Landesman–Lazer conditions for difference equations involving sublinear perturbations

We study the existence and uniqueness for discrete Neumann and periodic problems. We consider both ordinary and partial difference equations involving sublinear perturbations. All the proofs are based on reformulating these discrete problems as a general singular algebraic system. Firstly, we use variational techniques (specifically, the Saddle Point Theorem) and prove the existence result based on a type of Landesman–Lazer condition. Then we show that for a certain class of bounded nonlinearities this condition is even necessary and therefore, we specify also the cases in which there does not exist any solution. Finally, the uniqueness is discussed.     

A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk, and Winther includes a well-developed theory of finite element methods for Hodge–Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of a residual type for Arnold–Falk–Winther mixed finite element methods for Hodge–de Rham–Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by a unified treatment of the various Hodge–Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge–Laplacian.     

A numerical method for nonconvex multi-objective optimal control problems

A numerical method is proposed for constructing an approximation of the Pareto front of nonconvex multi-objective optimal control problems. First, a suitable scalarization technique is employed for the multi-objective optimal control problem. Then by using a grid of scalarization parameter values, i.e., a grid of weights, a sequence of single-objective optimal control problems are solved to obtain points which are spread over the Pareto front. The technique is illustrated on problems involving tumor anti-angiogenesis and a fed-batch bioreactor, which exhibit bang–bang, singular and boundary types of optimal control. We illustrate that the Bolza form, the traditional scalarization in optimal control, fails to represent all the compromise, i.e., Pareto optimal, solutions.     

A Linear Separability Criterion for Sets of Euclidean Space

We prove new theorems which describe a necessary and sufficient condition for linear (strong and non-strong) separability and inseparability of the sets in a finite-dimensional Euclidean space. We propose a universal measure for the thickness of the geometric margin (both the strong separation margin (separator) and the margin of unseparated points (pseudo-separator)) formed between the parallel generalized supporting hyperplanes of the two sets which are separated. The introduced measure allows comparing results of linear separation obtained by different techniques for both linearly separable and inseparable sets. An optimization program whose formulation provides a maximum thickness of the separator for the separable sets is considered. When the sets are inseparable, the same solver is guaranteed to construct a pseudo-separator with a minimum thickness. We estimate the distance between the convex and closed sets. We construct a cone of generalized support vectors for hyperplanes, each one of which linearly separates the considered sets. The interconnection of the problem of different types of linear separation of sets with some related problems is studied.     

The Policyholder’s static and dynamic decision making of life insurance and pension payments

We deal with the introduction of life insurance and pension decisions in the personal financial problem of optimal lifetime consumption of lifetime income. We introduce in Sect. 2 the classical notion of reserves and present well-known differential equations characterizing these. We start with the survival model and discuss also the case where pension saving takes place in a bank. We then analyze the disability model and the multistate model that are generalizations of the survival model. This structure is repeated in all sections of the paper. In Sect. 3 we introduce the notion of utility reserves that makes it possible to compare the different contracts offered by the insurance company, and present differential equations characterizing these. The utility reserve is the basis for static optimization of payment streams in Sect. 4 and for dynamic optimization of payment streams in Sect. 5. In particular in the case of dynamic optimization, the differential equation characterizing the utility reserve plays a crucial role since the so-called Hamilton–Jacobi–Bellman equation characterizing the optimal solution is based on it. Sections 2–5 are ended by a continued numerical example illustrating our findings for the survival model. We conclude by further remarks on generalizations and applications.     

Improved Results on Geometric Hitting Set Problems

We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P = NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are half-spaces in ?³ and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local-search algorithm which iterates over local improvements only.     

Dynamic reconfiguration in multigroup multicast routing under uncertainty

Motivated by high quality multimedia and remote collaborative environments, we solve the problem of multigroup multicast routing with a number of features important for real-world deployment of the interactive media technologies. Based on the ant colony optimization metaheuristic, our algorithm is the first to work with uncertain knowledge of underlying network capabilities and support on-the-fly media transcoding inside the multicast tree. New contributions of this work are described next. We present two extensions of the algorithm, which improve quality of the solution. We introduce an integrated approach to solution of the problem, which is effective for both original solving from scratch as well as for new dynamic reconfiguration of multicast tree, where minimum perturbation of existing solution is desired. Experiments show that our algorithm is not only successful in maintaining existing communication with low number of unnecessary disruptions, but also capable of keeping the multicast trees efficient.     

