Biconvex sets and optimization with biconvex functions: a survey and extensions

The problem of optimizing a biconvex function over a given (bi)convex or compact set frequently occurs in theory as well as in industrial applications, for example, in the field of multifacility location or medical image registration. Thereby, a function is called biconvex, if f(x,y) is convex in y for fixed x∈X, and f(x,y) is convex in x for fixed y∈Y. This paper presents a survey of existing results concerning the theory of biconvex sets and biconvex functions and gives some extensions. In particular, we focus on biconvex minimization problems and survey methods and algorithms for the constrained as well as for the unconstrained case. Furthermore, we state new theoretical results for the maximum of a biconvex function over biconvex sets. J. Gorski and K. Klamroth were partially supported by a grant of the German Research Foundation (DFG).     

Comparison of radii sets,entropy, QM methods,and sampling on MM‐PBSA,MM‐GBSA,and QM/MM‐GBSA ligand binding energies of F. tularensis enoyl‐ACP reductase (FabI)

To validate a method for predicting the binding affinities of FabI inhibitors, three implicit solvent methods, MM‐PBSA, MM‐GBSA, and QM/MM‐GBSA were carefully compared using 16 benzimidazole inhibitors in complex with Francisella tularensis FabI. The data suggests that the prediction results are sensitive to radii sets, GB methods, QM Hamiltonians, sampling protocols, and simulation length, if only one simulation trajectory is used for each ligand. In this case, QM/MM‐GBSA using 6 ns MD simulation trajectories together with GB^neck2, PM3, and the mbondi2 radii set, generate the closest agreement with experimental values (r² = 0.88). However, if the three implicit solvent methods are averaged from six 1 ns MD simulations for each ligand (called “multiple independent sampling”), the prediction results are relatively insensitive to all the tested parameters. Moreover, MM/GBSA together with GB^HCT and mbondi, using 600 frames extracted evenly from six 0.25 ns MD simulations, can also provide accurate prediction to experimental values (r² = 0.84). Therefore, the multiple independent sampling method can be more efficient than a single, long simulation method. Since future scaffold expansions may significantly change the benzimidazole's physiochemical properties (charges, etc.) and possibly binding modes, which may affect the sensitivities of various parameters, the relatively insensitive “multiple independent sampling method” may avoid the need of an entirely new validation study. Moreover, due to large fluctuating entropy values, (QM/)MM‐P(G)BSA were limited to inhibitors’ relative affinity prediction, but not the absolute affinity. The developed protocol will support an ongoing benzimidazole lead optimization program. © 2015 Wiley Periodicals, Inc.     

Optimal diffuse augmented atomic basis sets for extrapolation of the correlation energy

We seek correlation-consistent diffuse-augmented double-zeta and triple-zeta basis sets that perform optimally in extrapolating the correlation energy to the one-electron complete basis set limit, denoted oAVXZ and oAV(X + d)Z. The novel basis sets are method-dependent in that they are trained to perform optimally for the correlation energy at each specific level of theory. They are shown to yield accurate results in calculating both the energy and tensorial properties such as polarizabilities while not significantly altering the Hartree-Fock energy. Quantitatively, complete basis set limit (CBS)-/(oAVdZ,oAVtZ)-extrapolated correlation energies typically outperform, by 3- to 5-fold, the ones calculated with traditional ansatzes of similar flexibility. Attaining energies of CBS/(AVtZ,AVqZ) type or better accuracy, they frequently outperform expensive raw explicitly correlated ones. Promisingly, a limited test on CBS-extrapolated energies based on conventional basis sets has shown that they compare well even with extrapolated explicitly correlated ones. Calculated atomization and dissociation energies, molecular geometries, ionization potentials, and electron affinities also tend to outperform the ones obtained with traditional Dunning's ansatzes from which the new basis sets have been determined. The method for basis set generation is simple, and there is no reason of principle why the approach could not be adapted for handling other bases in the literature.     

Constructions of Sidon spaces and cyclic subspace codes

In this paper, we firstly construct several new kinds of Sidon spaces and Sidon sets by investigating some known results. Secondly, using these Sidon spaces, we will present a construction of cyclic subspace codes with cardinality τ · $\frac{{{q^n} - 1}}{{q - 1}}$ and minimum distance 2k−2, where τ is a positive integer. We furthermore give some cyclic subspace codes with size 2τ · $\frac{{{q^n} - 1}}{{q - 1}}$ and without changing the minimum distance 2k−2.     

Polytopes of high rank for the symmetric groups

In the Atlas of abstract regular polytopes for small almost simple groups by Leemans and Vauthier, the polytopes whose automorphism group is a symmetric group Sn of degree 5?n?9 are available. Two observations arise when we look at the results: (1) for n?5, the (n−1)-simplex is, up to isomorphism, the unique regular (n−1)-polytope having Sn as automorphism group and, (2) for n?7, there exists, up to isomorphism and duality, a unique regular (n−2)-polytope whose automorphism group is Sn. We prove that (1) is true for n≠4 and (2) is true for n?7. Finally, we also prove that Sn acts regularly on at least one abstract polytope of rank r for every 3?r?n−1.     

On the structure of the sumsets

Let A be a set of nonnegative integers. For h≥2, denote by hA the set of all the integers representable by a sum of h elements from A. In this paper, we prove that, if k≥3, and A={a₀,a₁,…,a_k−1} is a finite set of integers such that 0=a₀<a₁<?<a_k−1 and (a₁,…,a_k−1)=1, then there exist integers c and d and sets C⊆[0,c−2] and D⊆[0,d−2] such that hA=C∪[c,ha_k−1−d]∪(ha_k−1−D) for all . The result is optimal. This improves Nathanson’s result: h≥max{1,(k−2)(a_k−1−1)a_k−1}.     

On acyclic edge coloring of planar graphs without intersecting triangles

Let Δ denote the maximum degree of a graph. Fiam?ík first and then Alon et al. again conjectured that every graph is acyclically edge (Δ+2)-colorable. Even for planar graphs, this conjecture remains open. It is known that every triangle-free planar graph is acyclically edge (Δ+5)-colorable. This paper proves that every planar graph without intersecting triangles is acyclically edge (Δ+4)-colorable.     

Bases in Banach spaces of smooth functions on Cantor-type sets

We suggest a Schauder basis in Banach spaces of smooth functions and traces of smooth functions on Cantor-type sets. In the construction, local Taylor expansions of functions are used.