# Nonlinear Optimization and Adaptive Heuristics for Solving Irregular Object Packing Problems
###### Abstract  
We review and present several challenging model classes arising in the context of finding optimized object packings (OP). Except for the smallest and/or simplest general OP model instances, it is not possible to find their exact (closed-form) solution. Most OP problem instances become increasingly difficult to handle even numerically, as the number of packed objects increases. Specifically, here we consider classes of general OP problems that can be formulated in the framework of nonlinear optimization. Research experience demonstrates that--in addition to utilizing general-purpose nonlinear optimization solver engines--the insightful exploitation of problem-specific heuristics can improve the quality of numerical solutions. We discuss scalable OP problem classes aimed at packing general circles, spheres, ellipses, and ovals, with numerical (conjectured) solutions of non-trivial model instances. In addition to their practical relevance, these models and their various extensions can also serve as constrained global optimization test challenges.  
optimized object packings; general (irregular) packings; scalable models of packing circles, spheres, ellipses, and ovals; model implementations; nonlinear optimization; heuristics; illustrative numerical results +
Footnote †: journal:  
0000-0001-111  
Our article focuses on packing problems with a finite number of general (in principle, arbitrary) objects to be packed into various types of finite-size containers. Such models are often somewhat informally referred to as irregular packings. Irregular packing needs arise in many practical applications such as additive manufacturing, communication network design, crystallography, dashboard layout design, engineering design, facility dispersion, facility layout, facility location, fiber optic cabling (and other types of cabling) design, furniture making, garment and shoe manufacturing, material design, optimized (bin, cargo, container, pallet, vehicle) loading, and raw material cutting. Important and hard-to-handle additional considerations can be related to load configuration, balancing, loading preferences, sequencing, and scheduling aspects. Without going into further details here, consult, e.g., [PERSON] et al. [6], [PERSON] and [PERSON] [7], and [PERSON] [8], with extensive reference lists.  
Let us also mention some specific examples of OP applications. The contributing authors of the volume edited by [PERSON] and [PERSON] [9] discuss the packing of hazardous containers on ships, dynamic packings for datacenter resource management, the optimized packing of free-form objects in engineering design, non-standard packing problems arising in space engineering applications, cutting and packing problems with placement constraints, container loading using heuristics, the design of optimal LED streetlights, approximate packing approaches, robust designs for circle coverings of a square, the optimized packing of jobs in spatial scheduling, optimized object packings using quasi-\(\Phi\)-functions, graph-coloring models, and metaheuristics for packing applications.