New ODE Methods for Equality Constrained Optimization (1) — Equations

Abstract

To deal with equality constrained optimization problems (ECP), we introduce in this paper "(ECP)-equation", a class of new systems of ordinary differential equations for (ECP), containing a matrix parameter called (ECP)-direction matrix, which plays a central role in it, and a scalar parameter called (ECP)-rate factor. It is shown that by following the trajectory of the equation, a stationary point or hopefully a local solution can be located under very mild conditions. As examples, several schemes of (ECP)-direction matrices and (ECP)-rate factors are given to construct concrete forms of the (ECP)-equation, including almost all the existing projected gradient type versions as special cases. As will be shown in a subsequent paper where the implementation problems are considered in detail, application of an example of these forms results in encouraging performance in experiments.