Packing Multidimensional Spheres in an Optimized Hyperbolic Container

Abstract

The problem of packing multidimensional spheres in a container defined by a hyperbolic surface is introduced. The objective is to minimize the height of the hyperbolic container under non-overlapping and containment conditions for the spheres, considering minimal allowable distances between them. To the best of our knowledge, no mathematical models addressing optimized packing spheres in hyperbolic containers have been proposed before. Our approach is based on a space dimensionality reduction transformation. This transformation relies on projecting a multidimensional hyperboloid into a lower-dimensional space sequentially up to two-dimensional case. Employing the phi-function technique, packing spheres in the hyperbolic container is formulated as a nonlinear programming problem. The latter is solved using a model-based heuristic combined with a decomposition approach. Numerical results are presented for a wide range of parameters, i.e., space dimension, number of spheres, and metric characteristics of the hyperbolic container. The results demonstrate efficiency of the proposed modeling and solution approach highlighting new opportunities for packing problems within non-traditional geometries.

1. Introduction

Sphere packing, i.e., arranging spherical objects in a larger container under specific conditions, is a well-established area of research with both theoretical and practical significance. Its versatility has led to applications across diverse fields. Packing problems are generally categorized into two classes: open dimension problems (ODPs) and knapsack problems [1]. The ODP focuses on optimizing the dimensions of the container, while the knapsack problem aims to maximize either the number of packed objects or the total packed volume. Continuous nonlinear programming (NLP) models are commonly employed for ODPs, while mixed-integer NLP models are typically used for knapsack problems.

Our focus on irregular containers is motivated by several key considerations. Sphere packing within irregular containers has important applications in materials science, nanotechnology, and additive manufacturing [2,3]. In materials science, sphere packing helps to understand the arrangement of atoms and molecules, influencing properties such as strength, density, and thermal conductivity [4]. In nanotechnology, it aids in the design and fabrication of nanostructures and nanoparticles, optimizing their arrangement for specific functions [5]. In 3D printing and additive manufacturing, sphere packing principles are used to enhance material efficiency and structural integrity. Furthermore, topology optimization enables the creation of products with improved mechanical performance and reduced material consumption through computational analysis of optimal material distribution [6,7].

A von Mises stress-driven circle packing algorithm for generating conformal lattice structures in additive manufacturing is studied in [8]. This approach optimizes lattice density by varying circle sizes based on stress intensity, resulting in Voronoi or Delaunay-based lattices that enhance stiffness. A simplified truss model reduces computational cost in finite element analysis, allowing efficient optimization using genetic algorithms, achieving up to 40% weight reduction in aerospace components while maintaining structural integrity.

Sphere packing for porous structure creation using the Voronoi tessellation method is considered in [9] focusing on modeling of porosity and mechanical properties relevant to biomedical applications. Disordered sphere packings of two different sizes are discussed in [10] showing that such arrangements can achieve densities comparable to ordered packings. This provides a theoretical foundation for improving particle density and flowability in metal powders used for additive manufacturing.

Sphere packing has also been applied to biomedical scaffolds. Macroporosity formation in bone scaffolds is modeled in [11] by packing of poly(lactide-co-glycolide) spheres. Sphere packing in mechano-regulation models, specifically the Pauwels model, is presented in [12]. Here an elementary sphere suspended in mesenchyme tissue is subjected to compression, tension, and shear forces, inducing cell formation of collagen fibrils. This modeling approach simulates the mechanical environment experienced by cells within scaffolds, aiding the design of structures that mimic natural skeletal tissue. Likewise, [13] uses sphere packing to model scaffold architecture with controlled porosity, enabling precise regulation of pore size and targeted drug release. These sphere-packed scaffolds mimic the extracellular matrix of bone, enhancing their effectiveness in bone defect repair.

While sphere packing in regular containers (rectangles, circles, etc.) has been extensively studied, especially for 2D objects like circles [14], ovals [15], and ellipses [16], the study of irregular containers remains a topic of significant interest [17]. Amore et al. [18] proposed an algorithm for dense packing configurations of congruent disks in arbitrary domains, such as rectangles, ellipses, crosses, and cardioids. Their introduction of “image” disks overcomes limitations of previous methods and suggests potential extensions to three-dimensional irregular containers.

A notable example of irregular containers is the hyperboloid structure, which offers advantages such as high strength, reduced material use, resistance to wind loads, and aesthetic appeal. These properties make hyperboloid geometries suitable for towers, cooling towers, and other industrial applications. In architecture, hyperbolic paraboloid roofs enable efficient material placement for structural stability and cost reduction. Similarly, hyperbolic geometries are used in designing components such as satellite dishes and other aerodynamic structures [19].

Various approaches have been proposed for solving sphere packing problems, including heuristic, nonlinear programming, probabilistic, and hybrid methods.

Heuristic methods provide approximate solutions to packing problems. Genetic algorithms, inspired by natural selection, have demonstrated high efficiency in solving circle packing problems [14]. Ant colony optimization, inspired by the foraging behavior of ants, employs probabilistic strategies to find efficient paths through solution spaces and has been successfully used for multiscale sphere packing [20]. The simulated annealing technique emulates the cooling process in metallurgy to approximate global optima and has been effectively applied to dense packing of spheres in cylindrical containers [21].

Probabilistic methods, such as Monte Carlo simulation, use random sampling to approximate solutions—particularly beneficial for high-dimensional packing problems [22]. Stochastic programming, which models uncertainty in sphere sizes or container dimensions, has also been applied to sphere packing optimization [23].

Nonlinear programming (NLP) techniques provide precise formulations for complex geometric challenges, particularly in irregular containers and variable object-size contexts [24,25].

Finally, hybrid methods combine the strengths of multiple approaches to enhance solution quality. The energy landscape paving and gradient descent algorithm integrates energy landscape mapping with adaptive-step gradient descent and has proven effective for both circle and sphere packing [26]. The combination of genetic algorithms with simulated annealing improves search performance over individual methods [27], while tabu search and particle swarm optimization effectively solve circle and sphere packing problems with high packing densities [28]. Another hybrid method, employing beam search, has shown promise in optimizing sphere packing configurations [29].

In this paper we develop a novel modeling and solution approach for packing spheres in a hyperbolic container, leveraging new modeling tools that address the geometric and computational challenges posed by the hyperbolic shape.

The main contributions of the paper are as follows:

○

A new problem of packing multidimensional spheres into a minimum-height hyperbolic container is introduced.

○

Non-overlapping and containment conditions for packing multidimensional spheres in the hyperbolic container, considering minimal allowable distances, are presented, using a space dimensionality reduction transformation and the phi-function technique.

○

A novel mathematical model for packing multidimensional spheres in the optimized hyperbolic container is formulated.

○

A model-based heuristic combined with a decomposition approach is developed to solve the corresponding large-scale nonconvex optimization problem.

