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Gennadiy Averkov

Publications and source records attributed to Gennadiy Averkov.

At least 19 recordsLinked to original sources
arXiv2026

Shallower ReLU Network Representations via Exact Linear Algebra

We study the depth required by ReLU networks to exactly represent piecewise linear functions, focusing specifically on the maximum function. This problem has recently received significant attention in both the ML and TCS literature. We prove that $\max_n(x)=\max\{x_1,\ldots,x_n\}$ is exactly representable with two hidden layers for every $n\leq 12$. Previously, this was only known up to $n\leq5$ [Bakaev, Brunck, Hertrich, Stade, Yehudayoff, STOC'26]. We obtain our constructions through an exact computer-assisted search within a space of candidate solutions: After a symmetry reduction, we obtain a finite system of linear equations over $\mathbb{Q}$ such that any solution yields a valid representation of the maximum function. The resulting constructions have a structured first hidden layer, which enables recursive substitution into deeper networks. This yields an exact ReLU representation of $\max_n$ with at most $\lceil \log_6(n/2) \rceil+1$ hidden layers. Consequently, every continuous piecewise-linear function on $\mathbb{R}^d$ admits an exact representation with at most $\lceil\log_6((d+1)/2)\rceil+1$ hidden layers; in particular, two hidden layers suffice for $d\leq 11$. Again, these results improve upon [Bakaev, Brunck, Hertrich, Stade, Yehudayoff, STOC'26], who proved analogous logarithmic bounds with base three.

cs.LG↗
arXiv2026

On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube

We prove a log-submodularity-type inequality for zonoids in $\mathbb{R}^4$, extending the three-dimensional result of Fradelizi, Madiman, Meyer, and Zvavitch. More generally, we conjecture a log-submodularity-type inequality for zonoids in arbitrary dimension. This inequality admits several equivalent formulations in terms of volumes of coordinate projections as well as in terms of mixed volumes, thereby unifying several geometric perspectives. We reduce the conjectured inequality to a polynomial inequality whose variables are associated with the vertices of a hypercube and whose coefficients encode the volumes of 0/1 simplices. This reduction reveals unexpected connections between mixed volumes of zonoids, matroid theory, and real algebraic geometry.

math.MG↗
arXiv2026

Mixed volumes of zonoids and the absolute value of the Grassmannian

Zonoids are Hausdorff limits of zonotopes, while zonotopes are convex polytopes defined as the Minkowski sums of finitely many segments. We present a combinatorial framework that links the study of mixed volumes of zonoids (a topic that has applications in algebraic combinatorics) with the study of the absolute value of the Grassmannian, defined as the image of the Grassmannian under the coordinate-wise absolute value map. We use polyhedral computations to derive new families of inequalities for n zonoids in dimension d, when (n,d)=(6,2) and (6,3). Unlike the classical geometric inequalities, originating from the Brunn-Minkowski and Aleksandrov-Fenchel inequalities, the inequalities we produce have the special feature of being Minkowski linear in each of the n zonoids they involve.

math.CO↗
arXiv2026

The relaxation complexity of the standard simplex is logarithmic

For a set $X$ of integer points, the relaxation complexity $\operatorname{rc}(X)$ is the smallest number of facets of any polyhedron $P$ such that $P \cap \mathbb{Z}^d = X$. In this paper, we focus on the case where $X$ is the discrete standard simplex $Δ_d = \{\mathbf{0}, \mathbf{e}_1, \dots, \mathbf{e}_d\}$. We show that $\operatorname{rc}(Δ_d) = O(\log d)$ by an explicit, elementary construction. This improves upon the previously best-known upper bound $\operatorname{rc}(Δ_d) = O(d / \sqrt{\log d})$ due to Aprile, Averkov, Di Summa, and Hojny (2024) and matches an asymptotic lower bound by Averkov and Schymura (2022).

cs.DM↗
arXiv2026

Exact Flatness Constant for One-Point Convex Bodies and the Discrete Isominwidth Problem: The Planar Case

A variant of the flatness problem from integer programming is studied, in which one considers convex bodies in $\mathbb{R}^d$ with at most $k$ interior lattice points. The maximum lattice width of such a body is denoted by Flt(d,k) and it is related to the classical flatness constant as well as a conjectural dual version of Minkowski's convex body theorem due to Makai. Moreover, it is shown that Flt(2, 1) = 3, i.e., any planar convex body with at most one interior point has lattice width at most three. This leads to an isominwidth inequality for the lattice point enumerator of planar convex bodies.

math.MG↗
arXiv2026

Optimal Proximity Bound and Product Function Estimates in Integer Linear Programming

