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Dávid Papp

Publications and source records attributed to Dávid Papp.

At least 19 recordsLinked to original sources
arXiv2026

Dual certificates of primal cone membership

We discuss optimization problems over convex cones in which membership is difficult to verify directly. In the standard theory of duality, vectors in the dual cone $K^*$ are associated with separating hyperplanes and interpreted as certificates of non-membership in the primal cone $K$. Complementing this perspective, we develop easily verifiable certificates of membership in $K$ given by vectors in $K^*$. Assuming that $K^*$ admits an efficiently computable logarithmically homogeneous self-concordant barrier, every vector in the interior of $K$ is associated with a full-dimensional cone of efficiently verifiable membership certificates. Consequently, rigorous certificates can be computed using numerical methods, including interior-point algorithms. The proposed framework is particularly well-suited to optimization over low-dimensional linear images of higher dimensional cones: we argue that these problems can be solved by optimizing directly over the (low-dimensional) dual cone, circumventing the customary lifting that introduces a large number of auxiliary variables. As an application, we derive a novel closed-form formula for computing exact primal feasible solutions from suitable dual feasible solutions; as the dual solutions approach optimality, the computed primal solutions do so as well. To illustrate the generality of our approach, we show that the new certification scheme is applicable to virtually every tractable subcone of nonnegative polynomials commonly used in polynomial optimization (such as SOS, SONC, SAGE and SDSOS polynomials, among others), facilitating the computation of rigorous nonnegativity certificates using numerical algorithms.

math.OC↗
arXiv2025

Differentiating a Linear Recursive Sequence

Consider a sequence of real-valued functions of a real variable given by a homogeneous linear recursion with differentiable coefficients. We show that if the functions in the sequence are differentiable, then the sequence of derivatives also satisfies a homogeneous linear recursion whose order is at most double the order of original recursion. Similarly to the well-known operations that determine the elementwise sum and product of two linear recursive sequences, the coefficient functions of our recursion for the derivatives are easily computable from the original coefficient functions and their derivatives by direct manipulation of the coefficients of the characteristic polynomial of the recursion, without determining the roots. A simple application, computing linear recursions for derivatives of orthogonal polynomials, is presented.

math.FA↗
arXiv2025

Interior-point algorithms with full Newton steps for nonsymmetric convex conic optimization

We design and analyze primal-dual, feasible interior-point algorithms (IPAs) employing full Newton steps to solve convex optimization problems in standard conic form. Unlike most nonsymmetric cone programming methods, the algorithms presented in this paper require only a logarithmically homogeneous self-concordant barrier (LHSCB) of the primal cone, but compute feasible and $\varepsilon$-optimal solutions to both the primal and dual problems in $O(\sqrtν\log(1/\varepsilon))$ iterations, where $ν$ is the barrier parameter of the LHSCB; this matches the best known theoretical iteration complexity of IPAs for both symmetric and nonsymmetric cone programming. The definition of the neighborhood of the central path and feasible starts ensure that the computed solutions are compatible with the dual certificates framework of (Davis and Papp, 2022). Several initialization strategies are discussed, including two-phase methods that can be applied if a strictly feasible primal solution is available, and one based on a homogeneous self-dual embedding that allows the rigorous detection of large feasible or optimal solutions. In a detailed study of a classic, notoriously difficult, polynomial optimization problem, we demonstrate that the methods are efficient and numerically reliable. Although the standard approach using semidefinite programming fails for this problem with the solvers we tried, the new IPAs compute highly accurate near-optimal solutions that can be certified to be near-optimal in exact arithmetic.

math.OC↗
arXiv2024

The smallest mono-unstable, homogeneous convex polyhedron has at least 7 vertices

We prove that every homogeneous convex polyhedron with only one unstable equilibrium (known as a mono-unstable convex polyhedron) has at least $7$ vertices. Although it has been long known that no mono-unstable tetrahedra exist, and mono-unstable polyhedra with as few as $18$ vertices and faces have been constructed, this is the first nontrivial lower bound on the number of vertices for a mono-unstable polyhedron. There are two main ingredients in the proof. We first establish two types of relationships, both expressible as (non-convex) quadratic inequalities, that the coordinates of the vertices of a mono-unstable convex polyhedron must satisfy, taking into account the combinatorial structure of the polyhedron. Then we use numerical semidefinite optimization algorithms to compute easily and independently verifiable, rigorous certificates that the resulting systems of quadratic inequalities (5943 in total) are indeed inconsistent in each case.

math.MG↗
arXiv2024

A global evidence map of human well-being and biodiversity co-benefits and trade-offs of natural climate solutions