○

Numerical results are provided to demonstrate the efficiency of the proposed approach for a wide range of space dimensions, number of spheres, and metric characteristics of the hyperbolic container.

The rest of this paper is structured as follows: The problem formulation of packing spheres into a minimum-height hyperbolic container is presented in Section 2. Section 3 describes tools of mathematical modeling for the problem stated. A mathematical model of the problem and its basic characteristics are considered in Section 4. Solution schemes of packing spheres into a one-sheeted and two-sheeted parabolic containers are introduced in Section 5. Computational results are provided in Section 6. Section 7 concludes.

2. Problem Formulation

Placement objects. Let us consider a finite family of multidimensional spheres

$$S_j(v_j)=\{x=(x_1,x_2,\dots ,x_n)\in {ℝ}^n:\hspace{0.33em}\stackrel{n}{{\displaystyle \sum _{i=1}}}{(x_i-y_{ji})}^2-{(r_j)}^2\le 0\},j\in J=\{1,2,\dots ,m\},$$

where $v_j=(y_{j1},y_{j2},\dots ,y_{jn})$ is a vector of unknown centers of the spheres $S_j$ of radii $r_j$, $j\in J$.

A vector of variable centers of spheres $S_j(v_j),$ $j\in J,$ is denoted by $v=(v_1,v_2,\dots ,v_m)\in {ℝ}^{nm}$.

Containers. We further define two types of containers, formed by: (a) two-sheeted circular hyperbolic surfaces and (b) one-sheeted circular hyperbolic surfaces.

Let ${H^{\prime }}_n=\{x\in {ℝ}^n:-{\displaystyle \sum _{i=1}^{n-1}{b}^2{x}_i^2}+a^2{x}_n^2-a^2{b}^2\le 0\}$, $n\ge 2$ be a convex subset of ${ℝ}^n$ bounded by the two-sheeted hyperboloid surface with given parameters $a,b$, while $P^{\prime }(h)=\{X\in {ℝ}^n:x_n-h\le 0,-x_n+b\le 0\}$ be a layer of ${ℝ}^n$ with variable parameter $h>b$, i.e.,

$${ℂ}^{\prime }(h)=\{x\in {ℝ}^n:\hspace{0.33em}-\stackrel{n-1}{{\displaystyle \sum _{i=1}}}{b}^2{x}_i^2+a^2{x}_n^2-a^2{b}^2\le 0, x_n-h\le 0,-x_n+b\le 0\}.$$

(1)

We refer ${ℂ}^{\prime }(h)={H^{\prime }}_n\cap P^{\prime }(h)$ of variable $h$ as the first type of container (Figure 1a).

Let ${H^{″}}_n=\{x\in {ℝ}^n:{\displaystyle \sum _{i=1}^{n-1}{b}^2{x}_i^2}-a^2{x}_n^2-a^2{b}^2\le 0\}$, $n\ge 2$ be a convex subset of ${ℝ}^n$ bounded by the one-sheeted hyperboloid surface with given parameters $a,b$, while $P^{″}(h)=\{x\in {ℝ}^n:x_n-h\le 0,-x_n+\overline{h}\le 0\}$ be a layer of ${ℝ}^n$ with a given parameter $h_0<0$ and variable parameter $h>0$, i.e.,

$${ℂ}^{″}(h)=\{x\in {ℝ}^n:\stackrel{n-1}{{\displaystyle \sum _{i=1}}}{b}^2{x}_i^2-a^2{x}_n^2-a^2{b}^2\le 0, x_n-h\le 0,-x_n+\overline{h}\le 0\},$$

(2)

We refer ${ℂ}^{″}(h)={H^{″}}_n\cap P^{″}(h)$ of variable height $h$ as the second type of container (Figure 1b).

The feasibility conditions. Within this study the non-overlapping of each pair of spheres $S_j(v_j)$ and $S_k(v_k)$, i.e.,

$$\mathrm{int}{S}_j(v_j)\cap \mathrm{int}{S}_k(v_k)=\varnothing \mathrm{for} \mathrm{all} j<k,j,k\in J,$$

and the containment of each sphere $S_j(v_j)$ within the container, i.e.,

$$S_j(v_j)\subset ℂ(h)\iff \mathrm{int}{S}_j(v_j)\cap \overline{ℂ}(h)=\varnothing \mathrm{for} \mathrm{all} j\in J,\overline{ℂ}(h)=R^n\backslash \mathrm{int}ℂ(h),$$

are regarded as basic feasibility conditions. Here the notation $\mathrm{int}(\cdot )$ is used to denote the topological interior of the corresponding set $(\cdot )$ [30].

In addition, distance conditions may be given: a minimal allowable distance ${\rho }_{jk}$ between each pair of spheres $S_j(v_j)$ and $S_k(v_k)$, i.e.,

$$dist(S_j(v_j),S_k(v_k))\ge {\rho }_{jk}\mathrm{for} \mathrm{all} j<k,j,k\in J,$$

as well as a minimal allowable distance ${\rho }_j$ between each sphere $S_j(v_j)$ and the boundary of the container $ℂ(h)$, i.e.,

$$dist(S_j(v_j),\overline{ℂ}(h))\ge {\rho }_j\mathrm{for} \mathrm{all} j\in J.$$

Let us formulate a packing problem in the following setting.

Packing Multidimensional Spheres in an Optimized Hyperbolic Container (PSH). Arrange a given family of multidimensional spheres in a minimum-height hyperbolic container considering the feasibility conditions (non-overlapping, containment and/or distance constraints).

3. Tools of Mathematical Modeling

The construction of the feasibility constraints presented in this work is a generalization of the phi-function approach described in [31], which extends the Minkowski sum of two objects to multidimensional space to define their feasible placements. In our case, the problem of packing a multidimensional sphere inside a hyperboloid is reduced to a two-dimensional setting using a so-called space dimensionality reduction transformation. This transformation yields a 2D hyperboloid and 2D sphere with transformed coordinates, for which we construct special classes of continuous functions to describe the containment condition. Multidimensional packing problems arise, e.g., in project management [32], where objects correspond to projects consuming various resources, while a container represents resource limitations. The general formulation is considered to highlight that the proposed algorithmic approach is suitable for any dimension. To the best of our knowledge, no mathematical models for packing spheres in a hyperbolic container were considered before.

Let us introduce a function

$$f(x_p)=\mathrm{sgn}(x_1)\cdot ‖x_p‖\mathrm{for} p\ge 2,$$

(3)

where $\mathrm{sgn}(x)=\left\{\begin{array}{l}1\hspace{0.33em}\mathrm{if}\hspace{0.17em}x\ge 0\\ -1\hspace{0.33em}\mathrm{if}\hspace{0.17em}x<0\end{array}\right.$.

