Quadratic Optimization

We present a new algorithm for nonconvex bound-constrained quadratic optimization. In the strictly convex case, our method is equivalent to the state-of-the-art algorithm by Dostal and Schoberl [Comput. Optim. Appl., 30 (2005), pp. 23–43]. Unlike their method, however, we establish a convergence theory for our algorithm that holds even when the problems are nonconvex. This is achieved by carefully addressing the challenges associated with directions of negative curvature, in particular, those that may naturally arise when applying the conjugate gradient algorithm to an indefinite system of equations. Our presentation and analysis deal explicitly with both lower and upper bounds on the optimization variables, whereas the work by Dostal and Schoberl considers only strictly convex problems with lower bounds. To handle this generality, we introduce the reduced chopped gradient that is analogous to the reduced free gradient previously used. The reduced chopped gradient leads to a new condition that is used to determine when optimization over a given subspace should be terminated. This condition, although not equivalent, is motivated by a similar condition used by Dostal and Schoberl. Numerical results illustrate the superior performance of our method over commonly used solvers that employ gradient projection steps and subspace acceleration.

In our recent work (Caraballo et al. 2005) we presented preliminary results for subsonic cavity flow control using a reduced-order model based feedback control derived from experimental measurements. The model was developed using the Proper Orthogonal Decomposition of PIV images in conjunction with the Galerkin projection of the Navier- Stokes equations onto the resulting spatial eigenfunctions. A linear-quadratic optimal controller