Michael T. M. Emmerich

Michael T. M. Emmerich

The 2026 disproof of Erdős's unit-distance conjecture and Sawin's subsequent explicit quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes integer parameters whose choice is not fully optimized. This report starts from Sawin's nonlinear integer optimization problem and develops an open-source Python verification pipeline. The pipeline is first validated by reproducing Sawin's published parameter choice and is then applied to computationally improved certificates. We optimize and verify certificates involving sets of primes $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. The implementation is deliberately lean, so that all results can be replicated on standard hardware and the procedures can be extended. We compare a deterministic greedy heuristic, a tailored integer evolution strategy with two-sided geometric, or discrete-Laplace, integer mutation and repair operators for number-theoretic feasibility, and a two-parent discrete-recombination variant. Four certificate levels are reported: Sawin's published example with $δ=0.0141144286784982\ldots$, a greedy certificate with $δ=0.0151718056372133\ldots$, a tailored integer evolution strategy certificate with $R=6672416/100000$ and $δ=0.0152616610684193\ldots$, and a recombination variant with the same $R$ and $δ=0.0152628688170072\ldots$. Consequently, the best current certificate supports the cautious statement $u(n)>n^{1.0152}$ for arbitrarily large $n$. Beyond this unit-distance application, the work illustrates how randomized optimization heuristics can improve explicit certificates in pure mathematics and combinatorial geometry.

André H. Deutz, Michael T. M. Emmerich, Hao Wang

The problem of approximating the Pareto front of a multiobjective optimization problem can be reformulated as the problem of finding a set that maximizes the hypervolume indicator. This paper establishes the analytical expression of the Hessian matrix of the mapping from a (fixed size) collection of $n$ points in the $d$-dimensional decision space (or $m$ dimensional objective space) to the scalar hypervolume indicator value. To define the Hessian matrix, the input set is vectorized, and the matrix is derived by analytical differentiation of the mapping from a vectorized set to the hypervolume indicator. The Hessian matrix plays a crucial role in second-order methods, such as the Newton-Raphson optimization method, and it can be used for the verification of local optimal sets. So far, the full analytical expression was only established and analyzed for the relatively simple bi-objective case. This paper will derive the full expression for arbitrary dimensions ($m\geq2$ objective functions). For the practically important three-dimensional case, we also provide an asymptotically efficient algorithm with time complexity in $O(n\log n)$ for the exact computation of the Hessian Matrix' non-zero entries. We establish a sharp bound of $12m-6$ for the number of non-zero entries. Also, for the general $m$-dimensional case, a compact recursive analytical expression is established, and its algorithmic implementation is discussed. Also, for the general case, some sparsity results can be established; these results are implied by the recursive expression. To validate and illustrate the analytically derived algorithms and results, we provide a few numerical examples using Python and Mathematica implementations. Open-source implementations of the algorithms and testing data are made available as a supplement to this paper.

Michael T. M. Emmerich, Ksenia Pereverdieva, André Deutz

We study the computational complexity of exact cardinality-constrained minimum Riesz $s$-energy subset selection in finite metric spaces: given $n$ points, select $k<n$ points of minimum Riesz $s$-energy. The objective sums inverse-power pair interactions and therefore promotes well-separated subsets; as $s$ becomes large, it increasingly approaches a bottleneck criterion governed by the closest selected pair, linking it to minimum pairwise distance (MPD). Building on the general-metric NP-hardness result of Pereverdieva et al. (2025), we prove that NP-hardness persists for point sets in the Euclidean plane when $s$ is part of the input. In contrast, finite ultrametric spaces form an exact tractable regime: on rooted binary ultrametric trees with $n$ leaves, an optimal size-$k$ subset can be computed by dynamic programming in $O(nk^2)$ time. We also discuss the ordered one-dimensional Euclidean case, where the classical MPD objective admits simple dynamic programming, but the additive Riesz energy does not appear to allow the same state compression. Finally, we explain why one natural route to fixed-$s$ Euclidean hardness does not close: Fowler-style 3SAT gadgets, together with zeta-function bounds for far-field interactions, show why this approach still requires an exponent depending on $k$. Together, these results provide a compact complexity landscape for a natural diversity or dispersion objective, distinguishing Euclidean hardness, ultrametric tractability, and the ordered one-dimensional case.

