# Nonlinear Optimization and Adaptive Heuristics for Solving Irregular Object Packing Problems
## 5 Further Scalable Object Configuration Problems
### Estimating the Quality of Numerical Solutions  
In practical object-packing applications, a well-crafted, global scope search-based solution that incorporates problem-specific insight is typically quite acceptable. The illustrative results visually presented here and in our referenced works support this claim. A satisfactory numerical solution does not necessarily imply, however, that the solution found is within a guaranteed distance to the unknown best possible solution of some hard packing problem, even in a probabilistic sense (our preceding discussion related to the \"simplest\" GCP models illustrates this point). Therefore, estimating the quality of conjectured solutions is a relevant issue that we briefly address here.  
A simple \"brute force\" approach to obtaining solution estimates can be based on solving the same packing problem a number of times using some randomization mechanism and recording the best solution found as an estimate of the unknown solution. This general approach can be theoretically validated by standard stochastic convergence arguments (assuming that the sampling procedure is theoretically exhaustive in the decision variable space, and that the sample size can be arbitrarily increased as required). Randomized global optimization methods have been extensively studied: consult, e.g., [PERSON] [50], [PERSON] [65], [PERSON] and [PERSON] [66].  
Sophisticated extensions of the outlined global sampling approach can be based on applying the theory of extreme-order statistics and utilizing the sampling distribution of the optimum estimates. For details, we refer to [PERSON] and [PERSON] [66], [PERSON] and [PERSON] [67].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Square Region**} & \multicolumn{2}{c}{**Hexagonal Region**} \\ \cline{3-6}
**Values of \(p\), \(r\)** & **Test Case** & **Objective** & **Time (s)** & **Objective** & **Time (s)** \\ \hline \(p=3,r=1.5\) & 1 & 0.021305 & 8.4 & 0.028127 & 12.8 \\  & 2 & 0.070798 & 10.1 & 0.090017 & 13.4 \\  & 3 & 0.072242 & 15.2 & 0.090042 & 28.5 \\  & 4 & 0.023848 & 20.7 & 0.029633 & 28.1 \\  & 5 & 0.073014 & 17 & 0.090184 & 23.4 \\  & 6 & 0.065005 & 85.6 & 0.070549 & 78.5 \\ \(p=10,r=1.0\) & 1 & 0.025543 & 212.6 & 0.027923 & 228.4 \\  & 2 & 0.048605 & 277 & 0.052259 & 253.4 \\  & 3 & 0.0492 & 398 & 0.05289 & 520.6 \\  & 4 & 0.028027 & 347.7 & 0.029664 & 540.4 \\  & 5 & 0.050149 & 386.3 & 0.054299 & 515.5 \\  & 6 & 0.040088 & 1617.8 & 0.043147 & 2186.6 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Illustrative PDO model results.  
Figure 5: Two solved PDO model instances.  
In certain scalable models, optimality bounds can be based on considering the sequence of solved model instances (assuming the availability of sufficiently high-quality solutions). Let us also note that for certain scalable model types, regression models can be developed that support the estimation of the optimum value. This becomes especially useful for estimating the optimum value in higher-dimensional model instances that may be intractable by currently available software. To illustrate this approach, we refer to [PERSON] et al. [55].