We obtain an optimal proximity bound for integer linear programs in standard form max{cx: Ax=b, x nonnegative integer}, where A is an integer mxn matrix of rank m<n and b is an integer vector. Specifically, we show that the Euclidean distance from any optimal vertex solution of the LP relaxation to a nearest optimal integer solution is bounded by $\sqrt{\det(AA^t)}-1$ and that this estimate is asymptotically tight. We also derive bounds for the optimal integer solutions involving the product function $\prod_{i=1}^{n}(x_i+1)$ and discuss their applications in the knapsack setting.

math.OC↗
arXiv2026

Nonnegativity of signomials with Newton simplex over $\mathcal{A}$-convex sets

We study a class of signomials whose positive support is the set of vertices of a simplex and which may have several negative support points in the simplex. Various groups of authors have provided an exact characterization for the global nonnegativity of a signomial in this class in terms of circuit signomials and that characterization provides a tractable nonnegativity test. We generalize this characterization to the constrained nonnegativity over a set $X$ under an additional convexity precondition in the exponential moment space. This provides a tractable nonnegativity test over $X$ for the class in terms of a power cone program. Our proof methods rely on a variant of the convex cone of constrained SAGE signomials (sums of arithmetic-geometric exponentials) and the duality theory.

math.CO↗
arXiv2025

On the Expressiveness of Rational ReLU Neural Networks With Bounded Depth

To confirm that the expressive power of ReLU neural networks grows with their depth, the function $F_n = \max \{0,x_1,\ldots,x_n\}$ has been considered in the literature. A conjecture by Hertrich, Basu, Di Summa, and Skutella [NeurIPS 2021] states that any ReLU network that exactly represents $F_n$ has at least $\lceil\log_2 (n+1)\rceil$ hidden layers. The conjecture has recently been confirmed for networks with integer weights by Haase, Hertrich, and Loho [ICLR 2023]. We follow up on this line of research and show that, within ReLU networks whose weights are decimal fractions, $F_n$ can only be represented by networks with at least $\lceil\log_3 (n+1)\rceil$ hidden layers. Moreover, if all weights are $N$-ary fractions, then $F_n$ can only be represented by networks with at least $Ω( \frac{\ln n}{\ln \ln N})$ layers. These results are a partial confirmation of the above conjecture for rational ReLU networks, and provide the first non-constant lower bound on the depth of practically relevant ReLU networks.

cs.LG↗
arXiv2024

Sparse convex relaxations in polynomial optimization

We present a novel, general, and unifying point of view on sparse approaches to polynomial optimization. Solving polynomial optimization problems to global optimality is a ubiquitous challenge in many areas of science and engineering. Historically, different approaches on how to solve nonconvex polynomial optimization problems based on convex relaxations have been developed in different scientific communities. Here, we introduce the concept of monomial patterns. A pattern determines what monomials are to be linked by convex constraints in a convex relaxation of a polynomial optimization problem. This concept helps to understand existing approaches from different schools of thought, to develop novel relaxation schemes, and to derive a flexible duality theory, which can be specialized to many concrete situations that have been considered in the literature. We unify different approaches to polynomial optimization including polyhedral approximations, dense semidefinite relaxations, SONC, SAGE, and TSSOS in a self-contained exposition. We also carry out computational experiments to demonstrate the practical advantages of a flexible usage of pattern-based sparse relaxations of polynomial optimization problems.

math.OC↗
arXiv2023

Convex hulls of monomial curves, and a sparse positivstellensatz

Consider the closed convex hull $K$ of a monomial curve given parametrically as $(t^{m_1},\ldots,t^{m_n})$, with the parameter $t$ varying in an interval $I$. We show, using constructive arguments, that $K$ admits a lifted semidefinite description by $\mathcal{O}(d)$ linear matrix inequalities (LMIs), each of size $\left\lfloor \frac{n}{2} \right\rfloor +1$, where $d= \max \{m_1,\ldots,m_n\}$ is the degree of the curve. On the dual side, we show that if a univariate polynomial $p(t)$ of degree $d$ with at most $2k+1$ monomials is non-negative on $\mathbb{R}_+$, then $p$ admits a representation $p = t^0 σ_0 + \cdots + t^{d-k} σ_{d-k}$, where the polynomials $σ_0,\ldots,σ_{d-k}$ are sums of squares and $\operatorname{deg} (σ_i) \le 2k$. The latter is a univariate positivstellensatz for sparse polynomials, with non-negativity of $p$ being certified by sos polynomials whose degree only depends on the sparsity of $p$. Our results fit into the general attempt of formulating polynomial optimization problems as semidefinite problems with LMIs of small size. Such small-size descriptions are much more tractable from a computational viewpoint.

math.OC↗
arXiv2022

Plücker-type inequalities for mixed areas and intersection numbers of curve arrangements

