arXiv subjects

Anders Forsgren

Publications and source records attributed to Anders Forsgren.

24 records · Page 2Linked to original sources

arXiv2015

Primal and dual active-set methods for convex quadratic programming

Computational methods are proposed for solving a convex quadratic program (QP). Active-set methods are defined for a particular primal and dual formulation of a QP with general equality constraints and simple lower bounds on the variables. In the first part of the paper, two methods are proposed, one primal and one dual. These methods generate a sequence of iterates that are feasible with respect to the equality constraints associated with the optimality conditions of the primal-dual form. The primal method maintains feasibility of the primal inequalities while driving the infeasibilities of the dual inequalities to zero. The dual method maintains feasibility of the dual inequalities while moving to satisfy the primal inequalities. In each of these methods, the search directions satisfy a KKT system of equations formed from Hessian and constraint components associated with an appropriate column basis. The composition of the basis is specified by an active-set strategy that guarantees the nonsingularity of each set of KKT equations. Each of the proposed methods is a conventional active-set method in the sense that an initial primal- or dual-feasible point is required. In the second part of the paper, it is shown how the quadratic program may be solved as a coupled pair of primal and dual quadratic programs created from the original by simultaneously shifting the simple-bound constraints and adding a penalty term to the objective function. Any conventional column basis may be made optimal for such a primal-dual pair of shifted-penalized problems. The shifts are then updated using the solution of either the primal or the dual shifted problem. An obvious application of this approach is to solve a shifted dual QP to define an initial feasible point for the primal (or vice versa). The computational performance of each of the proposed methods is evaluated on a set of convex problems.

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arXiv2015

Robust generation of elementary flux modes

Elementary flux modes (EFMs) are vectors defined from a metabolic reaction network, giving the connections between substrates and products. EFMs-based metabolic flux analysis (MFA) estimates the flux over each EFM from external flux measurements through least-squares data fitting. In previous work we presented an optimization method of column generation type that facilitates EFMs-based MFA when the metabolic reaction network is so large that enumerating all EFMs is prohibitive. In this work we extend this model by including errors on measurements in a robust optimization framework. In the robust optimization problem, the least-squares data fitting is minimized subject to the error on each metabolite being as unfavourable as it can be, within a given interval. In general, inclusion of robustness may make the optimization problem significantly harder. However, we show that in our case the robust problem can be stated as a convex quadratic programming problem, i.e., of the same form as the original non-robust problem. Additionally, we demonstrate that the column-generation technique of the non-robust problem can be extended also to the robust problem. Furthermore, the option to indicate intervals on metabolites that are not measured is introduced in this column generation framework. The effect of including robustness in the model is evaluated in a case-study, which indicated that the solutions of our non-robust problems are in fact near-optimal also when robustness is considered. On the other hand, the addition of intervals on unmeasured metabolites resulted in a change of optimal solution. Implying that the inclusion of intervals on unmeasured metabolites is more important than the direct consideration of measurement errors, this despite up to 20% errors.

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arXiv2014

On Dynamically Generating Relevant Elementary Flux Modes in a Metabolic Network using Optimization

Elementary flux modes (EFMs) are pathways through a metabolic reaction network that connect external substrates to products. Using EFMs, a metabolic network can be transformed into its macroscopic counterpart, in which the internal metabolites have been eliminated and only external metabolites remain. In EFMs-based metabolic flux analysis (MFA) experimentally determined external fluxes are used to estimate the flux of each EFM. It is in general prohibitive to enumerate all EFMs for complex networks, since the number of EFMs increases rapidly with network complexity. In this work we present an optimization-based method that dynamically generates a subset of EFMs and solves the EFMs-based MFA problem simultaneously. The obtained subset contains EFMs that contribute to the optimal solution of the EFMs-based MFA problem. The usefulness of our method was examined in a case-study using data from a Chinese hamster ovary cell culture and two networks of varied complexity. It was demonstrated that the EFMs-based MFA problem could be solved at a low computational cost, even for the more complex network. Additionally, only a fraction of the total number of EFMs was needed to compute the optimal solution.

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arXiv2014

On solving symmetric systems of linear equations in an unnormalized Krylov subspace framework

In an unnormalized Krylov subspace framework for solving symmetric systems of linear equations, the orthogonal vectors that are generated by a Lanczos process are not necessarily on the form of gradients. Associating each orthogonal vector with a triple, and using only the three-term recurrences of the triples, we give conditions on whether a symmetric system of linear equations is compatible or incompatible. In the compatible case, a solution is given and in the incompatible case, a certificate of incompatibility is obtained. In particular, the case when the matrix is singular is handled. We also derive a minimum-residual method based on this framework and show how the iterates may be updated explicitly based on the triples, and in the incompatible case a minimum-residual solution of minimum Euclidean norm is obtained.

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arXiv2014

An elementary proof of linear programming optimality conditions without using Farkas' lemma

Although it is easy to prove the sufficient conditions for optimality of a linear program, the necessary conditions pose a pedagogical challenge. A widespread practice in deriving the necessary conditions is to invoke Farkas' lemma, but proofs of Farkas' lemma typically involve "nonlinear" topics such as separating hyperplanes between disjoint convex sets, or else more advanced LP-related material such as duality and anti-cycling strategies in the simplex method. An alternative approach taken previously by several authors is to avoid Farkas' lemma through a direct proof of the necessary conditions. In that spirit, this paper presents what we believe to be an "elementary" proof of the necessary conditions that does not rely on Farkas' lemma and is independent of the simplex method, relying only on linear algebra and a perturbation technique published in 1952 by Charnes. No claim is made that the results are new, but we hope that the proofs may be useful for those who teach linear programming.

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arXiv2014

On the connection between the conjugate gradient method and quasi-Newton methods on quadratic problems

It is well known that the conjugate gradient method and a quasi-Newton method, using any well-defined update matrix from the one-parameter Broyden family of updates, produce identical iterates on a quadratic problem with positive-definite Hessian. This equivalence does not hold for any quasi-Newton method. We define precisely the conditions on the update matrix in the quasi-Newton method that give rise to this behavior. We show that the crucial facts are, that the range of each update matrix lies in the last two dimensions of the Krylov subspaces defined by the conjugate gradient method and that the quasi-Newton condition is satisfied. In the framework based on a sufficient condition to obtain mutually conjugate search directions, we show that the one-parameter Broyden family is complete. A one-to-one correspondence between the Broyden parameter and the non-zero scaling of the search direction obtained from the corresponding quasi-Newton method compared to the one obtained in the conjugate gradient method is derived. In addition, we show that the update matrices from the one-parameter Broyden family are almost always well-defined on a quadratic problem with positive-definite Hessian. The only exception is when the symmetric rank-one update is used and the unit steplength is taken in the same iteration. In this case it is the Broyden parameter that becomes undefined.

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