The function (3) relies on the Euclidean norm $‖x_p‖={\displaystyle {\sum }_{i=1}^p{x}_i^2}$ to project coordinates of vector $x_p$ into a lower-dimensional space.

Then, we define containment conditions $S_j(v_j)\subset$$ℂ(h)\in \{{ℂ}^{\prime }(h),{ℂ}^{″}(h)\}$, using transformation (3).

3.1. Modeling Containment Constraint: $S_j(v_j)\subset {ℂ}^{\prime }(h)$

Firstly, we consider the relation $S_j(v_j)\subset$${H^{\prime }}_n$.

Let ${\Pi }_n$ be the hyperplane passing through axis $O{x}_n$ of ${H^{\prime }}_n$ and the center $v_j$ of the sphere $S_j(v_j)$.

In the sections ${\Pi }_n\cap {H^{\prime }}_n$ and ${\Pi }_n\cap S_{j(n)}(v_{j(n)})$, using transformation (3), we obtain the (n−1)D-sphere $S_{j(n-1)}(v_{j(n-1)})$ centered at $v_{j(n-1)}=(z_{j1},y_{j3},\dots ,y_{jn})$ with $z_{j1}=\mathrm{sgn}(y_{j1})\sqrt{y_{j1}^2+y_{j2}^2}$ and the (n−1)D-hyperboloid

$${H^{\prime }}_{n-1}=\{x\in {ℝ}^{n-1}:\hspace{0.33em}-b^2{z}_1^2-{\displaystyle {\sum }_{i=3}^{n-1}{b}^2{x}_i^2}+a^2{x}_n^2-a^2{b}^2\le 0\}$$

with $z_1=f(x_2)=\mathrm{sgn}(x_1)\sqrt{x_1^2+x_2^2}\in {H^{\prime }}_{n-1}$.

Then, we construct the hyperplane ${\Pi }_{n-1}$ passing through the axis $O{x}_n$ of $H_{n-1}$ and the center $v_{j(n-1)}$ of $S_{j(n-1)}(v_{j(n-1)})$.

In the sections ${\Pi }_{n-1}\cap {H^{\prime }}_{n-1}$ and ${\Pi }_{n-1}\cap S_{j(n-1)}(v_{j(n-1)})$, using transformation (3), we obtain the (n − 2) D-sphere $S_{j(n-2)}(v_{j(n-2)})$ of radius $r_j$ centered at $v_{j(n-2)}=(z_{j2},y_{j4},\dots ,y_{jn})$ with $z_{j2}=\mathrm{sgn}(y_{j1})\sqrt{y_{j1}^2+y_{j2}^2+y_{j3}^2}$ and the hyperboloid

$${H^{\prime }}_{n-2}=\{x\in {ℝ}^{n-2}:-\hspace{0.33em}{b}^2{z}_2^2-{\displaystyle \sum _{i=4}^{n-1}{b}^2{x}_i}+a^2{x}_n^2-a^2{b}^2\le 0\},$$

with $z_2=f(x_3)=\mathrm{sgn}(x_1)\sqrt{x_1^2+x_2^2+x_3^2}\in {H^{\prime }}_{n-2}$.

We continue the procedure until, in the sections ${\Pi }_3\cap {H^{\prime }}_3$ and ${\Pi }_3\cap S_{j3}(z_{j3},y_{j3})$, we obtain the 2D sphere $S_{j2}(z_{j(n-2)},y_{jn})$ of radius $r_j$ centered at $(z_{j(n-2)},y_{jn})$ with $z_{j(n-2)}=\mathrm{sgn}(y_{j1})\sqrt{{\displaystyle {\sum }_{i=1}^{n-1}{y}_{ji}^2}}$ and the 2D hyperboloid ${H^{\prime }}_2=\{X\in {ℝ}^2:\hspace{0.33em}-b^2{z}_{n-1}^2+a^2{x}_n^2-a^2{b}^2\le 0\}$ with $z_{n-2}=f(x_{n-1})=\mathrm{sgn}(x_1)\sqrt{{\displaystyle {\sum }_{i=1}^{n-1}{x}_i^2}}$.

To model the containment constraint, $S_j(v_j)\subset {H^{\prime }}_n$, we introduce an adjusted phi-function [31] for ${{\overline{H}}^{\prime }}_n$ and $S_j(v_j)$ where

$${{\overline{H}}^{\prime }}_n=R^n\backslash \mathrm{int}{H^{\prime }}_n=\{x\in {ℝ}^n:{\displaystyle \sum _{i=1}^{n-1}{b}^2{x}_i^2}-a^2{x}_n^2+a^2{b}^2\le 0\},$$

then $S_j(v_j)\subset {H^{\prime }}_n$$\iff \mathrm{int}{S}_j(v_j)\cap {{\overline{H}}^{\prime }}_n=\varnothing$.

An adjusted phi-function ${{\Phi }^{\prime }}_j(v_j)$ for $S_j(v_j)$ and ${\overline{H^{\prime }}}_n$ can be defined as follows:

Let us consider a function (see Appendix A for details)

$${{\phi }^{\prime }}_j(v_j,{\xi }_j)=-\mathrm{sgn}(y_{j1})\sqrt{{\displaystyle {\sum }_{i=1}^{n-1}{y}_{ji}^2}}\cdot \cos {\xi }_j+y_{jn}\sin {\xi }_j-\sqrt{b^2-(a^2-b^2){{\displaystyle \cos }}^2{\xi }_j}$$

Denote an approximation of a local minimum to the problem

$$\underset{{\xi }_j\in [{\beta }_1,{\beta }_2]}{{\displaystyle \min }}{{\phi }^{\prime }}_j(v_j,{\xi }_j)\mathrm{for} \mathrm{some} {v^{\prime }}_j$$

(4)

as $({v^{\prime }}_j,{\xi }_j^{*})$.

Then an adjusted phi-function ${{\Phi }^{\prime }}_j(v_j)$ can be derived in some neighborhood $U_{\epsilon }({v^{\prime }}_j)$ of the point ${v^{\prime }}_j$ in the following form:

$${{\Phi }^{\prime }}_j(v_j)={{\phi }^{\prime }}_j(v_j,{\xi }_j^{*})-r_j-{\rho }_j\mathrm{for} v_j\in U_{\epsilon }({v^{\prime }}_j)$$

(5)

Let us define $[{\beta }_1,{\beta }_2]$ in (4). Denote the angle between two asymptotes of hyperbola ${H^{\prime }}_2$ as $\alpha =2\arccos (b/\sqrt{a^2+b^2}).$ This means that two parallel tangents, ${\mathrm{\gamma }}_1$ to the boundary of ${H^{\prime }}_2$ and ${\mathrm{\gamma }}_2$ to the boundary of $S_{j2}(z_{j(n-1)},y_{jn})$, do exist if ${\xi }_j\in (\alpha /2, 2\pi -\alpha /2)$ (Figure 2).