Fractional-Order Legendre Functions and Their Application to Solve Fractional Optimal Control of Systems Described by Integro-differential Equations

In this paper, we introduce a set of functions called fractional-order Legendre functions (FLFs) to obtain the numerical solution of optimal control problems subject to the linear and nonlinear fractional integro-differential equations. We consider the properties of these functions to construct the operational matrix of the fractional integration. Also, we achieved a general formulation for operational matrix of multiplication of these functions to solve the nonlinear problems for the first time. Then by using these matrices the mentioned fractional optimal control problem is reduced to a system of algebraic equations. In fact the functions of the problem are approximated by fractional-order Legendre functions with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem converts to an optimization problem, which can then be solved numerically. The convergence of the method is discussed and finally, some numerical examples are presented to show the efficiency and accuracy of the method.     

Fubini Type Products for Densities and Liftings

In our former paper (Fund. Math. 166, 281–303, 2000) we discussed densities and liftings in the product of two probability spaces with good section properties analogous to that for measures and measurable sets in the Fubini Theorem. In the present paper we investigate the following more delicate problem: Let (Ω,Σ,μ) and (Θ,T,ν) be two probability spaces endowed with densities υ and τ, respectively. Can we define a density on the product space by means of a Fubini type formula \((\upsilon\odot\tau)(E)=\{(\omega,\theta):\omega\in\upsilon(\{\bar {\omega}:\theta\in\tau(E_{\bar{\omega}}\})\}\), for E measurable in the product, and the same for liftings instead of densities? We single out classes of marginal densities υ and τ which admit a positive solution in case of densities, where we have sometimes to replace the Fubini type product by its upper hull, which we call box product. For liftings the answer is in general negative, but our analysis of the above problem leads to a new method, which allows us to find a positive solution. In this way we solved one of the main problems of Musia?, Strauss and Macheras (Fund. Math. 166, 281–303, 2000).     

Finite element convergence for the Joule heating problem with mixed boundary conditions

We prove strong convergence of conforming finite element approximations to the stationary Joule heating problem with mixed boundary conditions on Lipschitz domains in three spatial dimensions. We show optimal global regularity estimates on creased domains and prove a priori and a posteriori bounds for shape regular meshes.     

Isoperimetric problem for exponential measure on the plane with $$\ell _1$$-metric

We give a solution to the isoperimetric problem for the exponential measure on the plane with the \(\ell _1\)-metric. As it turns out, among all sets of a given measure, the simplex or its complement (i.e. the ball in the \(\ell _1\)-metric or its complement) has the smallest boundary measure. The proof is based on a symmetrisation (along the sections of equal \(\ell _1\)-distance from the origin).     

Approximate homomorphisms on lattices

We prove two results concerning an Ulam-type stability problem for homomorphisms between lattices. One of them involves estimates by quite general error functions; the other deals with approximate (join) homomorphisms in terms of certain systems of lattice neighborhoods. As a corollary, we obtain a stability result for approximately monotone functions.     

An Inexact Modified Subgradient Algorithm for Primal-Dual Problems via Augmented Lagrangians

We consider a primal optimization problem in a reflexive Banach space and a duality scheme via generalized augmented Lagrangians. For solving the dual problem (in a Hilbert space), we introduce and analyze a new parameterized Inexact Modified Subgradient (IMSg) algorithm. The IMSg generates a primal-dual sequence, and we focus on two simple new choices of the stepsize. We prove that every weak accumulation point of the primal sequence is a primal solution and the dual sequence converges weakly to a dual solution, as long as the dual optimal set is nonempty. Moreover, we establish primal convergence even when the dual optimal set is empty. Our second choice of the stepsize gives rise to a variant of IMSg which has finite termination.     

