Ivo Bleylevens

Efficiency improvement in an nD systems approach to polynomial optimization

The problem of finding the global minimum of a so-called Minkowski-norm dominated polynomial can be approached by the matrix method of Stetter and Moller, which reformulates it as a large eigenvalue problem. A drawback of this approach is that the matrix involved is usually very large. However, all that is needed for modern iterative eigenproblem solvers is a routine which computes the action of the matrix on a given vector. This paper focuses on improving the efficiency of computing the action of the matrix on a vector. To avoid building the large matrix one can associate the system of first-order conditions with an nD system of difference equations. One way to compute the action of the matrix efficiently is by setting up a corresponding shortest path problem and solving it. It turns out that for large n the shortest path problem has a high computational complexity, and therefore some heuristic procedures are developed for arriving cheaply at suboptimal paths with acceptable performance.

Optimization of Compound Ranking for Structure‐Based Virtual Ligand Screening Using an Established FRED–Surflex Consensus Approach

The use of multiple target conformers has been applied successfully in virtual screening campaigns; however, a study on how to best combine scores for multiple targets in a hierarchic method that combines rigid and flexible docking is not available. In this study, we used a data set of 59 479 compounds to screen multiple conformers of four distinct protein targets to obtain an adapted and optimized combination of an established hierarchic method that employs the programs FRED and Surflex. Our study was extended and verified by application of our protocol to ten different data sets from the directory of useful decoys (DUD). We quantitated overall method performance in ensemble docking and compared several consensus scoring methods to improve the enrichment during virtual ligand screening. We conclude that one of the methods used, which employs a consensus weighted scoring of multiple target conformers, performs consistently better than methods that do not include such consensus scoring. For optimal overall performance in ensemble docking, it is advisable to first calculate a consensus of FRED results and use this consensus as a sub‐data set for Surflex screening. Furthermore, we identified an optimal method for each of the chosen targets and propose how to optimize the enrichment for any target.

An nD-systems approach to global polynomial optimization with an application to H 2 model order reduction

The problem of finding the global minimum of a multivariate polynomial can be approached by the matrix method of Stetter-Möller, which reformulates it as a large eigenvalue problem. The linear operators involved in this approach are studied using the theory of nD-systems. This supports the efficient application of iterative methods for solving eigenvalue problems such as Arnoldi methods and Jacobi-Davidson methods. This approach is demonstrated by an example which addresses optimal H2-model reduction of a linear dynamical model of order 10 to order 9.

On a Finiteness Result for the Number of Critical Points of the H2 Approximation Criterion

The long-standing open problem about whether the number of critical points in the H2 SISO real model order reduction problem is finite is answered in the positive in the case the transfer function of the to-be-reduced model has distinct poles (ie. only has poles of algebraic multiplicity one). This has important implications for various search methods for finding critical points or local minima of the criterion function for this model reduction problem. In fact more is shown namely that in a particular parametrization the number of complex solutions of the first order conditions of the H2 real model order reduction problem is finite and lies in a bounded set of which the bound can be determined from information about the to-be-reduced model. This implies that the H2 model order reduction problem can be solved in principle by constructive algebra methods (such as Groebner basis methods) in case the to-be-reduced model has distinct poles. It simplifies the methods for co-order three reduction as described in a companion paper.

On Algebraic and Linear Algebraic Aspects of Co-Order Three H2 Model Order Reduction

Abstract We present an algebraic method to compute a globally optimal H 2 approximation of order N – 3 to a given system of order N. First, the problem is formulated as a two-parameter polynomial eigenvalue problem with a special structure. To solve it, we apply and generalize algebraic techniques used in the computation of the Kronecker canonical form of a matrix pencil. Finiteness of the number of nontrivial solutions then allows the problem to be reduced to a one-parameter polynomial eigenvalue problem, which is solved with standard numerical methods. An example demonstrates the approach and provides a proof of principle.

PS6 - 4. Toward selective targeting of muscular AMPK: in search of a small molecule to change AMPK cellular localization

The energy-sensor AMP-activated protein kinase (AMPK) cycles between a glycogen-bound and a free state. The muscle-specific regulatory AMPKβ2 subunit carries a high affinity carbohydrate-binding module (CBM). Upon energy stress, such as exercise, AMPK localization at glycogen allows for rapid inhibition of glycogen synthesis, whereas cytosolic AMPK is capable of stimulating insulin-independent glucose uptake.

Polynomial optimization and a Jacobi-Davidson type method for commuting matrices

In this paper we introduce a new Jacobi-Davidson type eigenvalue solver for a set of commuting matrices, called JDCOMM, used for the global optimization of so-called Minkowski-norm dominated polynomials in several variables. The Stetter-Moller matrix method yields such a set of real non-symmetric commuting matrices since it reformulates the optimization problem as an eigenvalue problem. A drawback of this approach is that the matrix most relevant for computing the global optimum of the polynomial under investigation is usually large and only moderately sparse. However, the other matrices are generally much sparser and have the same eigenvectors because of the commutativity. This fact is used to design the JDCOMM method for this problem: the most relevant matrix is used only in the outer loop and the sparser matrices are exploited in the solution of the correction equation in the inner loop to greatly improve the efficiency of the method. Some numerical examples demonstrate that the method proposed in this paper is more efficient than approaches that work on the main matrix (standard Jacobi-Davidson and implicitly restarted Arnoldi), as well as conventional solvers for computing the global optimum, i.e., SOSTOOLS, GloptiPoly, and PHCpack.