A New Verified Optimization Technique for the "Packing Circles in a Unit Square" Problems

Abstract

Abstract. The paper presents a new verified optimization method for the problem of finding the densest packings of non-overlapping equal circles in a square. In order to provide reliable numerical results, the developed algorithm is based on interval analysis. As one of the most efficient parts of the algorithm, an interval-based version of a previous elimination procedure is introduced. This method represents the remaining areas still of interest as polygons fully calculated in a reliable way. Currently the most promising strategy of finding optimal circle packing configurations is to partition the original problem into subproblems. Still as a result of the highly increasing number of subproblems, earlier computer-aided methods were not able to solve problem instances where the number of circles was greater than 27. The present paper provides a carefully developed technique resolving this difficulty by eliminating large groups of subproblems together. As a demonstration of the capabilities of the new algorithm the problems of packing 28, 29, and 30 circles were solved within very tight tolerance values. Our verified procedure decreased the uncertainty in the location of the optimal packings by more than 700 orders of magnitude in all cases.Key words. interval arithmetic, branch-and-bound method, circle packing, optimality proof AMS subject classifications. 52C15, 52C26, 65G30, 65K05, 90C301. Introduction. The so-called 'optimal packing of equal circles into a square' problem class is an interesting part of geometrical optimization. Circle packing has several real-life applications, e.g. cutting out a given number of identical circle shaped objects from some kind of material with a minimal amount of waste. In addition, proving the optimality of a packing configuration is a serious theoretical challenge from both the mathematical and the computational points of view.The paper is organized as follows: in §2 we discuss the general definition of the problem and give a brief historical overview focusing mostly on those of the computeraided methods which form the basis of the present algorithm. Section 3 contains the basic definitions and properties of interval analysis. In §4 a general interval branchand-bound frame algorithm is introduced. After that, crucial parts of the B&B algorithm are discussed in detail, as those designed specifically for circle packing problems. As a specification step, §5 introduces a new elimination procedure based on a prior non-interval one known from the literature. In §6 the questions of finding global solutions are discussed and new methods for eliminating tile combinations are investigated. In §7 we propose an efficient way of handling occurrences of free (not fixed) circles in optimal packings. Finally, in §8 the results of the previous sections are applied by solving the packing problems of 28, 29, and 30 circles. The presented numerical results demonstrate how the algorithm works in the successive elimination steps.

Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming

“…. , n. This geometric problem has been extensively studied in the global optimization literature [6,13]. Via a well-known transformation the problem is equivalent to the "point packing" problem…”

“…Figure 5 gives the square roots of the solution values for the various relaxations, corresponding to bounds on the minimum distance bewteen two points. The "MAX" values correspond to high-precision estimates for the exact optimal values of PP obtained by verified computing techniques [6], available from http://packomania.com. It is worth noting that while these problems have some similarity with the sensor network problems considered in [11], the SDP relaxations for PP do not appear to be nearly as tight as those for the sensor problems.…”

Interval analysis and verification of mathematical models

“…A telling example for the capabilities of interval optimization methods is the results on circle packing problems. Mihály Csaba Markót [M04,MC05] was able to solve the problem cases of n = 28, 29, and 30, i.e. to find the configuration of n congruent nonoverlapping maximal circles fitting into the unit square.…”

Space-Filling Latin Hypercube Designs for Computer Experiments

“…Melissen (1997) gives a comprehensive overview of the historical developments and state-of-the-art research in this field. For the 2 -distance measure in the two-dimensional case, optimal solutions are known for n ≤ 30 and n = 36, see e.g., Kirchner and Wengerodt (1987), Peikert et al (1991), Nurmela and Östergård (1999), and Markót and Csendes (2005). Furthermore, many good approximating solutions have been found for n ≥ 31; see the Packomania website of Specht (2008).…”