Michael T. M. Emmerich

This paper studies preference-shaped expected improvement criteria for Bayesian multiobjective optimization. We consider two indicator families which are often used for similar algorithmic purposes, but which are geometrically different. The hypervolume indicator is based on a dystopian reference point and measures dominated volume in objective space. The R2 indicator is based on a utopian point and evaluates approximation sets through weighted Tchebycheff scalarization envelopes. The purpose of the paper is to make precise which preference transformations preserve exact computation, Pareto compatibility, and monotonicity properties, and which transformations change the underlying geometry. On the hypervolume side, we revisit canonical EHVI through the Deng representation, formulate product-density weighted EHVI in desirability coordinates, discuss cone-based EHVI as ordinary EHVI after a linear cone transformation, and separate these cases from truncated EHVI, where variance monotonicity may fail. On the R2 side, we prove that exact integral R2 improvement is not, in general, an ordinary objective-space weighted hypervolume. The obstruction is lower-dimensional: Lebesgue-density hypervolume cannot see certain boundary contributions that Tchebycheff scalarizations still detect. We then show that exact integral R2 improvement is exactly a scalarization-space volume, namely the measure of the Tchebycheff shadow between the incumbent scalarization envelope and the reference envelope. This representation yields finite-sum ER2I algorithms for discrete R2, quadrature methods for exact integral R2, and an achievement-space Gaussian surrogate formulation in which ER2I is an integral of scalar Gaussian expected improvements.

Michael T. M. Emmerich, Ksenia Pereverdieva, André H. Deutz

The Solow--Polasky diversity indicator (or magnitude) is a classical measure of diversity based on pairwise distances. It has applications in ecology, conservation planning, and, more recently, in algorithmic subset selection and diversity optimization. In this note, we investigate the computational complexity of selecting a subset of fixed cardinality from a finite set so as to maximize the Solow--Polasky diversity value. We prove that this problem is NP-hard in general metric spaces. The reduction is from the classical Independent Set problem and uses a simple metric construction containing only two non-zero distance values. Importantly, the hardness result holds for every fixed kernel parameter $θ_0>0$; equivalently, by rescaling the metric, one may fix the parameter to $1$ without loss of generality. A central point is that this is not a boilerplate reduction: because the Solow--Polasky objective is defined through matrix inversion, it is a nontrivial nonlinear function of the distances. Accordingly, the proof requires a dedicated strict-monotonicity argument for the specific family of distance matrices arising in the reduction; this strict monotonicity is established here for that family, but it is not assumed to hold in full generality. We also explain how the proof connects to continuity and monotonicity considerations for diversity indicators.

Michael T. M. Emmerich, Mahboubeh Nezhadmoghaddam, Jesús Guillermo Falcón Cardona

We study fixed-cardinality maximization of the inverse-matrix Solow--Polasky diversity, equivalently finite metric magnitude for the exponential kernel, on one-dimensional and ordered metric sets. The analysis starts from the known finite-line gap formula for the exponential kernel, which writes the excess inverse-matrix diversity as a sum of functions of consecutive gaps. Building on this formula, the main interval theorem proves that, for every $k\geq 2$, the unique maximizing $k$-point subset of $[0,1]$ is the equally spaced set. Thus the objective selects a uniform gap representation on the real line. A converse kernel proposition shows that, among normalized non-increasing distance kernels, requiring the corresponding adjacent-gap additive structure forces the exponential family. Further results transfer the interval theorem to ordered $\ell_1$ (L1, or Manhattan) curves by isometry: the maximizing sets are uniform in accumulated $\ell_1$ length. As a consequence, monotone biobjective Pareto fronts admit Solow--Polasky optimal finite approximations that are uniformly spaced in accumulated objective-space change, a natural representation when all parts of a continuous front should be covered. Examples, including a dense connected front and a finite disconnected ZDT3 front, illustrate how the continuous uniform-gap result appears on discrete candidate sets. Solow-Polasky diversity; diversity measures; finite metric magnitude; L1 distance; uniform spacing; Pareto-front approximation; multiobjective optimization; fixed-cardinality subset selection

Michael T. M. Emmerich

A nonsmooth set-gradient ascent method is developed for moving finite approximation sets toward the Pareto front in multiobjective optimization. The method optimizes layered set indicators: a base indicator is evaluated on successive nondomination layers, and the layer values are combined with rapidly decreasing weights. This gives ascent directions to nondominated and dominated points while preventing deeper layers from compensating for deterioration of the first front. Two base indicators are treated: the hypervolume indicator and the magnitude indicator of the dominated set, whose expansion over coordinate projections contains extent, projected-area, and volume terms. The scalar objectives are nonsmooth because nondomination layers change combinatorially and the active orthogonal-union geometry changes piecewise. On fixed strata, where layer assignments and active geometry remain unchanged, the indicators are piecewise smooth and chamberwise continuous. For the magnitude indicator, an exact gradient formula is derived as a linear combination of hypervolume gradients of projected shadow sets. Thus, for fixed objective dimension, magnitude gradients have the same asymptotic time complexity as hypervolume gradients. Lexicographic layer aggregation is related to a unary infinitesimal encoding. For finite-$ε$ surrogates, the main nonsmoothness mechanisms are isolated and chamberwise Lipschitz continuity on bounded sets is proved; a two-point counterexample shows that hard-layer scalarization is not globally continuous across layer switches. The theory motivates a projected finite-difference implementation with repulsion and recovery from stagnation. Numerical examples and reproducible code cover two- and three-objective settings, including objective-space tests, curved fronts, a supersphere benchmark, and traces comparing layered magnitude and hypervolume ascent.