Hereinafter, in all figures the green lines show the tangents to the sphere/container corresponding to different values of the parameter ${\xi }_j$. The blue lines represent asymptotes, while the red lines indicate rays from the sphere center to the extreme values of the search interval $[{\beta }_1,{\beta }_2]$ for the parameter ${\xi }_j$.

We consider the following four cases:

Case 1. If $(z_{j(n-1)},y_{jn})\in {H^{\prime }}_2$ and $z_{j(n-2)}\ge 0$, then $[{\beta }_1,{\beta }_2]=[\alpha /2,\pi /2]$.

Case 2. If $(z_{j(n-1)},y_{jn})\in {H^{\prime }}_2$ and $z_{j(n-2)}<0$, then $[{\beta }_1,{\beta }_2]=[\pi /2,\pi -\alpha /2]$.

Case 3. If $(z_{j(n-1)},y_{jn})\notin {H^{\prime }}_2$ and $z_{j(n-2)}\ge 0$, then $[{\beta }_1,{\beta }_2]=[\pi +\alpha /2,3\pi /2]$.

Case 4. If $(z_{j(n-1)},y_{jn})\notin {H^{\prime }}_2$ and $z_{j(n-2)}<0$, then $[{\beta }_1,{\beta }_2]=[3\pi /2,2\pi -\alpha /2]$.

Note that in (5)

$${{\phi }^{\prime }}_j(v_j,{\xi }_j)=-y_{j1}\cos {\xi }_j+y_{j2}\sin {\xi }_j-\sqrt{b^2-(a^2-b^2){{\displaystyle \cos }}^2{\xi }_j}, \mathrm{if} n=2,$$

and

$${{\phi }^{\prime }}_j(v_j,{\xi }_j)=-\mathrm{sgn}(y_{j1})\sqrt{y_{j1}^2+y_{j2}^2}\cos {\xi }_j+y_{j3}\sin {\xi }_j-\sqrt{b^2-(a^2+b^2){{\displaystyle \cos }}^2{\xi }_j}, \mathrm{if} n=3.$$

Based on (1), (4), and (5), an adjusted phi-function for $S_j(v_j)$ and $\overline{{ℂ}^{\prime }}(h)$ can be defined in the form

$${{\Psi }^{\prime }}_j(v_j,h)=\min \{{{\Phi }^{\prime }}_j(v_j), y_{jn}-b, h-y_{jn}-r_j-{\rho }_j\}$$

(6)

Therefore, ${{\Psi }^{\prime }}_j(v_j,h)\ge 0$ holds a given minimal allowable distance $\rho$ between a sphere $S_j(v_j)$ and the boundary of the container ${ℂ}^{\prime }(h)$.

Note that the inequality $\min \{{{\Phi }^{\prime }}_j(v_j), y_{jn}-b, h-y_{jn}-r_j-{\rho }_j\}\ge 0$ is equivalent to the inequality system

$$\left\{\begin{array}{l}{{\Phi }^{\prime }}_j(v_j)\ge 0\\ y_{jn}-b\ge 0\\ h-y_{jn}-r_j-{\rho }_j\ge 0\end{array}\right..$$

3.2. Modeling Containment Constraint: $S_j(v_j)\subset {ℂ}^{″}(h)$

Let there be a hyperboloid ${H^{″}}_n$ (2) and a sphere $S_j(v_j)$ (Figure 3). Then the complement of the hyperboloid ${H^{″}}_n$ to the whole space ${ℝ}^n$ has the form

$${\overline{H^{″}}}_n={ℝ}^n/\mathrm{int}{H^{″}}_n=\{x\in {ℝ}^n:-{\displaystyle \sum _{i=1}^{n-1}{b}^2{x}_i^2}+a^2{x}_n^2+a^2{b}^2\le 0\}.$$

A normalized phi-function for $S_j(v_j)$ and ${\overline{H^{″}}}_n$ using the approach applied in Section 3.1 (see Appendix B for details) takes the form

$${{\Phi }^{″}}_j(v_j,{\xi }_j)=\left\{\begin{array}{c}\min \{{{\psi }^{″}}_{1j}(v_j,{\xi }_j)-r_j, y_{j1},\sin {\xi }_j+a/\sqrt{a^2+b^2},-\sin {\xi }_j+a/\sqrt{a^2+b^2}\} \mathrm{if} y_{j1}⩾0\\ \min \{{{\psi }^{″}}_{2j}(v_j,{\xi }_j)-r_j,-y_{j1},\sin {\xi }_j+a/\sqrt{a^2+b^2},-\sin {\xi }_j+a/\sqrt{a^2+b^2}\} \mathrm{if} y_{j1}<0\end{array}\right.,$$

(7)

where

$$\begin{gathered} {\psi }_{1j}^{″}(v_j,{\xi }_j)=-\mathrm{sgn}(y_{j1})\sqrt{\stackrel{n-1}{{\displaystyle \sum _{i=1}}}{y}_{ji}^2}\cdot \cos {\xi }_j+y_{jn}\sin {\xi }_j+\sqrt{a^2-(a^2+b^2){{\displaystyle \sin }}^2{\xi }_j}, \\ {\psi }_{2j}^{″}(v_j,{\xi }_j)=\mathrm{sgn}(y_{j1})\sqrt{\stackrel{n-1}{{\displaystyle \sum _{i=1}}}{y}_{ji}^2}\cdot \cos {\xi }_j+y_{jn}\sin {\xi }_j+\sqrt{a^2-(a^2+b^2){{\displaystyle \sin }}^2{\xi }_j} \end{gathered}$$

See Appendix B for details.

Note that in (7)

$$\begin{gathered} {\psi }_{1j}^{″}(v_j,{\xi }_j)=-y_{j1}\cos {\xi }_j+y_{j2}\sin {\xi }_j+\sqrt{a^2-(a^2+b^2){{\displaystyle \sin }}^2{\xi }_j}, \\ {\psi }_{2j}^{″}(v_j,{\xi }_j)=y_{j1}\cos {\xi }_j+y_{j2}\sin {\xi }_j+\sqrt{a^2-(a^2+b^2){{\displaystyle \sin }}^2{\xi }_j} \end{gathered}$$

if n = 2, and

$$\begin{gathered} {\psi }_{1j}^{″}(v_j,{\xi }_j)=-\mathrm{sgn}(y_{j1})\sqrt{y_{j1}^2+y_{j2}^2}\cdot \cos {\xi }_j+y_{j3}\sin {\xi }_j+\sqrt{a^2-(a^2+b^2){{\displaystyle \sin }}^2{\xi }_j}, \\ {\psi }_{2j}^{″}(v_j,{\xi }_j)=\mathrm{sgn}(y_{j1})\sqrt{y_{j1}^2+y_{j2}^2}\cdot \cos {\xi }_j+y_{j3}\sin {\xi }_j+\sqrt{a^2-(a^2+b^2){{\displaystyle \sin }}^2{\xi }_j},\mathrm{if}n=3. \end{gathered}$$

Based on (2) and (7), the normalized phi-function for $S_j(v_j)$ and $\overline{{ℂ}^{″}}(h)$ (see Figure 2) can be presented as follows:

$${{\Psi }^{″}}_j(v_j,h,{\xi }_j)=\min \{{{\Phi }^{″}}_j(v_j,{\xi }_j),-h_0+y_{jn}-r_j, h-y_{jn}-r_j\}$$

(8)

Therefore, ${{\Psi }^{″}}_j(v_j,h,{\xi }_j)\ge {\rho }_j$ holds a given minimal allowable distance ${\rho }_j$ between a sphere $S_j(v_j)$ and the boundary of the container ${ℂ}^{″}(h)$.

