27,283 research outputs found
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
A parametric integer programming algorithm for bilevel mixed integer programs
We consider discrete bilevel optimization problems where the follower solves
an integer program with a fixed number of variables. Using recent results in
parametric integer programming, we present polynomial time algorithms for pure
and mixed integer bilevel problems. For the mixed integer case where the
leader's variables are continuous, our algorithm also detects whether the
infimum cost fails to be attained, a difficulty that has been identified but
not directly addressed in the literature. In this case it yields a ``better
than fully polynomial time'' approximation scheme with running time polynomial
in the logarithm of the relative precision. For the pure integer case where the
leader's variables are integer, and hence optimal solutions are guaranteed to
exist, we present two algorithms which run in polynomial time when the total
number of variables is fixed.Comment: 11 page
Inverse Optimization with Noisy Data
Inverse optimization refers to the inference of unknown parameters of an
optimization problem based on knowledge of its optimal solutions. This paper
considers inverse optimization in the setting where measurements of the optimal
solutions of a convex optimization problem are corrupted by noise. We first
provide a formulation for inverse optimization and prove it to be NP-hard. In
contrast to existing methods, we show that the parameter estimates produced by
our formulation are statistically consistent. Our approach involves combining a
new duality-based reformulation for bilevel programs with a regularization
scheme that smooths discontinuities in the formulation. Using epi-convergence
theory, we show the regularization parameter can be adjusted to approximate the
original inverse optimization problem to arbitrary accuracy, which we use to
prove our consistency results. Next, we propose two solution algorithms based
on our duality-based formulation. The first is an enumeration algorithm that is
applicable to settings where the dimensionality of the parameter space is
modest, and the second is a semiparametric approach that combines nonparametric
statistics with a modified version of our formulation. These numerical
algorithms are shown to maintain the statistical consistency of the underlying
formulation. Lastly, using both synthetic and real data, we demonstrate that
our approach performs competitively when compared with existing heuristics
Local strong maximal monotonicity and full stability for parametric variational systems
The paper introduces and characterizes new notions of Lipschitzian and
H\"olderian full stability of solutions to general parametric variational
systems described via partial subdifferential and normal cone mappings acting
in Hilbert spaces. These notions, postulated certain quantitative properties of
single-valued localizations of solution maps, are closely related to local
strong maximal monotonicity of associated set-valued mappings. Based on
advanced tools of variational analysis and generalized differentiation, we
derive verifiable characterizations of the local strong maximal monotonicity
and full stability notions under consideration via some positive-definiteness
conditions involving second-order constructions of variational analysis. The
general results obtained are specified for important classes of variational
inequalities and variational conditions in both finite and infinite dimensions
Computing semiparametric bounds on the expected payments of insurance instruments via column generation
It has been recently shown that numerical semiparametric bounds on the
expected payoff of fi- nancial or actuarial instruments can be computed using
semidefinite programming. However, this approach has practical limitations.
Here we use column generation, a classical optimization technique, to address
these limitations. From column generation, it follows that practical univari-
ate semiparametric bounds can be found by solving a series of linear programs.
In addition to moment information, the column generation approach allows the
inclusion of extra information about the random variable; for instance,
unimodality and continuity, as well as the construction of corresponding
worst/best-case distributions in a simple way
On the local stability of semidefinite relaxations
We consider a parametric family of quadratically constrained quadratic
programs (QCQP) and their associated semidefinite programming (SDP)
relaxations. Given a nominal value of the parameter at which the SDP relaxation
is exact, we study conditions (and quantitative bounds) under which the
relaxation will continue to be exact as the parameter moves in a neighborhood
around the nominal value. Our framework captures a wide array of statistical
estimation problems including tensor principal component analysis, rotation
synchronization, orthogonal Procrustes, camera triangulation and resectioning,
essential matrix estimation, system identification, and approximate GCD. Our
results can also be used to analyze the stability of SOS relaxations of general
polynomial optimization problems.Comment: 23 pages, 3 figure
Mathematical programs with complementarity constraints: convergence properties of a smoothing method
In this paper, optimization problems with complementarity constraints are considered. Characterizations for local minimizers of of Orders 1 and 2 are presented. We analyze a parametric smoothing approach for solving these programs in which is replaced by a perturbed problem depending on a (small) parameter . We are interested in the convergence behavior of the feasible set and the convergence of the solutions of for In particular, it is shown that, under generic assumptions, the solutions are unique and converge to a solution of with a rate . Moreover, the convergence for the Hausdorff distance , between the feasible sets of and is of order
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