Natural climate solutions (NCS) are critical for mitigating climate change through ecosystem-based carbon removal and emissions reductions. NCS implementation can also generate biodiversity and human well-being co-benefits and trade-offs ("NCS co-impacts"), but the volume of evidence on NCS co-impacts has grown rapidly across disciplines, is poorly understood, and remains to be systematically collated and synthesized. A global evidence map of NCS co-impacts would overcome key barriers to NCS implementation by providing relevant information on co-benefits and trade-offs where carbon mitigation potential alone does not justify NCS projects. We employ large language models to assess over two million articles, finding 257,266 relevant articles on NCS co-impacts. We analyze this large and dispersed body of literature using innovative machine learning methods to extract relevant data (e.g., study location, species, and other key variables), and create a global evidence map on NCS co-impacts. Evidence on NCS co-impacts has grown approximately ten-fold in three decades, although some of the most abundant evidence is associated with pathways that have less mitigation potential. We find that studies often examine multiple NCS pathways, indicating natural NCS pathway complements, and each NCS is often associated with two or more coimpacts. Finally, NCS co-impacts evidence and priority areas for NCS are often mismatched--some countries with high mitigation potential from NCS have few published studies on the broader co-impacts of NCS implementation. Our work advances and makes available novel methods and systematic and representative data of NCS co-impacts studies, thus providing timely insights to inform NCS research and action globally.

physics.soc-ph↗
arXiv2023

Rational Dual Certificates for Weighted Sums-of-Squares Polynomials with Boundable Bit Size

In (Davis and Papp, 2022), the authors introduced the concept of dual certificates of (weighted) sum-of-squares polynomials, which are vectors from the dual cone of weighted sums of squares (WSOS) polynomials that can be interpreted as nonnegativity certificates. This initial theoretical work showed that for every polynomial in the interior of a WSOS cone, there exists a rational dual certificate proving that the polynomial is WSOS. In this article, we analyze the complexity of rational dual certificates of WSOS polynomials by bounding the bit sizes of integer dual certificates as a function of parameters such as the degree and the number of variables of the polynomials, or their distance from the boundary of the cone. After providing a general bound, we explore several special cases, such as univariate polynomials nonnegative over the real line or a bounded interval, represented in different commonly used bases. We also provide an algorithm which runs in rational arithmetic and computes a rational certificate with boundable bit size for a WSOS lower bound of the input polynomial.

math.AG↗
arXiv2023

The smallest mono-unstable convex polyhedron with point masses has 8 faces and 11 vertices

In the study of monostatic polyhedra, initiated by John H. Conway in 1966, the main question is to construct such an object with the minimal number of faces and vertices. By distinguishing between various material distributions and stability types, this expands into a small family of related questions. While many upper and lower bounds on the necessary numbers of faces and vertices have been established, none of these questions has been so far resolved. Adapting an algorithm presented in (Bozóki et al., 2022), here we offer the first complete answer to a question from this family: by using the toolbox of semidefinite optimization to efficiently generate the hundreds of thousands of infeasibility certificates, we provide the first-ever proof for the existence of a monostatic polyhedron with point masses, having minimal number (V=11) of vertices (Theorem 3) and a minimal number (F=8) of faces. We also show that V=11 is the smallest number of vertices that a mono-unstable polyhedron can have in all dimensions greater than 1. (Corollary 6.)

math.MG↗
arXiv2022

Dual certificates and efficient rational sum-of-squares decompositions for polynomial optimization over compact sets

We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual cone certificates, which allows us to interpret vectors from the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a WSOS polynomial. Whereas conventional WSOS certificates are alternative representations of the polynomials they certify, dual certificates are distinct from the certified polynomials; moreover, each dual certificate certifies a full-dimensional convex cone of WSOS polynomials. As a result, rational WSOS certificates can be constructed from numerically computed dual certificates at little additional cost, without any rounding or projection steps applied to the numerical certificates. As an additional algorithmic application, we present an almost entirely numerical hybrid algorithm for computing the optimal WSOS lower bound of a given polynomial along with a rational dual certificate, with a polynomial-time computational cost per iteration and linear rate of convergence.

