A structured modified Newton approach for solving systems of nonlinear equations arising in interior-point methods for quadratic programming

Abstract

The focus in this work is on interior-point methods for inequality-constrained quadratic programs, and particularly on the system of nonlinear equations to be solved for each value of the barrier parameter. Newton iterations give high quality solutions, but we are interested in modified Newton systems that are computationally less expensive at the expense of lower quality solutions. We propose a structured modified Newton approach where each modified Jacobian is composed of a previous Jacobian, plus one low-rank update matrix per succeeding iteration. Each update matrix is, for a given rank, chosen such that the distance to the Jacobian at the current iterate is minimized, in both 2-norm and Frobenius norm. The approach is structured in the sense that it preserves the nonzero pattern of the Jacobian. The choice of update matrix is supported by results in an ideal theoretical setting. We also produce numerical results with a basic interior-point implementation to investigate the practical performance within and beyond the theoretical framework. In order to improve performance beyond the theoretical framework, we also motivate and construct two heuristics to be added to the method.

Polynomial worst-case iteration complexity of quasi-Newton primal-dual interior point algorithms for linear programming

“…Next we compute the new complementarity products obtained after a sequence of two steps, apply (11), and add and subtract the term α 2 X Ze to derive…”

“…Let us mention that we will also use the component-wise versions of Eqs. ( 9), (11) and (12). For example, in case of (12) this gives…”

“…Recently, Ek and Forsgren [11] presented a theoretical background and numerical experiments regarding a different kind of low rank updates. The proposed update is based on the Eckart-Young-Mirsky theorem, rather than on the secant equation satisfied by the Broyden update, and affects only the "third row" of matrix J (x k , λ k , z k ), related to the nonlinear part of F. Convex quadratic optimization problems were considered and local convergence was established for a simplified primal-dual interior point algorithm, but no complexity bound was provided.…”