x) the mxn Jacobian matrix of c(x), with rows the gradients of c(x). A point x∗ is feasible with respect to the constraints c(x) if c(x∗) ≥ 0, and infeasible if c(x∗) < 0. The constraint ci(x) is considered active at x if ci(x) = 0 and inactive if ci(x) > 0. Let A(x) be the submatrix of J(x) that includes only the gradients of the constraints active at x. Denote the nullspace matrix of A(x) by Z(x), so that the columns of Z(x) form a basis for the nullspace of A(x). Let the active set A(x) be the set of indices of those constraints active at x. x∗ is a local minimizer of NIP if it is feasible with respect to the constraints, and there exists a neighborhood of x∗ such that f(x∗) ≤ f(x) for all feasible points x in the neighborhood. If the inequality is strict for all feasible x 6= x∗, then x∗ is a strong local minimizer. Otherwise, x∗ is a weak local minimizer. Nonlinearly constrained problems have an abundance of diverse applications in engineering, finance, trajectory optimization, and many additional fields. The scope of scientific applications solved with nonlinearly constrained optimization software will only increase. No matter how well optimization algorithms perform, there will always be a demand for improvement as larger and more advanced mathematical models are developed. Nonlinearly constrained problems are considerably more difficult to solve than unconstrained or linearly constrained problems, partly due to the fact that obtaining and maintaining feasibility is itself an iterative procedure. Another difficulty results from the crucial curvature