math.OC↗
arXiv2021

alfonso: Matlab package for nonsymmetric conic optimization

We present alfonso, an open-source Matlab package for solving conic optimization problems over nonsymmetric convex cones. The implementation is based on the authors' corrected analysis of a primal-dual interior-point method of Skajaa and Ye. This method enables optimization over any convex cone as long as a logarithmically homogeneous self-concordant barrier is available for the cone or its dual. This includes many nonsymmetric cones, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Besides enabling the solution of problems which cannot be cast as optimization problems over a symmetric cone, it also offers performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. The worst-case iteration complexity of alfonso is the best known for non-symmetric cone optimization: $O(\sqrtν\log(1/ε))$ iterations to reach an $ε$-optimal solution, where $ν$ is the barrier parameter of the barrier function used in the optimization. alfonso can be interfaced with a Matlab function (supplied by the user) that computes the Hessian of a barrier function for the cone. For convenience, a simplified interface is also available to optimize over the direct product of cones for which a barrier function has already been built into the software. This interface can be easily extended to include new cones. Both interfaces are illustrated by solving linear programs. The oracle interface and the efficiency of alfonso are also demonstrated using a design of experiments problem in which the tailored barrier computation greatly decreases the solution time compared to using state-of-the-art conic optimization software.

math.OC↗
arXiv2019

Duality of sum of nonnegative circuit polynomials and optimal SONC bounds

Circuit polynomials are polynomials satisfying a number of conditions that make it easy to compute sharp and certifiable global lower bounds for them. Consequently, one may use them to find certifiable lower bounds for any polynomial by writing it as a sum of circuit polynomials with known lower bounds (if possible), in a fashion similar to the better-known sum-of-squares polynomials. Seidler and de Wolff recently showed that sums of nonnegative circuit (SONC) polynomials can be used to compute global lower bounds (called SONC bounds) for polynomials in this manner in polynomial time, as long as the polynomial is bounded from below and its support satisfies a nondegeneracy assumption. The quality of the SONC bound depends on the circuits used in the computation, but finding the set of circuits that yield the best attainable SONC bound among the astronomical number of candidate circuits is a non-trivial task that has not been addressed so far. In this paper we propose an efficient method to compute the optimal SONC lower bound by iteratively identifying the optimal circuits to use in the SONC bounding process. The method is based on a new proof of a recent result by Wang which states that (under the same nondegeneracy assumption) every SONC polynomial decomposes into SONC polynomials on the same support. Our proof, based on convex programming duality, removes the nondegeneracy assumption and motivates an algorithm that generates an optimal set of circuits and computes the corresponding SONC bound in a manner that is particularly attractive for sparse polynomials. The method is implemented and tested on a large set of sparse polynomial optimization problems with up to 40 unknowns, of degree up to 60, and up to 3000 monomials in the support. The results indicate that the method is efficient in practice. [abstract truncated for arXiv character limits]

math.OC↗
arXiv2019

On the notion of number in humans and machines

In this paper, we performed two types of software experiments to study the numerosity classification (subitizing) in humans and machines. Experiments focus on a particular kind of task is referred to as Semantic MNIST or simply SMNIST where the numerosity of objects placed in an image must be determined. The experiments called SMNIST for Humans are intended to measure the capacity of the Object File System in humans. In this type of experiment the measurement result is in well agreement with the value known from the cognitive psychology literature. The experiments called SMNIST for Machines serve similar purposes but they investigate existing, well known (but originally developed for other purpose) and under development deep learning computer programs. These measurement results can be interpreted similar to the results from SMNIST for Humans. The main thesis of this paper can be formulated as follows: in machines the image classification artificial neural networks can learn to distinguish numerosities with better accuracy when these numerosities are smaller than the capacity of OFS in humans. Finally, we outline a conceptual framework to investigate the notion of number in humans and machines.

cs.CV↗
arXiv2018

Sum-of-squares optimization without semidefinite programming

We propose a homogeneous primal-dual interior-point method to solve sum-of-squares optimization problems by combining non-symmetric conic optimization techniques and polynomial interpolation. The approach optimizes directly over the sum-of-squares cone and its dual, circumventing the semidefinite programming (SDP) reformulation which requires a large number of auxiliary variables. As a result, it has substantially lower theoretical time and space complexity than the conventional SDP-based approach. Although our approach avoids the semidefinite programming reformulation, an optimal solution to the semidefinite program can be recovered with little additional effort. Computational results confirm that for problems involving high-degree polynomials, the proposed method is several orders of magnitude faster than semidefinite programming.

math.OC↗
arXiv2018

Benchmarking Cognitive Abilities of the Brain with Computer Games

Most of the players have experienced the feeling of temporarily losing their character in a given gameplay situation when they cannot control the character, simply because they temporarily cannot see it. The main reasons for this feeling may be due to the interplay of the following factors: (1) the visual complexity of the game is unexpectedly increased compared with the previous time period as more and more game objects and effects are rendered on the display; (2) and/or the game is lagging; (3) and finally, it is also possible that the players have no sufficient experience with controlling the character. This paper focuses on the first reason. We have developed a benchmark program which allows its user to experience the feeling of losing character. While the user can control the character well the benchmark program will increase the visual complexity of the display. Otherwise, if the user lost the character then the program will decrease the complexity until the user will find the character again, and so on. The complexity is measured based on the number of changed pixels between two consecutive display images. Our measurements show that the average of bit per second values of losing and finding pairs describes the user well. The final goal of this research is to further develop our benchmark to a standard psychological test.

