An efficient optimization model and tabu search-based global optimization approach for continuous p-dispersion problem

TL;DR

This work tackles the continuous p-dispersion problem (CpDP) in non-convex multiply-connected regions by introducing a unified penalty-based optimization model that renders an almost-everywhere differentiable energy $E_D(X)$ and a tabu-search-based global optimizer (TSGO). By fixing a target distance $D$ and solving a sequence of unconstrained subproblems, the approach enables efficient use of gradient-based local solvers while enforcing containment and separation through penalty terms, including $l_{ij}$ and $O_{i0},O_{ik}$. The TSGO framework combines a vacancy-based insertion tabu search with a short MBH refinement and a SUMT-derived distance adjustment to progressively maximize $D$ while maintaining feasibility. Across benchmark instances for equal circle packing and point arrangement, TSGO consistently outperforms state-of-the-art methods, achieving high-precision solutions and new best-known configurations, with open-source code provided for replication and broader use.

Abstract

Continuous p-dispersion problems with and without boundary constraints are NP-hard optimization problems with numerous real-world applications, notably in facility location and circle packing, which are widely studied in mathematics and operations research. In this work, we concentrate on general cases with a non-convex multiply-connected region that are rarely studied in the literature due to their intractability and the absence of an efficient optimization model. Using the penalty function approach, we design a unified and almost everywhere differentiable optimization model for these complex problems and propose a tabu search-based global optimization (TSGO) algorithm for solving them. Computational results over a variety of benchmark instances show that the proposed model works very well, allowing popular local optimization methods (e.g., the quasi-Newton methods and the conjugate gradient methods) to reach high-precision solutions due to the differentiability of the model. These results further demonstrate that the proposed TSGO algorithm is very efficient and significantly outperforms several popular global optimization algorithms in the literature, improving the best-known solutions for several existing instances in a short computational time. Experimental analyses are conducted to show the influence of several key ingredients of the algorithm on computational performance.

Figures (11)

• Figure 1: Degree of constraint violation ($l_{ij}$) on the distance between two points $c_i$ and $c_j$, where the radius of two circles is $\frac{D}{2}$.
• Figure 2: The active edges and some representative foot points for the case that the dispersion point $c_i$ lies in the container, where the active edges are indicated in green and the circle is $B(c_i,D_b)$.
• Figure 3: The active edges and some representative foot points for the case that the dispersion point $c_i$ lies outside the container, where the active edges are indicated in green and the circle is $B(c_i,D_b)$.
• Figure 4: The active edges and some representative foot points for the case that the dispersion point $c_i$ lies in the hole $U_{k}$, where the active edges are indicated in green and the circle is $B(c_i,D_b)$.
• Figure 5: The active edges and some representative foot points for the case that the dispersion point $c_i$ lies outside the hole $hole_{k}$, where the active edges are indicated in green and the circle is $B(c_i,D_b)$.