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https://www.maths.ed.ac.uk/~gondzio/reports/ipmXXV.pdf

Interior point methods 25 years later

The helix is a ubiquitous motif for biopolymers. We propose a heuristic, entropically based model that predicts helix formation in a system of hard spheres and semiflexible tubes. We find that the entropy of the spheres is maximized when short stretches of the tube form a helix with a geometry close to that found in natural helices. Our model could be directly tested with wormlike micelles as the tubes, and the effect could be used to self-assemble supramolecular helices.

/pdf/entropically-driven-helix-formation-4g6kf6n18o.pdf

Entropically Driven Helix Formation

In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods and imposes restrictions on the way the preconditioner is computed. A two-step approach is used to design a preconditioner. First, the Newton equation system is regularized to guarantee better numerical properties and then it is preconditioned. The preconditioner is implicit, that is, its computation requires only matrix-vector multiplications with the original problem data. The method is therefore well-suited to problems in which matrices are not explicitly available and/or are too large to be stored in computer memory. Numerical properties of the approach are studied including the analysis of the conditioning of the regularized system and that of the preconditioned regularized system. The method has been implemented and preliminary computational results for small problems limited to 1 million of variables and 10 million of nonzero elements demonstrate the feasibility of the approach.

/pdf/matrix-free-interior-point-method-2c52fqvfx2.pdf

Matrix-free interior point method

The solution of KKT systems is ubiquitous in optimization methods and often dominates the computation time, especially when large-scale problems are considered. Thus, the effective implementation of such methods is highly dependent on the availability of effective linear algebra algorithms and software, that are able, in turn, to take into account specific needs of optimization. In this paper we discuss the mutual impact of linear algebra and optimization, focusing on interior point methods and on the iterative solution of the KKT system. Three critical issues are addressed: preconditioning, termination control for the inner iterations, and inertia control.

/pdf/on-mutual-impact-of-numerical-linear-algebra-and-large-scale-4cbblsx216.pdf

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

Iterative solvers appear to be very promising in the development of efficient software, based on Interior Point methods, for large-scale nonlinear optimization problems. In this paper we focus on the use of preconditioned iterative techniques to solve the KKT system arising at each iteration of a Potential Reduction method for convex Quadratic Programming. We consider the augmented system approach and analyze the behaviour of the Constraint Preconditioner with the Conjugate Gradient algorithm. Comparisons with a direct solution of the augmented system and with MOSEK show the effectiveness of the iterative approach on large-scale sparse problems.

On the iterative solution of KKT systems in potential reduction software for large-scale quadratic problems

In this paper, we present an interior-point algorithm for large and sparse convex quadratic programming problems with bound constraints. The algorithm is based on the potential reduction method and the use of iterative techniques to solve the linear system arising at each iteration. The global convergence properties of the potential reduction method are reassessed in order to take into account the inexact solution of the inner system. We describe the iterative solver, based on the conjugate gradient method with a limited-memory incomplete Cholesky factorization as preconditioner. Furthermore, we discuss some adaptive strategies for the fill-in and accuracy requirements that we use in solving the linear systems in order to avoid unnecessary inner iterations when the iterates are far from the solution. Finally, we present the results of numerical experiments carried out to verify the effectiveness of the proposed strategies. We consider randomly generated sparse problems without a special structure. Also, we compare the proposed algorithm with the MOSEK solver.

Interior-point solver for large-scale quadratic programming problems with bound constraints

We focus on the use of adaptive stopping criteria in iterative methods for KKT systems that arise in Potential Reduction methods for quadratic programming. The aim of these criteria is to relate the accuracy in the solution of the KKT system to the quality of the current iterate, to get computational efficiency. We analyze a stopping criterion deriving from the convergence theory of inexact Potential Reduction methods and investigate the possibility of relaxing it in order to reduce as much as possible the overall computational cost. We also devise computational strategies to face a possible slowdown of convergence when an insufficient accuracy is required.

Stopping criteria for inner iterations in inexact potential reduction methods: a computational study

We present two strategies for choosing a “hot” starting-point in the context of an infeasible potential reduction (PR) method for convex quadratic programming. The basic idea of both strategies is to select a preliminary point and to suitably scale it in order to obtain a starting point such that its nonnegative entries are sufficiently bounded away from zero, and the ratio between the duality gap and a suitable measure of the infeasibility is small. One of the two strategies is naturally suggested by the convergence theory of the PR method; the other has been devised to reduce the initial values of the duality gap and the infeasibility measure, with the objective of decreasing the number of PR iterations. Numerical experiments show that the second strategy generally performs better than the first, and both outperform a starting-point strategy based on the affine-scaling step.

Marco D'Apuzzo

Papers

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

On the iterative solution of KKT systems in potential reduction software for large-scale quadratic problems

Interior-point solver for large-scale quadratic programming problems with bound constraints

Stopping criteria for inner iterations in inexact potential reduction methods: a computational study

Starting-point strategies for an infeasible potential reduction method