33,035 research outputs found
On the degeneracy phenomenon for nonlinear optimal control problems with higher index state constraints
Relatório Técnico do Núcleo de Investigação Officina Mathematica.Necessary conditions of optimality (NCO) play an important role in
optimization problems. They are the major tool to select a set of
candidates to minimizers. In optimal control theory, the NCO
appear in the form of a Maximum Principle (MP). For certain
optimal control problems with state constraints, it might happen
that the MP are unable to provide useful information --- the set of
all admissible solutions coincides with the set of candidates that
satisfy the MP. When this happens, the MP is said to degenerate. In
the recent years, there has been some literature on fortified forms
of the MP in such way that avoid degeneracy. These fortified forms
involve additional hypotheses --- Constraint Qualifications.
Whenever the state constraints have higher index (i.e. their first
derivative with respect to time does not depend on control), the
current constraint qualifications are not adequate. So, the main
purpose here is fortify the maximum principle for optimal control
problems with higher index constraints, for which there is a need to
develop new constraint qualifications. The results presented here are
a generalization to nonlinear problems of a previous work.The financial support from Projecto FCT POSC/EEA-SRI/61831/2004 is
gratefully acknowledged
An improved multi-parametric programming algorithm for flux balance analysis of metabolic networks
Flux balance analysis has proven an effective tool for analyzing metabolic
networks. In flux balance analysis, reaction rates and optimal pathways are
ascertained by solving a linear program, in which the growth rate is maximized
subject to mass-balance constraints. A variety of cell functions in response to
environmental stimuli can be quantified using flux balance analysis by
parameterizing the linear program with respect to extracellular conditions.
However, for most large, genome-scale metabolic networks of practical interest,
the resulting parametric problem has multiple and highly degenerate optimal
solutions, which are computationally challenging to handle. An improved
multi-parametric programming algorithm based on active-set methods is
introduced in this paper to overcome these computational difficulties.
Degeneracy and multiplicity are handled, respectively, by introducing
generalized inverses and auxiliary objective functions into the formulation of
the optimality conditions. These improvements are especially effective for
metabolic networks because their stoichiometry matrices are generally sparse;
thus, fast and efficient algorithms from sparse linear algebra can be leveraged
to compute generalized inverses and null-space bases. We illustrate the
application of our algorithm to flux balance analysis of metabolic networks by
studying a reduced metabolic model of Corynebacterium glutamicum and a
genome-scale model of Escherichia coli. We then demonstrate how the critical
regions resulting from these studies can be associated with optimal metabolic
modes and discuss the physical relevance of optimal pathways arising from
various auxiliary objective functions. Achieving more than five-fold
improvement in computational speed over existing multi-parametric programming
tools, the proposed algorithm proves promising in handling genome-scale
metabolic models.Comment: Accepted in J. Optim. Theory Appl. First draft was submitted on
August 4th, 201
Controlled diffusion processes
This article gives an overview of the developments in controlled diffusion
processes, emphasizing key results regarding existence of optimal controls and
their characterization via dynamic programming for a variety of cost criteria
and structural assumptions. Stochastic maximum principle and control under
partial observations (equivalently, control of nonlinear filters) are also
discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A recursively feasible and convergent Sequential Convex Programming procedure to solve non-convex problems with linear equality constraints
A computationally efficient method to solve non-convex programming problems
with linear equality constraints is presented. The proposed method is based on
a recursively feasible and descending sequential convex programming procedure
proven to converge to a locally optimal solution. Assuming that the first
convex problem in the sequence is feasible, these properties are obtained by
convexifying the non-convex cost and inequality constraints with inner-convex
approximations. Additionally, a computationally efficient method is introduced
to obtain inner-convex approximations based on Taylor series expansions. These
Taylor-based inner-convex approximations provide the overall algorithm with a
quadratic rate of convergence. The proposed method is capable of solving
problems of practical interest in real-time. This is illustrated with a
numerical simulation of an aerial vehicle trajectory optimization problem on
commercial-of-the-shelf embedded computers
Optimal control for the thin-film equation: Convergence of a multi-parameter approach to track state constraints avoiding degeneracies
We consider an optimal control problem subject to the thin-film equation
which is deduced from the Navier--Stokes equation. The PDE constraint lacks
well-posedness for general right-hand sides due to possible degeneracies; state
constraints are used to circumvent this problematic issue and to ensure
well-posedness, and the rigorous derivation of necessary optimality conditions
for the optimal control problem is performed. A multi-parameter regularization
is considered which addresses both, the possibly degenerate term in the
equation and the state constraint, and convergence is shown for vanishing
regularization parameters by decoupling both effects. The fully regularized
optimal control problem allows for practical simulations which are provided,
including the control of a dewetting scenario, to evidence the need of the
state constraint, and to motivate proper scalings of involved regularization
and numerical parameters
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