RBFOpt: an open-source library for black-box optimization with costly function evaluations

We consider the problem of optimizing an unknown function given as an oracle over a mixed-integer box-constrained set. We assume that the oracle is expensive to evaluate, so that estimating partial derivatives by finite differences is impractical. In the literature, this is typically called a black-box optimization problem with costly evaluation. This paper describes the solution methodology implemented in the open-source library RBFOpt, available on COIN-OR. The algorithm is based on the Radial Basis Function method originally proposed by Gutmann (J Glob Optim 19:201–227, 2001.  https://doi.org/10.1023/A:1011255519438), which builds and iteratively refines a surrogate model of the unknown objective function. The two main methodological contributions of this paper are an approach to exploit a noisy but less expensive oracle to accelerate convergence to the optimum of the exact oracle, and the introduction of an automatic model selection phase during the optimization process. Numerical experiments show that RBFOpt is highly competitive on a test set of continuous and mixed-integer nonlinear unconstrained problems taken from the literature: it outperforms the open-source solvers included in our comparison by a large amount, and performs slightly better than a commercial solver. Our empirical evaluation provides insight on which parameterizations of the algorithm are the most effective in practice. The software reviewed as part of this submission was given the Digital Object Identifier (DOI)  https://doi.org/10.5281/zenodo.597767.     

QSDPNAL: a two-phase augmented Lagrangian method for convex quadratic semidefinite programming

In this paper, we present a two-phase augmented Lagrangian method, called QSDPNAL, for solving convex quadratic semidefinite programming (QSDP) problems with constraints consisting of a large number of linear equality and inequality constraints, a simple convex polyhedral set constraint, and a positive semidefinite cone constraint. A first order algorithm which relies on the inexact Schur complement based decomposition technique is developed in QSDPNAL-Phase I with the aim of solving a QSDP problem to moderate accuracy or using it to generate a reasonably good initial point for the second phase. In QSDPNAL-Phase II, we design an augmented Lagrangian method (ALM) wherein the inner subproblem in each iteration is solved via inexact semismooth Newton based algorithms. Simple and implementable stopping criteria are designed for the ALM. Moreover, under mild conditions, we are able to establish the rate of convergence of the proposed algorithm and prove the R-(super)linear convergence of the KKT residual. In the implementation of QSDPNAL, we also develop efficient techniques for solving large scale linear systems of equations under certain subspace constraints. More specifically, simpler and yet better conditioned linear systems are carefully designed to replace the original linear systems and novel shadow sequences are constructed to alleviate the numerical difficulties brought about by the crucial subspace constraints. Extensive numerical results for various large scale QSDPs show that our two-phase algorithm is highly efficient and robust in obtaining accurate solutions. The software reviewed as part of this submission was given the DOI (Digital Object Identifier)  https://doi.org/10.5281/zenodo.1206980.     

$$C_0$$-semigroups associated with uniquely ergodic Kantorovich modifications of operators

We extend to the context of \(L^p\) spaces and \(C_0\)-semigroups of operators our previous results from Heilmann and Ra?a (Positivity 21:897–910, 2017.  https://doi.org/10.1007/s11117-016-0441-1), concerning the eigenstructure and iterates of uniquely ergodic Kantorovich modifications of linking operators.     

On the second boundary value problem for a class of fully nonlinear flows II

This article is a continuation of an earlier work (Huang and Ye in Int Math Res Not, 2017.  https://doi.org/10.1093/imrn/rnx278), where the long time existence and convergence for some special cases of parabolic type special Lagrangian equations were given. The long time existence and convergence of the flow are obtained for all cases in this article. In particular, we can prescribe the second boundary value problems for a family of special Lagrangian graphs.     

Convergence Theory for Preconditioned Eigenvalue Solvers in a Nutshell

Preconditioned iterative methods for numerical solution of large matrix eigenvalue problems are increasingly gaining importance in various application areas, ranging from material sciences to data mining. Some of them, e.g., those using multilevel preconditioning for elliptic differential operators or graph Laplacian eigenvalue problems, exhibit almost optimal complexity in practice; i.e., their computational costs to calculate a fixed number of eigenvalues and eigenvectors grow linearly with the matrix problem size. Theoretical justification of their optimality requires convergence rate bounds that do not deteriorate with the increase of the problem size. Such bounds were pioneered by E. D’yakonov over three decades ago, but to date only a handful have been derived, mostly for symmetric eigenvalue problems. Just a few of known bounds are sharp. One of them is proved in doi: 10.1016/S0024-3795(01)00461-X for the simplest preconditioned eigensolver with a fixed step size. The original proof has been greatly simplified and shortened in doi: 10.1137/080727567 by using a gradient flow integration approach. In the present work, we give an even more succinct proof, using novel ideas based on Karush–Kuhn–Tucker theory and nonlinear programming.     

