11,727 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
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Second-Derivative SQP Methods for Large-Scale Nonconvex Optimization
The class of stabilized sequential quadratic programming (SQP) methods for nonlinearly constrained optimization solve a sequence of related quadratic programming (QP) subproblems formed from a two-norm penalized quadratic model of the Lagrangian function subject to shifted, linearized constraints. While these methods have been shown to exhibit superlinear local convergence even when the constraint Jacobian is rank deficient at the solution, they generally have no global convergence theory. To address this issue, primal-dual SQP methods (pdSQP) employ a certain primal-dual augmented Lagrangian merit function and solve a subproblem that involves the minimization of a quadratic model of the merit function subject to simple bound constraints. The model of the merit function is constructed so that the resulting primal-dual subproblem is equivalent to the stabilized SQP subproblem. When used in conjunction with a flexible line-search, the merit function guarantees convergence from any starting point, while the connection with the stabilized subproblem allows pdSQP to retain the superlinear local convergence that is characteristic of stabilized SQP methods.A new dynamic convexification framework is developed that is applicable for nonconvex general standard form, stabilized, and primal-dual bound-constrained QP subproblems. Dynamic convexification involves three distinct stages: pre-convexification, concurrent convexification and post-convexification. New techniques are derived and analyzed for the implicit modification of symmetric indefinite factorizations and for the imposition of temporary artificial constraints, both of which are suitable for pre-convexification. Concurrent convexification works synchronously with the active-set method used to solve the subproblem, and computes minimal modifications needed to ensure that the QP iterates are uniformly bounded. Finally, post-convexification defines an implicit modification that ensures the solution of the subproblem yields a descent direction for the merit function.A new exact second-derivative primal-dual SQP method (dcpdSQP) is formulated for large-scale nonconvex optimization. Convergence analysis is presented that demonstrates guaranteed global convergence. Extensive numerical testing indicates that the performance of the proposed method is comparable or better than conventional full convexification while significantly reducing the number of factorizations required
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
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