Abstract In this work computer-assisted optimality proofs are given for the problems of finding the densest packings of 31, 32, and 33 non-overlapping equal circles in a square. In a study of 2005, a fully interval arithmetic based global optimization method was introduced for the problem class, solving the cases 28, 29, 30. Until now, these were the largest problem instances solved on a computer. Using the techniques of that paper, the estimated solution time for the next three cases would have been 3–6 CPU months. In the present paper this former method is improved in both its local and global search phases. We discuss a new interval-based polygon representation of the core local method for eliminating suboptimal regions, which has a simpler implementation, easier proof of correctness, and faster behaviour than the former one. Furthermore, a modified strategy is presented for the global phase of the search, including improved symmetry filtering and tile pattern matching. With the new method the cases $$n=31,32,33$$ n = 31 , 32 , 33 have been solved in 26, 61, and 13 CPU hours, giving high precision enclosures for all global optimizers and the optimum value. After eliminating the hardware and compiler improvements since the former study, the new proof technique became roughly about 40–100 times faster than the previous one. In addition, the new implementation is suitable for solving the next few circle packing instances with similar computational effort.

This paper considers the task of finding the smallest circle into which one can pack a fixed number of non-overlapping unit squares that are free to rotate. Due to the rotation angles, the packing of unit squares into a container is considerably harder to solve than their circle packing counterparts. Therefore, optimal arrangements were so far proved to be optimal only for one or two unit squares. By a computer-assisted method based on interval arithmetic techniques, we solve the case of three squares and find rigorous enclosures for every optimal arrangement of this problem. We model the relation between the squares and the circle as a constraint satisfaction problem (CSP) and found every box that may contain a solution inside a given upper bound of the radius. Due to symmetries in the search domain, general purpose interval methods are far too slow to solve the CSP directly. To overcome this difficulty, we split the problem into a set of subproblems by systematically adding constraints to the center of each square. Our proof requires the solution of 6, 43 and 12 subproblems with 1, 2 and 3 unit squares respectively. In principle, the method proposed in this paper generalizes to any number of squares.

GLOBAL is a multi-start type stochastic method for bound constrained global optimization problems. Its goal is to find the best local minima that are potentially global. For this reason it involves a combination of sampling, clustering, and local search. The role of clustering is to reduce the number of local searches by forming groups of points around the local minimizers from a uniformly sampled domain and to start few local searches in each of those groups. We evaluate the performance of the GLOBAL algorithm on the BBOB 2009 noiseless testbed, containing problems which reflect the typical difficulties arising in real-world applications. The obtained results are also compared with those obtained form the simple multi-start procedure in order to analyze the effects of the applied clustering rule. An improved parameterization is introduced in the GLOBAL method and the performance of the new procedure is compared with the performance of the MATLAB GlobalSearch solver by using the BBOB 2010 test environment.

This paper presents an optimization procedure conceived to design parallel mechanisms (PMs) with legs of constant and/or variable length connected, at their endpoints, to a fixed base and a movable platform through universal and spherical joints, respectively. In the proposed procedure, the first natural frequency of the mechanism is the objective function to be maximized. The optimization problem is formulated by using dimensionless variables in order to identify the optimal geometry independently of mechanism size, platform density, and leg cross-sectional area and material. As a case study, the procedure is employed to find the optimal geometry of a 2-DOF spherical PM to be used as an orienting device in future space missions.