Note that the inequality

$${{\Psi }^{″}}_j(v_j,h,{\xi }_j)=\min \{{{\Phi }^{″}}_j(v_j,{\xi }_j),-h_0+y_{jn}-r_j, h-y_{jn}-r_j\}\ge {\rho }_j$$

is equivalent to the following inequality system

$$\left\{\begin{array}{l}{{\Phi }^{″}}_j(v_j,{\xi }_j)-{\rho }_j\ge 0\\ -h_0+y_{jn}-r_j-{\rho }_j\ge 0 \\ h-y_{jn}-r_j-{\rho }_j\ge 0\end{array}\right..$$

3.3. Modeling Non-Overlapping Constraint for Two Spheres $S_j(v_j)$ and $S_k(v_k)$

An adjusted phi-function for two multidimensional spheres $S_j(v_j)$ and $S_k(v_k)$ has the form

$${\varphi }_{jk}(v_j,v_k)={‖v_j-v_k‖}^2-{(r_j+r_k+{\rho }_{jk})}^2\ge 0, j<k,j,k\in J.$$

Note that ${\varphi }_{jk}(v_j,v_k)\ge 0$ provides the distance constraint $dist(S_j(v_j),S_k(v_k))\ge {\rho }_{jk}$.

In particular, ${\varphi }_{jk}(v_j,v_k)\ge 0\iff \mathrm{int}{S}_j(v_j)\cap \mathrm{int}{S}_k(v_k)=\varnothing$ for ${\rho }_{jk}=0$.

4. Mathematical Models

A mathematical model of the PSH problem for ${ℂ}^{\prime }(h)$ can be formulated as the following optimization problem:

$$\underset{(v,h)\in W^{\prime }\subset {ℝ}^{nm+1}}{\min }h,$$

(9)

where

$$W^{\prime }=\{(v,h): {\varphi }_{jk}(v_j,v_k)\ge 0, j<k,j,k\in J,{{\Psi }^{\prime }}_j(v_j,h)\ge 0,j\in J,h\ge 0\},$$

(10)

where inequality ${\varphi }_{jk}(v_j,v_k)\ge 0$ holds allowable distance ${\rho }_{jk}$ between each pair of spheres $S_j(v_j)$ and $S_k(v_k)$, while inequality ${{\Psi }^{\prime }}_j(v_j,h)\ge 0$ ensures an arrangement of each sphere $S_j(v_j)$ inside the container ${ℂ}^{\prime }(h)$ considering allowable distance ${\rho }_j$.

The number of inequalities that form the feasible region $W^{\prime }$ (10) is $0.5m(m-1)+3m+1$, while the number of the problem variables is $nm+1$.

A mathematical model of the PSH problem for ${ℂ}^{″}(h)$ can be stated as the following optimization problem:

$$\underset{(v,h,\xi )\in W^{″}\subset {ℝ}^{(n+1)m+1}}{\min }h,$$

(11)

where

$$W^{″}=\{(v,h,\xi ): {\varphi }_{jk}(v_j,v_k)\ge 0, j<k,j,k\in J,{{\Psi }^{″}}_j(v_j,h,{\xi }_j)-{\rho }_j\ge 0,j\in J,h\ge 0\},$$

(12)

where $\xi =({\xi }_1,\dots ,{\xi }_m)$, inequality ${\varphi }_{jk}(v_j,v_k)\ge 0$ holds allowable distance ${\rho }_{jk}$ between each pair of spheres $S_j(v_j)$ and $S_k(v_k)$, while inequality ${{\Psi }^{″}}_j(v_j,h,{\xi }_j)-{\rho }_j\ge 0$ ensures an arrangement of a sphere $S_j(v_j)$ in the container ${ℂ}^{″}(h)$ considering allowable distance ${\rho }_j$.

The number of inequalities that form the feasible region $W^{″}$(12) is $0.5m(m-1)+3m+1$, while the number of the problem variables is $(n+1)m+1$.

We describe model-based solution approaches for both problems (9)–(10) and (11)–(12) in the next section.

5. Solution Algorithms

5.1. Packing Spheres into the ${ℂ}^{\prime }(h)$ Container

The following solution approach to the problem (9) and (10) is proposed.

Firstly, we provide the container height $h=h^{(0)}$ guaranteeing a placement of all spheres $S_j(v_j)$, $j\in J,$ fully inside the container ${ℂ}^{\prime }(h)$ subject to feasibility conditions. Then, we use Algorithm A1 (see Appendix C) to generate the sphere centers $v_j^{(0)}\in ℂ\prime (h^{(0)}),$ $j\in J,$ ensuring the minimal allowable distances ${\rho }_{kj}$ between each pair of spheres $S_j$ and $S_k$, as well as the minimal allowable distance ${\rho }_j$ between each sphere $S_j$ and the boundary of the container ${ℂ}^{\prime }(h)$.

Further, we apply the feasible direction method combined with the ε-active inequalities strategy [33,34], using the iterative procedure

$$Z^{(\tau +1)}:=Z^{(\tau )}+t^{(\tau +1)}{\Delta }^{(\tau +1)},$$

where $Z^{(\tau )}=(v^{(\tau )},h^{(\tau )})\in W$ such that $F(Z^{(\tau +1)})<F(Z^{(\tau )})$, $\tau =0,1,2,\dots$.

Here, ${\Delta }^{(\tau +1)}$ is the search direction at the iteration $\tau +1$. It is computed based on the feasible direction method as a solution of a linear programming problem and provides a direction in which the objective function decreases while maintaining feasibility with respect to the ε-active constraints. The value of $t^{(\tau +1)}$ is the step size along the direction ${\Delta }^{(\tau +1)}$. It is chosen to ensure feasibility preservation, meaning the new point must remain within the feasible region defined by all constraints, ensuring that no inequality is violated during the update. This guarantees that the movement of each sphere respects both non-overlap and containment conditions throughout the optimization process. The point $Z^{(0)}$ is generated meeting feasibility of the problem (9) and (10), using Algorithm A1 (see Appendix C).