Serre’s modularity conjecture (I)

This paper is the first part of a work which proves Serre’s modularity conjecture. We first prove the cases \(p\not=2\) and odd conductor, and p=2 and weight 2, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see Khare and Wintenberger (Invent. Math., doi: 10.1007/s00222-009-0206-6, 2009). We then reduce the general case to a modularity statement for 2-adic lifts of modular mod 2 representations. This statement is now a theorem of Kisin (Invent. Math., doi: 10.1007/s00222-009-0207-5, 2009).     

A note “On <Emphasis Type="Italic">ss</Emphasis>-quasinormal or weakly <Emphasis Type="Italic">s</Emphasis>-permutably embedded subgroups of finite groups”

We point out some gaps in the paper (Monatsh Math doi: 10.1007/s00605-016-0877-1, 2016) by Kong and Guo, and give new brief proofs of the main results.     

Bakry–Émery curvature and diameter bounds on graphs

We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry–Émery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet–Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from Fathi and Shu (Bernoulli 24(1):672–698, 2018) and Horn et al. (J für die reine und angewandte Mathematik (Crelle’s J), 2017,  https://doi.org/10.1515/crelle-2017-0038) and solve a conjecture from Cushing et al. (Bakry–Émery curvature functions of graphs, 2016).     

The linear complexity of a class of binary sequences with optimal autocorrelation

Binary sequences with optimal autocorrelation and large linear complexity have important applications in cryptography and communications. Very recently, a class of binary sequences of period 4p with optimal autocorrelation was proposed by interleaving four suitable Ding–Helleseth–Lam sequences (Des. Codes Cryptogr.,  https://doi.org/10.1007/s10623-017-0398-5), where p is an odd prime with \(p \equiv 1(\bmod 4)\). The objective of this paper is to determine the minimal polynomial and the linear complexity of this class of binary optimal sequences via a sequence polynomial approach. It turns out that this class of sequences has quite good linear complexity.     

An Arc Search Infeasible Interior-Point Algorithm for Symmetric Optimization Using a New Wide Neighborhood

In this paper, we propose an infeasible interior-point algorithm for symmetric optimization problems using a new wide neighborhood and estimating the central path by an ellipse. In contrast of most interior-point algorithms for symmetric optimization which search an \(\varepsilon\)-optimal solution of the problem in a small neighborhood of the central path, our algorithm searches for optimizers in a new wide neighborhood of the ellipsoidal approximation of central path. The convergence analysis of the algorithm is shown and it is proved that the iteration bound of the algorithm is \(O ( r\log\varepsilon^{-1} ) \) which improves the complexity bound of the recent proposed algorithm by Liu et al. (J. Optim. Theory Appl., 2013,  https://doi.org/10.1007/s10957-013-0303-y) for symmetric optimization by the factor \(r^{\frac{1}{2}}\) and matches the currently best-known iteration bound for infeasible interior-point methods.     

Deformations of Charged Axially Symmetric Initial Data and the Mass–Angular Momentum–Charge Inequality

We show how to reduce the general formulation of the mass–angular momentum–charge inequality, for axisymmetric initial data of the Einstein–Maxwell equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. It is also shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass, angular momentum, and charge. This extends previous work by the authors (Cha and Khuri, Ann Henri Poincaré, doi: 10.1007/s00023-014-0332-6, arXiv:1401.3384, 2014), in which the role of charge was omitted. Lastly, we improve upon the hypotheses required for the mass–angular momentum–charge inequality in the maximal case.     

Modelling and optimization of least-cost water distribution networks with multiple supply sources and users

The proper allocation of water resources is a very important practical problem in the field of water network planning. Optimization models that are expeditious and easy to use for all stakeholders of the sector play an important role for water resource management. The present work resumes and reviews a least-cost optimization model proposed by our group (Maiolo and Pantusa in Water Sci Tech-W Sup.  https://doi.org/10.2166/ws.2015.114, 2016), able to design a water distribution network with multiple supply sources and multiple users. This approach requires of solving an optimization problem based on a nonlinear objective function which is proportional to the cost of the water distribution network. The cost of pre-existing pipelines is considered null. A more realistic scenario, able to consider the maximum flow rate allowed for existing sources-users connections, is considered here. In order to illustrate the usefulness and flexibility of the proposed approach, an application of the model to the real case of the province of Croton, Southern Italy, is presented.     

Super duality and crystal bases for quantum ortho-symplectic superalgebras II

Let \(\mathcal {O}^\mathrm{int}_q(m|n)\) be a semisimple tensor category of modules over a quantum ortho-symplectic superalgebra of type B, C, D introduced in Kwon (Int Math Res Not, 2015. doi: 10.1093/imrn/rnv076). It is a natural counterpart of the category of finitely dominated integrable modules over a quantum group of type B, C, D from a viewpoint of super duality. Continuing the previous work on type B and C (Kwon in Int Math Res Not, 2015. doi: 10.1093/imrn/rnv076), we classify the irreducible modules in \(\mathcal {O}^\mathrm{int}_q(m|n)\) and prove the existence and uniqueness of their crystal bases in case of type D. A new combinatorial model of classical crystals of type D is introduced, whose super analog gives a realization of crystals for the highest weight modules in \(\mathcal {O}^\mathrm{int}_q(m|n)\).     