Michael T. M. Emmerich

We investigate \emph{magnitude} as a new unary and strictly Pareto-compliant quality indicator for finite approximation sets to the Pareto front in multiobjective optimization. Magnitude originates in enriched category theory and metric geometry, where it is a notion of size or point content for compact metric spaces and a generalization of cardinality. For dominated regions in the \(\ell_1\) box setting, magnitude is close to hypervolume but not identical: it contains the top-dimensional hypervolume term together with positive lower-dimensional projection and boundary contributions. This paper gives a first theoretical study of magnitude as an indicator. We consider multiobjective maximization with a common anchor point. For dominated sets generated by finite approximation sets, we derive an all-dimensional projection formula, prove weak and strict set monotonicity on finite unions of anchored boxes, and thereby obtain weak and strict Pareto compliance. Unlike hypervolume, magnitude assigns positive value to boundary points sharing one or more coordinates with the anchor point, even when their top-dimensional hypervolume contribution vanishes. We then formulate projected set-gradient methods and compare hypervolume and magnitude on biobjective and three-dimensional simplex examples. Numerically, magnitude favors boundary-including populations and, for suitable cardinalities, complete Das--Dennis grids, whereas hypervolume prefers more interior-filling configurations. Computationally, magnitude reduces to hypervolume on coordinate projections; for fixed dimension this yields the same asymptotic complexity up to a factor \(2^d-1\), and in dimensions two and three \(Θ(n\log n)\) time. These results identify magnitude as a mathematically natural and computationally viable alternative to hypervolume for finite Pareto front approximations.

Michael T. M. Emmerich

We study exact fixed-cardinality Solow--Polasky diversity subset selection on ordered finite $\ell_1$ sets, with monotone biobjective Pareto fronts and their higher-dimensional staircase analogues as central applications. Solow--Polasky diversity was introduced in biodiversity conservation, whereas the same inverse-matrix expression appears in metric geometry as magnitude: for a finite metric space $(X,d)$ with exponential similarity matrix $Z_{ij}=e^{-q d(x_i,x_j)}$, the quantity $\1^\top Z^{-1}\1$ is the magnitude of the scaled finite metric space $(X,qd)$ whenever the weighting is defined by the inverse matrix. Thus, in this finite exponential-kernel setting, Solow--Polasky diversity and magnitude are mathematically the same object viewed through different motivations. Building on the linear-chain magnitude formula of Leinster and Willerton, we provide a detailed proof of the scaled consecutive-gap identity $ \SP(X)=1+\sum_r \tanh(qg_r/2), $ where the $g_r$ are the gaps between consecutive selected points. We then prove an exact Bellman-recursion theorem for maximizing this value over all subsets of a prescribed cardinality, yielding an $O(kn^2)$ dynamic program for an ordered $n$-point candidate set and subset size $k$. Finally, we prove ordered $\ell_1$ reductions showing that the same algorithm applies to monotone biobjective Pareto-front approximations and, more generally, to finite coordinatewise monotone $\ell_1$ staircases in $\R^d$. These are precisely the ordered $\ell_1$ chains for which the Manhattan metric becomes a line metric along the chosen order, so the one-dimensional dynamic program applies without modification. Keywords: Dynamic Programming, Solow--Polasky Diversity, Complexity Theory, Multiobjective Optimization, Pareto front, Magnitude

Michael T. M. Emmerich, Ksenia Pereverdieva, André H. Deutz

We prove that, for every fixed $θ_0>0$, selecting a subset of prescribed cardinality that maximizes the Solow--Polasky diversity indicator is NP-hard for finite point sets in $\mathbb{R}^2$ with the Euclidean metric, and therefore also for finite point sets in $\mathbb{R}^d$ for every fixed dimension $d \ge 2$. This strictly strengthens our earlier NP-hardness result for general metric spaces by showing that hardness persists under the severe geometric restriction to the Euclidean plane. At the same time, the Euclidean proof technique is different from the conceptually easier earlier argument for arbitrary metric spaces, and that general metric-space construction does not directly translate to the Euclidean setting. In the earlier proof one can use an exact construction tailored to arbitrary metrics, essentially exploiting a two-distance structure. In contrast, such an exact realization is unavailable in fixed-dimensional Euclidean space, so the present reduction requires a genuinely geometric argument. Our Euclidean proof is based on two distance thresholds, which allow us to separate yes-instances from no-instances by robust inequalities rather than by the exact construction used in the general metric setting. The main technical ingredient is a bounded-box comparison lemma for the nonlinear objective $\mathbf{1}^{\top}Z^{-1}\mathbf{1}$, where $Z_{ij}=e^{-θ_0 d(x_i,x_j)}$. This lemma controls the effect of perturbations in the pairwise distances well enough to transfer the gap created by the reduction. The reduction is from \emph{Geometric Unit-Disk Independent Set}. We present the main argument in geometric form for finite subsets of $\mathbb{R}^2$, with an appendix supplying the bit-complexity details needed for polynomial-time reducibility.