The core steps of our approach to solving the PSH problem for ${ℂ}^{\prime }(h)$ are presented below.

Step 1. Take $h:=h^{(0)}$, $\tau :=0$.

Step 2. Use Algorithm A1 to generate the sphere centers $v_j^{(0)},$ $j\in J,$ ensuring $v_j^{(0)}\in ℂ\prime (h^{(0)}),$ $j\in J$, and feasibility conditions of the problem (9) and (10).

Step 3. Calculate the value of the adjusted phi-function ${{\Phi }^{\prime }}_j(v_j^{(\tau )})$ (5) by solving the corresponding optimization problem (4), using a bisection search, i.e., ${{\Phi }^{\prime }}_j(v_j^{(\tau )})={\phi }^{\prime }(v_j^{(\tau )},{\xi }_j^{*})-r_j$, for all $j\in J$.

Step 4. Construct a set ${W^{\prime }}_{\epsilon }^{(\tau )}$ of ε-active inequalities in (10) of the form ${{\Psi }^{\prime }}_j(v_j^{(\tau )},h^{(\tau )})-{\rho }_j\le \epsilon$, $y_{nj}^{(\tau )}-b-{\rho }_j\le \epsilon ,$ $h^{(\tau )}-y_{nj}^{(\tau )}-r_j-{\rho }_j\le \epsilon$ for $j\in J$ and ${\varphi }_{jk}(v_j^{(\tau )},v_k^{(\tau )})\le \epsilon$ for $j<k,j,k\in J$.

Step 5. Calculate a feasible descent vector ${\Delta }^{(\tau +1)}$ for the problem

$$\underset{(v,h)\in {W^{\prime }}_{\epsilon }^{(\tau )}}{\min }h,$$

using an approach described in [34].

Step 6. Set $t^{(\tau +1)}:=1$.

Step 7. Set $Z^{(\tau +1)}:=Z^{(\tau )}+t^{(\tau +1)}{\Delta }^{(\tau +1)}$.

Step 8. If $Z^{(\tau +1)}\notin W^{\prime }$, then set $t^{(\tau +1)}:=t^{(\tau +1)}/2$ and go to Step 7; otherwise, go to Step 9.

Step 9. If $‖Z^{(\tau +1)}-Z^{(\tau )}‖<{10}^{-5}$, then stop the algorithm; otherwise, set $\tau :=\tau +1$, and go to Step 3.

5.2. Packing Spheres into the ${ℂ}^{″}(h)$ Container

Let us consider a solution strategy for the problem (11) and (12) for $ℂ(h)={ℂ}^{″}(h)$.

Firstly, we set sufficiently large $h=h^{(0)}$ so that all spheres $S_j(v_j),$ $j\in J,$ can fit the container ${ℂ}^{″}(h)$ keeping feasibility conditions.

Taking $Z^{(0)}=(v^{(0)},h^{(0)},{\xi }^{(0)})$ with ${\xi }_j^{(0)}\in$ $\left[-a/\sqrt{a^2+b^2},a/\sqrt{a^2+b^2}\right]$ as a feasible starting point to the problem (11) and (12), we derive functions ${{\Phi }^{″}}_j(v_j,{\xi }_j)$, $j\in J,$ defined in (7). Then, we apply an iterative decomposition technique [35] to solve the problem (11) and (12). At each iteration τ only a subset of the solution space (12) is considered. This subset includes feasibility constraints only for those pairs of spheres that are arranged no farther than a predefined distance from each other, as well as between the spheres and the boundary of the container, subject to the restriction that the movement of each sphere is limited to a local neighborhood. This localized movement control preserves feasibility and allows the algorithm to efficiently explore the solution space.

To accomplish this, a sequence of the following nonlinear programming problems is solved:

$$\underset{Y=(v,h,\xi )\subset {W^{″}}^{(\tau )}}{\min }h, \tau =0,1,2,\dots ,$$

(13)

where

$$\begin{array}{l}{W^{″}}^{(\tau )}=\{Y=(v,h,\xi )\in {ℝ}^{(n+1)m+1}: {\varphi }_{jk}(v_j,v_k)\ge 0, (k,j)\in J^{(\tau )},\\ {{\Psi }^{″}}_j(v_j,h,{\xi }_j)-{\rho }_j\ge 0, j\in I^{(\tau )},h\ge 0, \Vert v_j-v_j^{(\tau )}\Vert \le d^{(\tau )}, y_{j1}⩾0 (\mathrm{or} -y_{j1}>0), j\in J\}.\end{array}$$

(14)

In (14),

$$\begin{gathered} J^{(\tau )}=\{J\times J|k<j|‖v_j^{(\tau )}-v_k^{(\tau )}‖\le 2d^{(\tau )}+\rho \}, \\ {I}^{(\tau )}=\{j\in J|{{\Psi }^{″}}_j(v_j^{(\tau )},h^{(\tau )},{\xi }_j^{(\tau )})\le d^{(\tau )}+\rho \}, \end{gathered}$$

$d^{(\tau )}=\underset{i\in J}{\max }{r}_i>0$ is a decomposition parameter, which serves as a balance between the number of NLP subproblems to be solved and the number of nonlinear inequalities involved in a feasible set of each of the NLP subproblems.

Each constraint for variable $y_{j1}$ is either $y_{j1}⩾0$ or $-y_{j1}>0$ depending on ${{\Phi }^{″}}_j(v_j,{\xi }_j)$ defined in (7). The choice between these two inequalities depends on the value of $y_{j1}^{(\tau )}$. If $y_{j1}^{(\tau )}>0$, then the constraint $y_{j1}⩾0$ is applied. If $y_{j1}^{(\tau )}<0$, then the constraint $-y_{j1}>0$ is chosen. If $y_{j1}^{(\tau )}=0$, then the constraint is reversed compared to the previous step to allow switching between the two opposite inequalities.

A starting point $Y^{(0)}=(v^{(0)},h^{(0)},{\xi }^{(0)})$ is generated meeting feasibility conditions of the problem (11) and (12), using Algorithm A2 (see Appendix C).

Then a local minimum point $Y^{*(\tau )}=(v^{*(\tau )},h^{*(\tau )},{\xi }^{*(\tau )})$ of the problem (13) and (14) is computed, using an algorithm, presented below.

The core steps of our algorithm to solve the PSH problem for ${ℂ}^{″}(h)$ are as follows.

Step 1. Take $h:=h^{(0)}$, $\tau :=0$.

Step 2. If $\tau =0$, then go to Step 4; otherwise, go to Step 3.

Step 3. Derive ${{\Phi }^{″}}_j(v_j,{\xi }_j),j\in J$ defined by (7).