cs.HC↗
arXiv2018

On "A Homogeneous Interior-Point Algorithm for Non-Symmetric Convex Conic Optimization"

In a recent paper, Skajaa and Ye proposed a homogeneous primal-dual interior-point method for non-symmetric conic optimization. The authors showed that their algorithm converges to $\varepsilon$-accuracy in $O(\sqrtν\log \varepsilon^{-1})$ iterations, where $ν$ is the complexity parameter associated with a barrier function for the primal cone, and thus achieves the best-known iteration complexity for this class of problems. However, an earlier result from the literature was used incorrectly in the proofs of two intermediate lemmas in that paper. In this note, we propose new proofs of these results, allowing the same complexity bound to be established.

math.OC↗
arXiv2017

Optimization of spatiotemporally fractionated radiotherapy treatments with bounds on the achievable benefit

Spatiotemporal fractionation schemes, that is, treatments delivering different dose distributions in different fractions, may lower treatment side effects without compromising tumor control. This is achieved by hypofractionating parts of the tumor while delivering approximately uniformly fractionated doses to the healthy tissue. Optimization of such treatments is based on biologically effective dose (BED), which leads to computationally challenging nonconvex optimization problems. Current optimization methods yield only locally optimal plans, and it has been unclear whether these are close to the global optimum. We present an optimization model to compute rigorous bounds on the normal tissue BED reduction achievable by such plans. The approach is demonstrated on liver tumors, where the primary goal is to reduce mean liver BED without compromising other treatment objectives. First a uniformly fractionated reference plan is computed using convex optimization. Then a nonconvex quadratically constrained quadratic programming model is solved to local optimality to compute a spatiotemporally fractionated plan that minimizes mean liver BED subject to the constraints that the plan is no worse than the reference plan with respect to all other planning goals. Finally, we derive a convex relaxation of the second model in the form of a semidefinite programming problem, which provides a lower bound on the lowest achievable mean liver BED. The method is presented on 5 cases with distinct geometries. The computed spatiotemporal plans achieve 12-35% mean liver BED reduction over the reference plans, which corresponds to 79-97% of the gap between the reference mean liver BEDs and our lower bounds. This indicates that spatiotemporal treatments can achieve substantial reduction in normal tissue BED, and that local optimization provides plans that are close to realizing the maximum potential benefit.

physics.med-ph↗
arXiv2017

Semi-infinite programming using high-degree polynomial interpolants and semidefinite programming

In a common formulation of semi-infinite programs, the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval. If this function is sufficiently closely approximable by a polynomial or a rational function, then the semi-infinite program can be reformulated as an equivalent semidefinite program. Solving this semidefinite program is challenging if the polynomials involved are of high degree, due to numerical difficulties and bad scaling arising both from the polynomial approximations and from the fact that the semidefinite programming constraints coming from the sum-of-squares representation of nonnegative polynomials are badly scaled. We combine rational function approximation techniques and polynomial programming to overcome these numerical difficulties, using sum-of-squares interpolants. Specifically, it is shown that the conditioning of the reformulations using sum-of-squares interpolants does not deteriorate with increasing degrees, and problems involving sum-of-squares interpolants of hundreds of degrees can be handled without difficulty. The proposed reformulations are sufficiently well scaled that they can be solved easily with every commonly used semidefinite programming solver, such as SeDuMi, SDPT3, and CSDP. Motivating applications include convex optimization problems with semi-infinite constraints and semidefinite conic inequalities, such as those arising in the optimal design of experiments. Numerical results align with the theoretical predictions; in the problems considered, available memory was the only factor limiting the degrees of polynomials, to approximately 1000.

math.OC↗
arXiv2016

Generating nested quadrature formulas for general weight functions with known moments

We revisit the problem of extending quadrature formulas for general weight functions, and provide a generalization of Patterson's method for the constant weight function. The method can be used to compute a nested sequence of quadrature formulas for integration with respect to any continuous probability measure on the real line with finite moments. The advantages of the method include that it works directly with the moments of the underlying distribution, and that for distributions with rational moments the existence of the formulas can be verified by exact rational arithmetic.

math.NA↗