Any collection of $n$ compact convex planar sets $K_1,\dots, K_n$ defines a vector of ${n\choose 2}$ mixed areas $V(K_i,K_j)$ for $1\leq i<j\leq n$. We show that for $n\geq 4$ these numbers satisfy certain Plücker-type inequalities. Moreover, we prove that for $n=4$ these inequalities completely describe the space of all mixed area vectors $(V(K_i,K_j) : 1\leq i<j\leq 4)$. For arbitrary $n\geq 4$ we show that this space has a semialgebraic closure of full dimension. As an application, we obtain an inequality description for the smallest positive homogeneous set containing the configuration space of intersection numbers of quadruples of tropical curves.

math.CO↗
arXiv2022

The role of rationality in integer-programming relaxations

For a finite set $X \subset \mathbb{Z}^d$ that can be represented as $X = Q \cap \mathbb{Z}^d$ for some polyhedron $Q$, we call $Q$ a relaxation of $X$ and define the relaxation complexity $rc(X)$ of $X$ as the least number of facets among all possible relaxations $Q$ of $X$. The rational relaxation complexity $rc_\mathbb{Q}(X)$ restricts the definition of $rc(X)$ to rational polyhedra $Q$. In this article, we focus on $X = Δ_d$, the vertex set of the standard simplex, which consists of the null vector and the standard unit vectors in $\mathbb{R}^d$. We show that $rc(Δ_d) \leq d$ for every $d \geq 5$. That is, since $rc_{\mathbb{Q}}(Δ_d)=d+1$, irrationality can reduce the minimal size of relaxations. This answers an open question posed by Kaibel and Weltge (Lower bounds on the size of integer programs without additional variables, Mathematical Programming, 154(1):407-425, 2015). Moreover, we prove the asymptotic statement $rc(Δ_d) \in O(\frac{d}{\sqrt{\log(d)}})$, which shows that the ratio $rc(Δ_d)/rc_{\mathbb{Q}}(Δ_d)$ goes to $0$, as $d\to \infty$.

math.OC↗
arXiv2022

Efficient MIP Techniques for Computing the Relaxation Complexity

The relaxation complexity rc(X) of the set of integer points X contained in a polyhedron is the minimal number of inequalities needed to formulate a linear optimization problem over X without using auxiliary variables. Besides its relevance in integer programming, this concept has interpretations in aspects of social choice, symmetric cryptanalysis, and machine learning. We employ efficient mixed-integer programming techniques to compute a robust and numerically more practical variant of the relaxation complexity. Our proposed models require row or column generation techniques and can be enhanced by symmetry handling and suitable propagation algorithms. Theoretically, we compare the quality of our models in terms of their LP relaxation values. The performance of those models is investigated on a broad test set and is underlined by their ability to solve challenging instances that could not be solved previously.

math.OC↗
arXiv2021

Convexification of box-constrained polynomial optimization problems via monomial patterns

Convexification is a core technique in global polynomial optimization. Currently, there are two main approaches competing in theory and practice: the approach of nonlinear programming and the approach based on positivity certificates from real algebra. The former are comparatively cheap from a computational point of view, but typically do not provide tight relaxations with respect to bounds for the original problem. The latter are typically computationally expensive, but do provide tight relaxations. We embed both kinds of approaches into a unified framework of monomial relaxations. We develop a convexification strategy that allows to trade off the quality of the bounds against computational expenses. Computational experiments show very encouraging results.

math.OC↗
arXiv2021

A remark on the approximation of non-negative polynomials by SONC polynomials

A SONC polynomial is a sum of finitely many non-negative circuit polynomials, whereas a non-negative circuit polynomial is a non-negative polynomial whose support is a simplicial circuit. We show that there exist non-negative polynomials that cannot be uniformly approximated by SONC polynomials arbitrarily well.

math.OC↗
arXiv2021

Computational Aspects of Relaxation Complexity: Possibilities and Limitations

The relaxation complexity $\mathrm{rc}(X)$ of the set of integer points $X$ contained in a polyhedron is the smallest number of facets of any polyhedron $P$ such that the integer points in $P$ coincide with $X$. It is a useful tool to investigate the existence of compact linear descriptions of $X$. In this article, we derive tight and computable upper bounds on $\mathrm{rc}_{\mathbb{Q}}(X)$, a variant of $\mathrm{rc}(X)$ in which the polyhedra $P$ are required to be rational, and we show that $\mathrm{rc}(X)$ can be computed in polynomial time if $X$ is 2-dimensional. Further, we investigate computable lower bounds on $\mathrm{rc}(X)$ with the particular focus on the existence of a finite set $Y \subseteq \mathbb{Z}^d$ such that separating $X$ and $Y \setminus X$ allows us to deduce $\mathrm{rc}(X) \geq k$. In particular, we show for some choices of $X$ that no such finite set $Y$ exists to certify the value of $\mathrm{rc}(X)$, providing a negative answer to a question by Weltge (2015). We also obtain an explicit formula for $\mathrm{rc}(X)$ for specific classes of sets $X$ and present the first practically applicable approach to compute $\mathrm{rc}(X)$ for sets $X$ that admit a finite certificate.

math.OC↗