arXiv subjects

Christian Günther

Publications and source records attributed to Christian Günther.

10 recordsLinked to original sources

arXiv2026

Bishop-Phelps Type Scalarization for Vector Optimization in Real Topological-Linear Spaces

It is well-known that scalarization techniques (e.g., in the sense of Gerstewitz; Kasimbeyli; Pascoletti and Serafini; Zaffaroni) are useful for generating (weakly, properly) efficient solutions of vector optimization problems. One recognized approach is the conic scalarization method in vector optimization in real normed spaces proposed by Kasimbeyli (2010, SIAM J Optim 20), which is based on augmented dual cones and Bishop-Phelps type (norm-linear) scalarizing functions. In this paper, we present new results on cone separation in real topological-linear spaces by using Bishop-Phelps type separating cones / separating seminorm-linear functions. Moreover, we study some extensions of known scalarization results in vector optimization (in the sense of Eichfelder; Gerstewitz; Jahn; Kasimbeyli; Pascoletti and Serafini). On this basis, we propose a Bishop-Phelps type scalarization method for vector optimization problems in real topological-linear spaces, which is based on Bishop-Phelps type cone-representing and cone-monotone scalarizing functions (e.g., Gerstewitz scalarizing functions or seminorm-linear scalarizing functions). Thus, our method also extends Kasimbeyli's conic scalarization method from real normed spaces to real topological-linear spaces. Within this framework, we derive new Bishop-Phelps type scalarization results for the concepts of weak efficiency and different types of proper efficiency.

math.OC↗

arXiv2026

Proper efficiency results in vector optimisation in real linear-topological spaces based on vectorial penalisation

In this paper, we are dealing with constrained vector optimisation problems where the objective function acts between real linear-topological spaces. Our aim is to study the relationships between the sets of properly efficient solutions to constrained and unconstrained vector optimisation problems under certain cone convexity assumptions on the objective function using a vectorial penalisation approach.

math.OC↗

arXiv2025

Symmetric and Non-Symmetric Cone Separation via Bishop-Phelps Cones in Normed Spaces

In this paper, we study relationships between symmetric and non-symmetric separation of (not necessarily convex) cones by using separating cones of Bishop-Phelps type in real normed spaces. Besides extending some known results for the non-symmetric cone separation approach, we propose a new symmetric cone separation approach and establish cone separation results for it by using some cone separation results obtained for the non-symmetric cone separation approach twice (by swapping the roles of the cones). In addition to specifically emphasizing the results for the convex case, we also present some existence results for (bounded) convex bases of convex cones. Finally, we highlight some applications of symmetric and non-symmetric cone separation in optimization.

math.OC↗

arXiv2025

Necessary conditions for approximate solutions of vector and set optimization problems with variable domination structure

We consider vector and set optimization problems with respect to variable domination structures given by set-valued mappings acting between the preimage space and the image space of the objective mapping, as well as by set-valued mappings with the same input and output space, that coincides with the image space of the objective mapping. The aim of this paper is to derive necessary conditions for approximately nondominated points of problems with a single-valued objective function, employing an extension of Ekeland's Variational Principle for problems with respect to variable domination structures in terms of generalized differentiation in the sense of Mordukhovich. For set-valued objective mappings, we derive necessary conditions for approximately nondominated points of problems with variable domination structure taking into account the incompatibility between openness and optimality and a directional openness result for the sum of set-valued maps. We describe the necessary conditions for approximately nondominated points of set optimization problems with variable domination structure in terms of the limiting (Mordukhovich) generalized differentiation objects.