Step 4. Generate the spheres’ centers $v_j^{(0)}\in {ℂ}^{″}(h^{(0)}),$$j\in J$, using Algorithm A2 Step 5. Solve the problem (13) and (14) starting from the point $Y^{(\tau )}=(v^{(\tau )},h^{(\tau )},{\xi }^{(\tau )})$ and compute a local minimum point $Y^{*(\tau )}=(v^{*(\tau )},h^{*(\tau )},{\xi }^{*(\tau )})$ using the local solver IPOPT [36].

Step 6. If $‖Y^{(\tau )*}-Y^{(\tau )}‖<{10}^{-5}$, then stop the algorithm with $Y^{*(\tau )}=(v^{*(\tau )},h^{*(\tau )},{\xi }^{*(\tau )})$; otherwise, set $Y^{(\tau +1)}:=Y^{(\tau )*}$, $\tau :=\tau +1$ and return to Step 5.

6. Numerical Examples

The computational experiment was conducted on a personal computer with the following technical specifications: Intel^® Core™ i5-5300U CPU @ 2.30 GHz, 8.00 GB RAM, x64-based processor. The implementation was carried out using the C++ programming language with OpenGL 4.6 for 3D graphics rendering.

We consider two groups of instances to solve the PSH problem for packing different number m of n-dimensional spheres, with varying dimensions n, into (a) the ${ℂ}^{\prime }(h)$ container and (b) the ${ℂ}^{″}(h)$ container. Input and output data are provided in

Table 1. Input and output data for ${ℂ}^{\prime }(h)$.

Instance #1 input	$n=2$	$a=3$	$b=6$	$m=300$
$r_j=0.592,$$j=1,\dots ,12$, $r_j=0.612,$ $j=13,\dots ,60$,  $r_j=0.68,$$j=61,\dots ,108,$$r_j=0.747$,  $j=109,\dots ,156,$ $r_j=0.76$,  $j=157,\dots ,204$,  $r_j=0.807$,  $j=205,\dots ,252,$ $r_j=0.845$,  $j=253,\dots ,300$
Instance #1 output	$h^{*}=$$37.007$		time 814 s		Optimized packing
Instance #2 input	$n=2$	$a=3$	$b=6$	$m=100$
${\rho }_{jk}=1,1\le j<k\le 100,$ ${\rho }_j=0.5, j=1,\dots ,100$ $r_j=0.527,$$j=1,\dots ,10$,  $r_j=0.564,$ $j=11,\dots ,20$,  $r_j=0.566,$$j=21,\dots ,30,$$r_j=0.592$,  $j=31,\dots ,40,$ $r_j=0.612$,  $j=41,\dots ,50$,  $r_j=0.680$,  $j=51,\dots ,60,$ $r_j=0.747$,  $j=61,\dots ,70$,  $r_j=0.760$,  $j=71,\dots ,80$,  $r_j=0.807$,  $j=81,\dots ,90$,  $r_j=0.845$,  $j=91,\dots ,100$
Instance #2 output	$h^{*}=$$34.5592$		time 157 s		Optimized packing
Instance #3 input	$n=2$	$a=3$	$b=6$	$m=100$
${\rho }_j=r_{\max }-r_j,$ ${\rho }_{jk}=r_{\max }-r_j+0.01\cdot k$ $r_j=0.7,$$j=1,\dots ,25$,  $r_j=1,$ $j=26,\dots ,50$,  $r_j=1.2,$$j=51,\dots ,75,$ $r_j=1.5,$$j=76,\dots ,100.$ where $r_{\max }=\max \{r_1,r_2,\dots ,r_m\}$
Instance #3 output	$h^{*}=$$50.1068$		time 35 s		Optimized packing
Instance #4 input	$n=3$	$a=2$	$b=5$	$m=200$
$r_j=0.564$ ,  $j=1,\dots ,8$ ,  $r_j=0.566$ ,  $j=9,\dots ,32$ ,  $r_j=0.592,$$j=33,\dots ,56$ ,  $r_j=0.612$ ,  $j=57,\dots ,80$ ,  $r_j=0.68$ ,  $j=81,\dots ,104$ ,  $r_j=0.747$ ,  $j=105,\dots ,128$ ,  $r_j=0.76$ ,  $j=129,\dots ,152$ ,  $r_j=0.807$ ,  $j=153,\dots ,176$ ,  $r_j=0.845$ ,  $j=177,\dots ,200$
Instance #4 output	$h^{*}=16.1158$		time 924 s		Optimized packing
Instance #5 input	$n=4$	$a=2$	$b=5$	$m=200$
$r_j=0.564,$$j=1,\dots ,8$ ,  $r_j=0.566$ ,  $j=9,\dots ,32$ ,  $r_j=0.592$ ,  $j=33,\dots ,56$ ,  $r_j=0.612$ ,  $j=57,\dots ,80$ ,  $r_j=0.68$ ,  $j=81,\dots ,104$ ,  $r_j=0.747$ ,  $j=105,\dots ,128$ ,  $r_j=0.76$ ,  $j=129,\dots ,152$ ,  $r_j=0.807$ ,  $j=153,\dots ,176$ ,  $r_j=0.845$ ,  $j=177,\dots ,200$
Instance #5 output	$h^{*}=$$15.3686$		time 1478 s

and

Table 2. Input and output data for ${ℂ}^{″}(h)$.

