This project develops scalable
algorithms and supporting convergence
theory for large-scale nonlnear
optimization problems motivated by real-time
dynamic optimization. Real-time
dynamic optimization involves the
solution of parametric optimization
problems in which the problem data
is driven by a dynamic system that
we seek to control or drive towards an optimal
manifold. Applications include
predictive control and state estimation
(data assimilation). A key issue arising
in this framework and that limits the
use of off-the-shelf solvers is computational
latency: if the optimization
problem cannot be solved at a frequency
comparable to the system dynamics, the
system will destabilize. We investigate
approaches that have the potential of
achieving manifold tracking
stability in O(n) operations and
thus support model updates in the
minimum possible time. For such
approaches, given enough computational
resources to reduce the compute time
below the update time, tracking
stability will be achieved. A particular
emphasis is on DO problems arising from
NMPC and data assimilation of real-time
operations of energy systems, such as
the power grid, power plants, and
environmental applications, which poses
stringent constraints on the time to
solution.

To give an order of magnitude
estimate of the complexity of instances
of problems of interest,
the power grid of the Midwest region
includes 10,000
transmission nodes, 1,000 generation
nodes, and 5,000 demand nodes. Such
complexity can easily generate dynamic
optimization problems with more than one
million state and control variables.
Another large-scale
application is the one of large-scale
coordinated HVAC (heating, ventilation,
and air conditioning) building control
in urban systems,
currently contemplated in many major US
cities.

Research:

Sponsored by the Department
of Energy (DOE) Office of Science, we have
been using generalized equation
theory, we have derived conditions
under which a single
sequential-quadratic-programming (SQP)
can achieve stable tracking of the
optimal manifold (see Figure 1). We
demonstrated that the manifold tracking
error can be made of second order with
respect to the time step even if the
active-set changes in time and even
if the SQP iteration is performed
inexactly. This result was later
generalized by Dontchev and Rockafellar in
which again, using generalized equation
theory, they show that a corrector
SQP step can further reduce the tracking
error to fourth order with respect to the
time step.

We have also developed an augmented
lagrangian (AL) scheme coupled to a
projected Gauss-Seidel strategy (see
Figure 2) to enable inexact SQP
iterates and fast detection of activity.
The strategy works well in practice but
the AL regularization degrades the
tracking error to first order. These
results motivated further work in NLP
algorithms. We have developed an exact
differentiable penalty framework,
which has many interesting properties of
interest in large-scale parametric
optimization. We believe that this work
has huge implications to the NLP field as
it enables fast detection of activity,
warm-starts, inertia detection, and the
use of iterative linear algebra. In
addition, we have shown that the exact
penalty can detect minima of the
optimization problem without introducing
of spurious points. We are currently
investigation approaches to deal with
ill-conditionning induced by the penalty
function.
Collaborators

Zavala, V. M. and Anitescu, M.
Scalable
Nonlinear Programming Via Exact
Differentiable Penalty Functions and
Trust-Region Newton Methods.
Under Review, 2013. [pdf]

Zavala, V. M.; and Anitescu, M.
Real-Time
Nonlinear Optimization as a
Generalized Equation.
SIAM J. Control and Optimization,
48, pp. 5444-5454, 2010. [pdf]

Robbins,
B.; and Zavala, V.M. Convergence
Analysis of a Parallel Newton Scheme
for Dynamic Power Grid Simulations.International
Workshop on High Performance
Computing, Networking and
Analytics for the Power Grid,
2011. [pdf]

Zavala, V. M.; and
Anitescu, M. Achieving
Higher Frequencies in Large-Scale
Nonlinear Model Predictive Control.
IEEE CDC, 2010. [pdf]

Figure 1. Stable Manifold Solution Tracking Using
Exact Penalty Function

Figure 2. Fast Active-Set Detection Using
Projected Gauss-Seidel for Different Problems