7,204 research outputs found
OSQP: An Operator Splitting Solver for Quadratic Programs
We present a general-purpose solver for convex quadratic programs based on
the alternating direction method of multipliers, employing a novel operator
splitting technique that requires the solution of a quasi-definite linear
system with the same coefficient matrix at almost every iteration. Our
algorithm is very robust, placing no requirements on the problem data such as
positive definiteness of the objective function or linear independence of the
constraint functions. It can be configured to be division-free once an initial
matrix factorization is carried out, making it suitable for real-time
applications in embedded systems. In addition, our technique is the first
operator splitting method for quadratic programs able to reliably detect primal
and dual infeasible problems from the algorithm iterates. The method also
supports factorization caching and warm starting, making it particularly
efficient when solving parametrized problems arising in finance, control, and
machine learning. Our open-source C implementation OSQP has a small footprint,
is library-free, and has been extensively tested on many problem instances from
a wide variety of application areas. It is typically ten times faster than
competing interior-point methods, and sometimes much more when factorization
caching or warm start is used. OSQP has already shown a large impact with tens
of thousands of users both in academia and in large corporations
A Parametric Non-Convex Decomposition Algorithm for Real-Time and Distributed NMPC
A novel decomposition scheme to solve parametric non-convex programs as they
arise in Nonlinear Model Predictive Control (NMPC) is presented. It consists of
a fixed number of alternating proximal gradient steps and a dual update per
time step. Hence, the proposed approach is attractive in a real-time
distributed context. Assuming that the Nonlinear Program (NLP) is
semi-algebraic and that its critical points are strongly regular, contraction
of the sequence of primal-dual iterates is proven, implying stability of the
sub-optimality error, under some mild assumptions. Moreover, it is shown that
the performance of the optimality-tracking scheme can be enhanced via a
continuation technique. The efficacy of the proposed decomposition method is
demonstrated by solving a centralised NMPC problem to control a DC motor and a
distributed NMPC program for collaborative tracking of unicycles, both within a
real-time framework. Furthermore, an analysis of the sub-optimality error as a
function of the sampling period is proposed given a fixed computational power.Comment: 16 pages, 9 figure
A Duality-Based Approach for Distributed Optimization with Coupling Constraints
In this paper we consider a distributed optimization scenario in which a set
of agents has to solve a convex optimization problem with separable cost
function, local constraint sets and a coupling inequality constraint. We
propose a novel distributed algorithm based on a relaxation of the primal
problem and an elegant exploration of duality theory. Despite its complex
derivation based on several duality steps, the distributed algorithm has a very
simple and intuitive structure. That is, each node solves a local version of
the original problem relaxation, and updates suitable dual variables. We prove
the algorithm correctness and show its effectiveness via numerical
computations
An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the AC Optimal Power Flow
A novel trust region method for solving linearly constrained nonlinear
programs is presented. The proposed technique is amenable to a distributed
implementation, as its salient ingredient is an alternating projected gradient
sweep in place of the Cauchy point computation. It is proven that the algorithm
yields a sequence that globally converges to a critical point. As a result of
some changes to the standard trust region method, namely a proximal
regularisation of the trust region subproblem, it is shown that the local
convergence rate is linear with an arbitrarily small ratio. Thus, convergence
is locally almost superlinear, under standard regularity assumptions. The
proposed method is successfully applied to compute local solutions to
alternating current optimal power flow problems in transmission and
distribution networks. Moreover, the new mechanism for computing a Cauchy point
compares favourably against the standard projected search as for its activity
detection properties
A duality-based approach for distributed min-max optimization with application to demand side management
In this paper we consider a distributed optimization scenario in which a set
of processors aims at minimizing the maximum of a collection of "separable
convex functions" subject to local constraints. This set-up is motivated by
peak-demand minimization problems in smart grids. Here, the goal is to minimize
the peak value over a finite horizon with: (i) the demand at each time instant
being the sum of contributions from different devices, and (ii) the local
states at different time instants being coupled through local dynamics. The
min-max structure and the double coupling (through the devices and over the
time horizon) makes this problem challenging in a distributed set-up (e.g.,
well-known distributed dual decomposition approaches cannot be applied). We
propose a distributed algorithm based on the combination of duality methods and
properties from min-max optimization. Specifically, we derive a series of
equivalent problems by introducing ad-hoc slack variables and by going back and
forth from primal and dual formulations. On the resulting problem we apply a
dual subgradient method, which turns out to be a distributed algorithm. We
prove the correctness of the proposed algorithm and show its effectiveness via
numerical computations.Comment: arXiv admin note: substantial text overlap with arXiv:1611.0916
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
Reduced Memory Footprint in Multiparametric Quadratic Programming by Exploiting Low Rank Structure
In multiparametric programming an optimization problem which is dependent on
a parameter vector is solved parametrically. In control, multiparametric
quadratic programming (mp-QP) problems have become increasingly important since
the optimization problem arising in Model Predictive Control (MPC) can be cast
as an mp-QP problem, which is referred to as explicit MPC. One of the main
limitations with mp-QP and explicit MPC is the amount of memory required to
store the parametric solution and the critical regions. In this paper, a method
for exploiting low rank structure in the parametric solution of an mp-QP
problem in order to reduce the required memory is introduced. The method is
based on ideas similar to what is done to exploit low rank modifications in
generic QP solvers, but is here applied to mp-QP problems to save memory. The
proposed method has been evaluated experimentally, and for some examples of
relevant problems the relative memory reduction is an order of magnitude
compared to storing the full parametric solution and critical regions
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