On upper approximations of Pareto fronts

In one of our earlier works, we proposed to approximate Pareto fronts to multiobjective optimization problems by two-sided approximations, one from inside and another from outside of the feasible objective set, called, respectively, lower shell and upper shell. We worked there under the assumption that for a given problem an upper shell exists. As it is not always the case, in this paper we give some sufficient conditions for the existence of upper shells. We also investigate how to constructively search infeasible sets to derive upper shells. We approach this issue by means of problem relaxations. We formally show that under certain conditions some subsets of lower shells to relaxed multiobjective optimization problems are upper shells in the respective unrelaxed problems. Results are illustrated by a numerical example representing a small but real mechanical problem. Practical implications of the results are discussed.     

Illiquid financial market models and absence of arbitrage

We study financial market models with different liquidity effects. In the first part of this paper, we extend the short-term price impact model introduced by Rogers and Singh (2007) to a general semimartingale setup. We show the convergence of the discrete-time into the continuous-time modeling framework when trading times approach each other. In the second part, arbitrage opportunities in illiquid economies are considered, in particular a modification of the feedback effect model of Bank and Baum (2004). We demonstrate that a large trader cannot create wealth at no risk within this framework. Here we have to assume that the price process is described by a continuous semimartingale.     

Posynomial Parametric Geometric Programming with Interval Valued Coefficient

The article presents solution procedure of geometric programming with imprecise coefficients. We have considered problems with imprecise data as a form of an interval in nature. Many authors have solved the imprecise problem by geometric programming technique in a different way. In this paper, we introduce parametric functional form of an interval number and then solve the problem by geometric programming technique. The advantage of the present approach is that we get optimal solution of the objective function directly without solving equivalent transformed problems. Numerical examples are presented to support of the proposed approach.     

An empirical evaluation of walk-and-round heuristics for mixed integer linear programs

Feasibility pump is a general purpose technique for finding feasible solutions of mixed integer programs. In this paper we report our computational experience on using geometric random walks and a random ray approach to provide good points for the feasibility pump. Computational results on MIPLIB2003 and COR@L test libraries show that the walk-and-round approach improves the upper bounds of a large number of test problems when compared to running the feasibility pump either at the optimal solution or the analytic center of the continuous relaxation. In our experiments the hit-and-run walk (a specific type of random walk strategy) started from near the analytic center is generally better than other random search approaches, when short walks are used. The performance may be improved by expanding the feasible region before walking. Although the upper bound produced in the geometric random walk approach are generally inferior than the best available upper bounds for the test problems, we managed to prove optimality of three test problems which were considered unsolved in the COR@L benchmark library (though the COR@L bounds available to us seem to be out of date).     

Eta products of weight one,and decomposition of primes in non-abelian extensions

This article is a slightly extended and revised version of a conference talk at “Arithmetik an der A7” in Würzburg, June 23rd, 2017. We present a conjecture on the coincidence of Hecke theta series of weight 1 on three distinct quadratic fields. Then we discuss a special instance of the Deligne–Serre Theorem, implying that the decomposition of prime numbers in a certain extension of the rationals is governed by the coefficients of the eta product \(\eta^{2}(z)\).     

Existence and continuity of solution trajectories of generalized equations with application in electronics

We consider a special form of parametric generalized equations arising from electronic circuits with AC sources and study the effect of perturbing the input signal on solution trajectories. Using methods of variational analysis and strong metric regularity property of an auxiliary map, we are able to prove the regularity properties of the solution trajectories inherited by the input signal. Furthermore, we establish the existence of continuous solution trajectories for the perturbed problem. This can be achieved via a result of uniform strong metric regularity for the auxiliary map.     

A note on blocks of skeleton tolerances

We provide a full characterization of lattices which can be blocks of the skeleton tolerance relation of a finite lattice. Moreover, we formulate a necessary condition for a lattice to be such a block in the case of finite distributive lattices with at most k-dimensional maximal boolean intervals.