CONVEX-OPTIMIZATION VIA FEEDBACKS

Three dynamical systems are associated with a problem of convex optimi zation in a finite-dimensional space. For system trajectories x(t), th e ratios x(t)/t are, respectively, (i) solution tracking (staying with in the solution set X-0), (ii) solution abandoning (reaching X-0 as ti me t goes back to the initial instant), and (iii) solution approaching (approaching X-0 as time t goes to infinity). The systems represent a closed control system with appropriate feedbacks. In typical cases, t he structure of the trajectories is simple enough. For instance, for a problem of quadratic programming with linear and box constraints, sol ution-approaching dynamics are described by a piecewise-linear ODE wit h a finite number of polyhedral domains of linearity. Finding the orde r of visiting these domains yields an analytic resolution of the origi nal problem; a detailed analysis is given for a particular example. A discrete-time approach is outlined.