Instance #6 input	$n=4$	$a=2$	$b=5$	$h_0=-3$	$m=200$
$r_j=0.627$ ,  $j=1,\dots ,40$$r_j=0.664$ ,  $j=41,\dots ,61$ ,  $r_j=0.666$ ,  $j=62,\dots ,84$ ,  $r_j=0.712$ ,  $j=85,\dots ,104$ ,  $r_j=0.7612$ ,  $j=105,\dots ,123$ ,  $r_j=0.766$ ,  $j=124,\dots ,139$ ,  $r_j=0.784$ ,  $j=140,\dots ,162$ ,  $r_j=1.04$ ,  $j=163,\dots ,183$ ,  $r_j=1.1$ ,  $j=184\dots ,200$
Instance #6 output	$h^{*}=$$22.1659$			time 480 s		Optimized packing
Instance #7 input	$n=3$	$a=3$	$b=4.5$	$h_0=-5$	$m=300$
$r_j=0.527$ ,  $j=1,\dots ,60$ ,  $r_j=0.566$ ,  $j=61,\dots ,120$ ,  $r_j=0.892$ ,  $j=121,\dots ,180$ ,  $r_j=0.9612$ ,  $j=181,\dots ,240$ ,  $r_j=0.964$ ,  $j=241,\dots ,300$
Instance #7 output	$h^{*}=$$11.5952$			time 1490 s		Optimized packing
Instance #8 input	$n=3$	$a=4$	$b=5$	$h_0=-3$	$m=100$
${\rho }_{jk}=0.4,1\le j<k\le 100,$ ${\rho }_j=0.2, j=1,\dots ,100$ $r_j=0.527$,  $j=1,\dots ,18$,  $r_j=0.566$,  $j=19,\dots ,36$,  $r_{37}=0.8566$,  $r_{38}=0.866$,  $r_{39}=0.872$,  $r_j=0.892$,  $j=40,\dots ,57$,  $r_{58}=0.920$,  $r_j=0.9612$,  $j=59,\dots ,77$,  $r_j=0.964$,  $j=78,\dots ,95$,  $r_{96}=0.97$,  $r_{97}=1.12$,  $r_{98}=1.15$,  $r_{99}=1.27$,  $r_{100}=1.4$
Instance #8 output	$h^{*}=$$8.0642$			time 397 s		Optimized packing
Instance #9 input	$n=3$	$a=4$	$b=5$	$h_0=-3$ $m=100$
${\rho }_j=0.1\cdot r_j\cdot ((j+1)\mathrm{mod}5)$ ${\rho }_{jk}=0.1\cdot r_j\cdot ((j+k+1)\mathrm{mod}5)$ $r_j=0.527,$$j=1,\dots ,18$,  $r_j=0.566,$ $j=19,\dots ,36$,  $r_j=0.892,$$j=37,\dots ,54,$$r_j=0.9612,$$j=55,\dots ,73,$ $r_j=0.964,$$j=74,\dots ,91,$$r_{92}=0.97,$$r_{93}=0.92,$ $r_{94}=0.872,$$r_{95}=0.8566,$$r_{96}=0.872,$$r_{97}=1.12,$ $r_{98}=1.15,$$r_{99}=1.27,$$r_{100}=1.4$.
Instance #9 output	$h^{*}=$$6.5382$		time 64 s			Optimized packing
Instance #10 input	$n=5$	$a=2$	$b=3$	$h_0=-2$	$m=50$
${\mathrm{r}}_{\mathrm{j}}=0.527$ ,  $\mathrm{j}=1,\dots ,8$ ,  ${\mathrm{r}}_{\mathrm{j}}=0.566$ ,  $\mathrm{j}=9,\dots ,16$ ,  ${\mathrm{r}}_{17}=0.856$ ,  ${\mathrm{r}}_{18}=0.866$ ,  ${\mathrm{r}}_{19}=0.872$ ,  ${\mathrm{r}}_{\mathrm{j}}=0.892$ ,  $\mathrm{j}=20,\dots ,27$ ,  ${\mathrm{r}}_{28}=0.92$ ,  ${\mathrm{r}}_{\mathrm{j}}=0.9612$ ,  $\mathrm{j}=29,\dots ,37$ ,  ${\mathrm{r}}_{\mathrm{j}}=0.964$ ,  $\mathrm{j}=38,\dots ,45$ ,  ${\mathrm{r}}_{46}=0.97$ ,  ${\mathrm{r}}_{47}=1.12$ ,  ${\mathrm{r}}_{48}=1.15$ ,  ${\mathrm{r}}_{49}=1.27$ ,  ${\mathrm{r}}_{50}=1.4$
Instance #10 output	$h^{*}=$$4.6199$			time 987 s

. For each problem instance the tables indicate input data: dimension $n$ of the problem, parameters $a,b,\overline{h}$ of the container, the total number m of spheres and their radii $r_i$, and values of the distance parameters ${\rho }_{jk},{\rho }_j$ (where applicable). The tables also present an output of the algorithm: the best value of the container’s height $h^{*}$ and a corresponding total CPU time.

To the best of our knowledge, there are no computational results on optimized packing spheres in a hyperbolic container. We sincerely hope that the solutions presented here to illustrate the proposed algorithmic approach can serve as a benchmark for future research.

To characterize computational time savings achieved by decomposition, several problem instances were solved with/without using the decomposition approach for the value of the decomposition parameter d = 1.5. The following number of spheres, m = 50, 100, 200, 300, and dimensions n = 3, 4, 5 were considered. For each problem instance the CPU time savings are defined as (CPU time without decomposition)/(CPU time with decomposition) and indicated in parentheses after the number of spheres. Notation ($-$) indicates that the solution of the corresponding instance without decomposition was not obtained within the three-hour CPU time limit. The results can be summarized as follows: n = 3, m = 50(4), 100(10), 200(50), 300(200); n = 4, m = 50(3), 100(9), 200(40), 300($-$); and n = 5, m = 50(2.5), 100(8), 200(3.5), 300($-$). We can see that the effect of the decomposition approach for n = 3 significantly increases with the number of spheres. However, for higher dimensions n = 4,5 decomposition savings are not so high. We can expect that the reason is that the kissing number [37] is increased for higher dimensions, thus increasing the number of active constraints in pairwise restrictions of nonoverlapping and complicating the local optimization.

7. Conclusions

A novel modeling and solution approach for packing multidimensional spheres in a minimum-height hyperbolic container is proposed. The most challenging part of the corresponding mathematical model is formulating containment conditions for spheres in an n-dimensional hyperboloid. Using the phi-function approach these conditions are defined explicitly, and a corresponding large-scale nonlinear programming problem is formulated. To cope with large dimensions, a decomposition approach is used in combination with a nonlinear optimization technique. Numerical results demonstrate efficiency of the proposed approach for different problem parameters.

The proposed solution approach is focused on obtaining a local solution of the main model. Using optimization techniques to enhance solutions, such as a multistart strategy, a jump algorithm, or a lattice-based algorithm (see, e.g., [34,38]), is a challenging area for future research. Note that the main mathematical models (9), (10) and (11), (12) are exact in the sense that they contain all optimal solutions of the corresponding problems. However, the proposed solution approach, being universal and serving for any dimension of the problem, is in fact a model-based heuristic aimed at obtaining reasonable feasible solutions. Thus, constructing more efficient exact and heuristic solution approaches for the main models is an interesting direction for future research. Solutions obtained by the proposed technique can serve for benchmarking and comparison. Since global optimality of the solutions is not guaranteed, deriving lower bounds for the optimal height $h^{*}$ is critical to estimate suboptimality. Some results in this direction are on the way.

The assumption that all objects have spherical shapes is rather restrictive and is not fulfilled in many real-world applications. Revising the main models to consider polyhedral objects is necessary to make the models more realistic.

Although the main models are valid for any parameters a, b of the hyperbolic container, it would be interesting to analyze sensitivity of the proposed solution approach to these parameters, e.g., in terms of computational time and the quality of solutions.

It is interesting to explore the optimized packing of spheres, defined by arbitrary norms [39], and irregular objects composed of a union of spheres [40]. Thus, packing more general shapes [41,42,43,44] can be studied without significant changes in the original model. Some results in this direction are on the way.

In many applications the objects must be arranged as distant from each other as possible, thus giving rise to the concept of the sparsest packing [45]. This is especially important when the packing problem corresponds to designing hollowing structures, e.g., in additive manufacturing. Incorporating these sparsest packing conditions in the general model is an interesting area for future research.