arXiv subjects

Christian Bingane

Publications and source records attributed to Christian Bingane.

10 recordsLinked to original sources

arXiv2023

Maximal perimeter and maximal width of a convex small polygon

A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ sides are unknown when $s \ge 4$. In this paper, we propose an approach to construct convex small $n$-gons of large perimeter and large width when $n=2^s$ with $s\ge 2$. Assuming the existence of an axis of symmetry, a convex small $n$-gon is described as a composition of $n/2$ and both its perimeter and its width are given as functions of a single variable. By selecting the composition that minimizes the violation of a cycle constraint by a particular solution, the $n$-gons constructed outperform the best $n$-gons found in the literature. For example, for $n=64$, the perimeter and the width obtained are within $10^{-22}$ and $10^{-12}$ of the maximal perimeter and the maximal width, respectively. From our results, it appears that Mossinghoff's conjecture on the diameter graph of a convex small $2^s$-gon with maximal perimeter is not true when $s \ge 4$.

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arXiv2022

Tight bounds on the maximal perimeter and the maximal width of convex small polygons

A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ vertices are not known when $s \ge 4$. In this paper, we construct a family of convex small $n$-gons, $n=2^s$ and $s\ge 3$, and show that the perimeters and the widths obtained cannot be improved for large $n$ by more than $a/n^6$ and $b/n^4$ respectively, for certain positive constants $a$ and $b$. In addition, assuming that a conjecture of Mossinghoff is true, we formulate the maximal perimeter problem as a nonlinear optimization problem involving trigonometric functions and, for $n=2^s$ with $3 \le s\le 7$, we provide global optimal solutions.

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arXiv2022

Largest small polygons: A sequential convex optimization approach

A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m\ge 7$. Finding the largest small $n$-gon for a given number $n\ge 3$ can be formulated as a nonconvex quadratically constrained quadratic optimization problem. We propose to solve this problem with a sequential convex optimization approach, which is an ascent algorithm guaranteeing convergence to a locally optimal solution. Numerical experiments on polygons with up to $n=128$ sides suggest that the optimal solutions obtained are near-global. Indeed, for even $6 \le n \le 12$, the algorithm proposed in this work converges to known global optimal solutions found in the literature.

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arXiv2022

Tight bounds on the maximal perimeter of convex equilateral small polygons

A small polygon is a polygon that has diameter one. The maximal perimeter of a convex equilateral small polygon with $n=2^s$ sides is not known when $s \ge 4$. In this paper, we construct a family of convex equilateral small $n$-gons, $n=2^s$ and $s \ge 4$, and show that their perimeters are within $O(1/n^4)$ of the maximal perimeter and exceed the previously best known values from the literature. In particular, for the first open case $n=16$, our result proves that Mossinghoff's equilateral hexadecagon is suboptimal.

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arXiv2022

Small polygons with large area

A polygon is \textit{small} if it has unit diameter. The maximal area of a small polygon with a fixed number of sides $n$ is not known when $n$ is even and $n\geq14$. We determine an improved lower bound for the maximal area of a small $n$-gon for this case. The improvement affects the $1/n^3$ term of an asymptotic expansion; prior advances affected less significant terms. This bound cannot be improved by more than $O(1/n^3)$. For $n=6$, $8$, $10$, and $12$, the polygon we construct has maximal area.

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arXiv2022

Tight bounds on the maximal area of small polygons: Improved Mossinghoff polygons

A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m \ge 7$. In this paper, we construct, for each $n=2m$ and $m\ge 3$, a small $n$-gon whose area is the maximal value of a one-variable function. We show that, for all even $n\ge 6$, the area obtained improves by $O(1/n^5)$ that of the best prior small $n$-gon constructed by Mossinghoff. In particular, for $n=6$, the small $6$-gon constructed has maximal area.

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arXiv2022

The equilateral small octagon of maximal width

A small polygon is a polygon of unit diameter. The maximal width of an equilateral small polygon with $n=2^s$ vertices is not known when $s \ge 3$. This paper solves the first open case and finds the optimal equilateral small octagon. Its width is approximately $3.24\%$ larger than the width of the regular octagon: $\cos(π/8)$. In addition, the paper proposes a family of equilateral small $n$-gons, for $n=2^s$ with $s\ge 4$, whose widths are within $O(1/n^4)$ of the maximal width.

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arXiv2021

CONICOPF: Conic relaxations for AC optimal power flow computations

Computational speed and global optimality are key needs for practical algorithms for the optimal power flow problem. Two convex relaxations offer a favorable trade-off between the standard second-order cone and the standard semidefinite relaxations for large-scale meshed networks in terms of optimality gap and computation time: the tight-and-cheap relaxation (TCR) and the quadratic convex relaxation (QCR). We compare these relaxations on 60 PGLib-OPF test cases with up to 1,354 buses under three operating conditions and show that TCR dominates QCR on all 20 typical (TYP) test cases, on 18 out of 20 active power increase (API), and 12 out of 20 small angle difference (SAD). Selected state-of-the-art conic relaxations are implemented in the new MATLAB-based package CONICOPF available on GitHub.

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arXiv2019

Tight-and-cheap conic relaxation for the AC optimal power flow problem

The classical alternating current optimal power flow problem is highly nonconvex and generally hard to solve. Convex relaxations, in particular semidefinite, second-order cone, convex quadratic, and linear relaxations, have recently attracted significant interest. The semidefinite relaxation is the strongest among them and is exact for many cases. However, the computational efficiency for solving large-scale semidefinite optimization is lower than for second-order cone optimization. We propose a conic relaxation obtained by combining semidefinite optimization with the reformulation-linearization technique, commonly known as RLT. The proposed relaxation is stronger than the second-order cone relaxation and nearly as tight as the standard semidefinite relaxation. Computational experiments using standard test cases with up to 6515 buses show that the time to solve the new conic relaxation is up to one order of magnitude lower than for the chordal relaxation, a semidefinite relaxation technique that exploits the sparsity of power networks.

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arXiv2019

Tight-and-cheap conic relaxation for the optimal reactive power dispatch problem

The optimal reactive power dispatch (ORPD) problem is an alternating current optimal power flow (ACOPF) problem where discrete control devices for regulating the reactive power, such as shunt elements and tap changers, are considered. The ORPD problem is modelled as a mixed-integer nonlinear optimization problem and its complexity is increased compared to the ACOPF problem, which is highly nonconvex and generally hard to solve. Recently, convex relaxations of the ACOPF problem have attracted a significant interest since they can lead to global optimality. We propose a tight conic relaxation of the ORPD problem and show that a round-off technique applied with this relaxation leads to near-global optimal solutions with very small guaranteed optimality gaps, unlike with the nonconvex continuous relaxation. We report computational results on selected MATPOWER test cases with up to 3375 buses.

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