math.OC↗

arXiv2024

Damage Location in Mechanical Structures by Multi-Objective Pattern Search

We propose a multi-objective global pattern search algorithm for the task of locating and quantifying damage in flexible mechanical structures. This is achieved by identifying eigenfrequencies and eigenmodes from measurements and matching them against the results of a finite element simulation model, which leads to a nonsmooth nonlinear bi-objective parameter estimation problem. A derivative-free optimization algorithm is required since the problem is nonsmooth and also because complex mechanical simulation models are often solved using commercial black-box software. Moreover, the entire set of non-dominated solutions is of interest to practitioners. Most solution approaches published to date are based on meta-heuristics such as genetic algorithms. The proposed multi-objective pattern-search algorithm provides a mathematically well-founded alternative. It features a novel sorting procedure that reduces the complexity in our context. Test runs on two experimental structures with multiple damage scenarios are used to validate the approach. The results demonstrate that the proposed algorithm yields accurate damage locations and requires moderate computational resources. From the engineer's perspective it represents a promising tool for structural health monitoring.

math.OC↗

arXiv2024

Nonlinear Strict Cone Separation Theorems in Real Normed Spaces

In this paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone / conical surface in real (reflexive) normed spaces. In essence, we follow the nonlinear and nonsymmetric separation approach developed by Kasimbeyli (2010, SIAM J. Optim. 20), which is based on augmented dual cones and Bishop-Phelps type (normlinear) separating functions. Compared to Kasimbeyli's separation theorem, we formulate our theorems for the separation of two cones under weaker conditions (concerning convexity and closedness requirements) with respect to the involved cones. By a new characterization of the algebraic interior of augmented dual cones in real normed spaces, we are able to establish relationships between our cone separation results and the results derived by Kasimbeyli (2010, SIAM J. Optim. 20) and by Garcia-Castano, Melguizo-Padial and Parzanese (2023, Math. Meth. Oper. Res. 97).

math.FA↗

arXiv2023

Nonlinear Cone Separation Theorems in Real Topological Linear Spaces

The separation of two sets (or more specific of two cones) plays an important role in different fields of mathematics such as variational analysis, convex analysis, convex geometry, optimization. In the paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone / conical surface in real (topological) linear spaces. Basically, we follow the separation approach by Kasimbeyli (2010, SIAM J. Optim. 20) based on augmented dual cones and Bishop-Phelps type (normlinear) separating functions. Classical separation theorems for convex sets are the key tool for proving our main nonlinear cone separation theorems.

math.FA↗

arXiv2019

Sequence pairs with asymptotically optimal aperiodic correlation

The Pursley-Sarwate criterion of a pair of finite complex-valued sequences measures the collective smallness of the aperiodic autocorrelations and the aperiodic crosscorrelations of the two sequences. It is known that this quantity is always at least 1 with equality if and only if the sequence pair is a Golay pair. We exhibit pairs of complex-valued sequences whose entries have unit magnitude for which the Pursley-Sarwate criterion tends to 1 as the sequence length tends to infinity. Our constructions use different carefully chosen Chu sequences.

cs.IT↗

arXiv2016

Merit factors of polynomials derived from difference sets

The problem of constructing polynomials with all coefficients $1$ or $-1$ and large merit factor (equivalently with small $L^4$ norm on the unit circle) arises naturally in complex analysis, condensed matter physics, and digital communications engineering. Most known constructions arise (sometimes in a subtle way) from difference sets, in particular from Paley and Singer difference sets. We consider the asymptotic merit factor of polynomials constructed from other difference sets, providing the first essentially new examples since 1991. In particular we prove a general theorem on the asymptotic merit factor of polynomials arising from cyclotomy, which includes results on Hall and Paley difference sets as special cases. In addition, we establish the asymptotic merit factor of polynomials derived from Gordon-Mills-Welch difference sets and Sidelnikov almost difference sets, proving two recent conjectures.

math.CO↗

arXiv2016

$L^q$ norms of Fekete and related polynomials

A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erdős, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small $L^q$ norm on the complex unit circle. We consider the Fekete polynomials \[ f_p(z)=\sum_{j=1}^{p-1}(j\mid p)\,z^j, \] where $p$ is an odd prime and $(\,\cdot\mid p)$ is the Legendre symbol (so that $z^{-1}f_p(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of $L^q$ and $L^2$ norm of $f_p(z)$ when $q$ is an even positive integer and $p\to\infty$. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the $L^4$ norm of these polynomials.

math.NT↗