Borwein–Preiss vector variational principle

This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi: 10.1006/jmaa.2000.6813). We introduce and characterize two seemingly new natural concepts of \(\varepsilon \)-minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function.     

Connectedness of Solution Sets of Strong Vector Equilibrium Problems with an Application

In this paper, the connectedness of solution set of a strong vector equilibrium problem in a finite dimensional space, is investigated. Firstly, a nonconvex separation theorem is given, that is, a neither open nor closed set and a compact subset in a finite dimensional space can be strictly separated by a sublinear and strongly monotone function. Secondly, in terms of the nonconvex separation theorem, the union relation between the solution set of the strong vector equilibrium problem and the solution sets of a series of nonlinear scalar problems, is established. Under suitable assumptions, the connectedness and the path connectedness of the solution set of the strong vector equilibrium problem are obtained. In particular, we solve partly the question related to the path connectedness of the solution set of the strong vector equilibrium problem. The question is proposed by Han and Huang in the reference (J Optim Theory Appl, 2016.  https://doi.org/10.1007/s10957-016-1032-9). Finally, as an application, we apply the main results to derive the connectedness of the solution set of a linear vector optimization problem.     

Error bounds for mixed integer nonlinear optimization problems

We introduce a-posteriori and a-priori error bounds for optimality and feasibility of a point generated as the rounding of an optimal point of the NLP relaxation of a mixed-integer nonlinear optimization problem. Our analysis mainly bases on the construction of a tractable approximation of the so-called grid relaxation retract. Under appropriate Lipschitz assumptions on the defining functions, we thereby generalize and slightly improve results for the mixed-integer linear case from Stein (Mathematical Programming, 2015, doi: 10.1007/s10107-015-0872-7). In particular, we identify cases in which the optimality and feasibility errors tend to zero at an at least linear rate for increasingly refined meshes.     

A remark on the result of Radu Miculescu and Alexandru Mihail

In this note, we show that the assumption of continuity considered in the recent result of Miculescu and Mihail (J Fixed Point Theory Appl, doi: 10.1007/s11784-017-0411-7, 2017) can be relaxed further. We also observe that the power convex contraction introduced by Miculescu and Mihail (see condition (1.1)) provides one more solution to the open question of Rhoades (Contemp Math 72:233–245, 1988] regarding existence of a contractive definition which is strong enough to generate a fixed point but does not force the mapping to be continuous at the fixed point.     

Hedging with temporary price impact

We consider the problem of hedging a European contingent claim in a Bachelier model with temporary price impact as proposed by Almgren and Chriss (J Risk 3:5–39, 2001). Following the approach of Rogers and Singh (Math Financ 20:597–615, 2010) and Naujokat and Westray (Math Financ Econ 4(4):299–335, 2011), the hedging problem can be regarded as a cost optimal tracking problem of the frictionless hedging strategy. We solve this problem explicitly for general predictable target hedging strategies. It turns out that, rather than towards the current target position, the optimal policy trades towards a weighted average of expected future target positions. This generalizes an observation of Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b) from their homogenous Markovian optimal investment problem to a general hedging problem. Our findings complement a number of previous studies in the literature on optimal strategies in illiquid markets as, e.g., Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b), Naujokat and Westray (Math Financ Econ 4(4):299–335, 2011), Rogers and Singh (Math Financ 20:597–615, 2010), Almgren and Li (Option hedging with smooth market impact. Preprint, 2015), Moreau et al. (Math Financ. doi: 10.1111/mafi.12098, 2015), Kallsen and Muhle-Karbe (High-resilience limits of block-shaped order books. Preprint, 2014), Guasoni and Weber (Mathematical Financ. doi: 10.1111/mafi.12099, 2015a; Nonlinear price impact and portfolio choice. Preprint, 2015b), where the frictionless hedging strategy is confined to diffusions. The consideration of general predictable reference strategies is made possible by the use of a convex analysis approach instead of the more common dynamic programming methods.     

Sample selection models for count data in R

We provide a detailed hands-on tutorial for the R package SemiParSampleSel (version 1.5). The package implements selection models for count responses fitted by penalized maximum likelihood estimation. The approach can deal with non-random sample selection, flexible covariate effects, heterogeneous selection mechanisms and varying distributional parameters. We provide an overview of the theoretical background and then demonstrate how SemiParSampleSel can be used to fit interpretable models of different complexity. We use data from the German Socio-Economic Panel survey (SOEP v28, 2012. doi: 10.5684/soep.v28) throughout the tutorial.