Michael T. M. Emmerich

We study the optimal placement of $k$ ordered points on the unit interval for the bounded pair potential \[ K_q(d)=e^{-d^q}, \qquad q>0. \] The family interpolates between strongly cusp-like kernels for $0<q<1$, the threshold kernel $e^{-d}$, and the flatter Gaussian-type regime $q>1$. Our emphasis is on the transition from collision-free minimizers to endpoint-collapsed minimizers. We reformulate the problem in gap variables, record convexity, symmetry, and the Karush-Kuhn-Tucker conditions, and give a short proof that collisions are impossible for $0<q<1$. At the threshold $q=1$ we recover the endpoint-clustering law for $e^{-d}$, while for $q>1$ we identify critical exponents $q_k$ beyond which interior points are no longer optimal. For odd $k$ we derive the exact universal value \[ q_{2m+1} = \frac{\log(1/(-\log((1+e^{-1})/2)))}{\log 2} \approx 1.396363475, \] and for even $k=4,6,\dots,20$ we compute the numerical transition values \[ \begin{aligned} &q_4\approx 1.062682507,\quad q_6\approx 1.155601329,\quad q_8\approx 1.206132611,\quad q_{10}\approx 1.238523533,\\ &q_{12}\approx 1.261308114,\quad q_{14}\approx 1.278305167,\quad q_{16}\approx 1.291510874,\quad q_{18}\approx 1.302082885,\\ &q_{20}\approx 1.310744185. \end{aligned} \] We also include comparison tables and diagrams for the kernels $e^{-\sqrt d}$, $e^{-d}$, and $e^{-d^2}$, briefly relate the bounded family to the singular Riesz kernel $d^{-s}$, and identify the $q\to 0^+$ limit with the Fekete/Chebyshev--Lobatto configuration on $[0,1]$.

Yali Wang, Bas van Stein, Michael T. M. Emmerich et al.

A customized multi-objective evolutionary algorithm (MOEA) is proposed for the multi-objective flexible job shop scheduling problem (FJSP). It uses smart initialization approaches to enrich the first generated population, and proposes various crossover operators to create a better diversity of offspring. Especially, the MIP-EGO configurator, which can tune algorithm parameters, is adopted to automatically tune operator probabilities. Furthermore, different local search strategies are employed to explore the neighborhood for better solutions. In general, the algorithm enhancement strategy can be integrated with any standard EMO algorithm. In this paper, it has been combined with NSGA-III to solve benchmark multi-objective FJSPs, whereas an off-the-shelf implementation of NSGA-III is not capable of solving the FJSP. The experimental results show excellent performance with less computing budget.

Jiaqi Zhao, Vitor Basto Fernandes, Licheng Jiao et al.

Finding a good classifier is a multiobjective optimization problem with different error rates and the costs to be minimized. The receiver operating characteristic is widely used in the machine learning community to analyze the performance of parametric classifiers or sets of Pareto optimal classifiers. In order to directly compare two sets of classifiers the area (or volume) under the convex hull can be used as a scalar indicator for the performance of a set of classifiers in receiver operating characteristic space. Recently, the convex hull based multiobjective genetic programming algorithm was proposed and successfully applied to maximize the convex hull area for binary classification problems. The contribution of this paper is to extend this algorithm for dealing with higher dimensional problem formulations. In particular, we discuss problems where parsimony (or classifier complexity) is stated as a third objective and multi-class classification with three different true classification rates to be maximized. The design of the algorithm proposed in this paper is inspired by indicator-based evolutionary algorithms, where first a performance indicator for a solution set is established and then a selection operator is designed that complies with the performance indicator. In this case, the performance indicator will be the volume under the convex hull. The algorithm is tested and analyzed in a proof of concept study on different benchmarks that are designed for measuring its capability to capture relevant parts of a convex hull. Further benchmark and application studies on email classification and feature selection round up the analysis and assess robustness and usefulness of the new algorithm in real world settings.