Penalty functions and two-step selection procedure based DIRECT-type algorithm for constrained global optimization

Abstract

Applied optimization problems often include constraints. Although the well-known derivative-free global-search DIRECT algorithm performs well solving box-constrained global optimization problems, it does not naturally address constraints. In this article, we develop a new algorithm DIRECT-GLce for general constrained global optimization problems incorporating two-step selection procedure and penalty function approach in our recent DIRECT-GL algorithm. The proposed algorithm effectively explores hyper-rectangles with infeasible centers which are close to boundaries of feasibility and may cover feasible regions. An extensive experimental investigation revealed the potential of the proposed approach compared with other existing DIRECT-type algorithms for constrained global optimization problems, including important engineering problems.

Appendices

Appendix A: The mathematical formulations of engineering problems

NASA speed reducer design problem (Liu et al. 2017; Ray and Liew 2003).

The overall weight subject to constraints on bending stress of the gear teeth, surface stress, and transverse deflections of the shafts and stresses in the shafts is minimized. This problem has 7 design variables and 11 constraints. The optimization problem is formulated as following:

$$ \begin{array}{ll} \min & f(\mathbf{x}) = 0.7854x_{1}{x_{2}^{2}}(3.3333{x_{3}^{2}} + 14.9334x_{3} - 43.0934) \\ & -1.508x_{1}({x_{6}^{2}}+{x_{7}^{2}})+ 7.4777({x_{6}^{3}}+{x_{7}^{3}}) \\ & + 0.7854(x_{4}{x_{6}^{2}}+x_{5}{x_{7}^{2}}) \\ \mathrm{ s.t.} & g_{1}(\mathbf{x}) = \frac{27}{x_{1}{x_{2}^{2}}x_{3}}-1 \leq \mathbf{0},~ g_{2}(\mathbf{x})=\frac{397.5}{x_{1}{x_{2}^{2}}{x_{3}^{2}}} - 1 \leq \mathbf{0}, \\ & g_{3}(\mathbf{x})=\frac{1.93{x_{4}^{3}}}{x_{2}x_{3}{x_{6}^{4}}} - 1 \leq \mathbf{0},~ g_{4}(\mathbf{x})=\frac{1.93{x_{5}^{3}}}{x_{2}x_{3}{x_{7}^{4}}} - 1 \leq \mathbf{0},\\ & g_{5}(\mathbf{x})= \frac{((\frac{745x_{4}}{x_{2}x_{3}})^{2}+ 16.9 \times 10^{6})^{0.5}}{110{x_{6}^{3}}} - 1 \leq \mathbf{0}, \\ & g_{6}(\mathbf{x})=\frac{((\frac{745x_{5}}{x_{2}x_{3}})^{2}+ 157.5 \times 10^{6})^{0.5}}{85{x_{7}^{3}}} - 1 \leq \mathbf{0}, \\ & g_{7}(\mathbf{x})=\frac{x_{2}x_{3}}{40} - 1 \leq \mathbf{0},~ g_{8}(\mathbf{x})= \frac{5x_{2}}{x_{1}} - 1 \leq \mathbf{0}, \\ & g_{9}(\mathbf{x})=\frac{x_{1}}{12x_{2}} - 1 \leq \mathbf{0},~ g_{10}(\mathbf{x})=\frac{1.5x_{6}+ 1.9}{x_{4}} - 1 \leq \mathbf{0}, \\ & g_{11}(\mathbf{x})=\frac{1.1x_{7}+ 1.9}{x_{5}} - 1 \leq \mathbf{0} \end{array} $$

where 2.6 ≤ x₁ ≤ 3.6, 0.7 ≤ x₂ ≤ 0.8, 17 ≤ x₃ ≤ 28, 7.3 ≤ x₄ ≤ 8.3, 7.8 ≤ x₅ ≤ 8.3, 2.9 ≤ x₆ ≤ 3.9, 5 ≤ x₇ ≤ 5.5.

Table 10 Key characteristics of the optimization test problems with equality constraints

Pressure vessel design problem (Kazemi et al. 2011; Liu et al. 2017).

The total cost of material, forming, and welding of a cylindrical vessel is minimized. This problem has four design variables and six constraints The optimization problem formulated as following:

$$ \begin{array}{ll} \min & f(\mathbf{x})= 0.6224x_{1}x_{3}x_{4}+ 1.7781x_{2}{x_{3}^{2}}+ 3.1661{x_{1}^{2}}x_{4} \\ & + 19.84{x_{1}^{2}}x_{3}\\ \mathrm{ s.t.} & g_{1}(\mathbf{x})= -x_{1}+ 0.0193x_{3} \leq \mathbf{0}, \\ & g_{2}(\mathbf{x})= -x_{2}+ 0.00954x_{3} \leq \mathbf{0}, \\ & g_{3}(\mathbf{x})=- \pi {x_{3}^{2}}x_{4}- \frac{4}{3} \pi {x_{3}^{3}}+ 1296000 \leq \mathbf{0}, \\ & g_{4}(\mathbf{x})= x_{4}-240 \leq \mathbf{0},~ g_{5}(\mathbf{x})= 1.1-x_{1} \leq \mathbf{0}, \\ & g_{6}(\mathbf{x})= 0.6-x_{2} \leq \mathbf{0} \end{array} $$

where 1 ≤ x₁ ≤ 1.375, 0.625 ≤ x₂ ≤ 1, 25 ≤ x₃ ≤ 150, 25 ≤ x₄ ≤ 240.

Tension/compression spring design problem (Kazemi et al. 2011; Liu et al. 2017).

The weight subject to constraints on minimum deflection, shear stress, surge frequency, and limits on outside diameter is minimized. This problem has three design variables and four constraints. The optimization problem formulated as following:

$$ \begin{array}{ll} \min & f(\mathbf{x})={x_{1}^{2}}x_{2}(x_{3}+ 2)\\ \mathrm{ s.t.} & g_{1}(\mathbf{x})= 1- \frac{{x_{2}^{3}}x_{3}}{71875{x_{1}^{4}}} \leq \mathbf{0}, \\ & g_{2}(\mathbf{x})= \frac{x_{2}(4x_{2}-x_{1})}{12566{x_{1}^{3}}(x_{2}-x_{1})}+ \frac{2.46}{12566{x_{1}^{2}}}-1 \leq \mathbf{0}, \\ & g_{3}(\mathbf{x})= 1- \frac{140.54x_{1}}{x_{3}{x_{2}^{2}}} \leq \mathbf{0},~ g_{4}(\mathbf{x})= \frac{x_{1}+x_{2}}{1.5}-1 \leq \mathbf{0} \end{array} $$

where 0.05 ≤ x₁ ≤ 0.2, 0.25 ≤ x₂ ≤ 1.3, 2 ≤ x₃ ≤ 15.

Three-bar truss design problem (Liu et al. 2017; Ray and Liew 2003).

The volume subject to stress constraints is minimized. This problem has two design variables and three constraints. The optimization problem formulated as following:

$$ \begin{array}{ll} \min & f(\mathbf{x})= 100(2 \sqrt{2} x_{1}+x_{2})\\ \mathrm{ s.t.} & g_{1}(\mathbf{x})= \frac{\sqrt{2} x_{1}+x_{2}}{\sqrt{2} {x_{1}^{2}}+ 2x_{1}x_{2}}2-2 \leq \mathbf{0}, \\ & g_{2}(\mathbf{x})= \frac{x_{2}}{\sqrt{2} {x_{1}^{2}}+ 2x_{1}x_{2}}2-2 \leq \mathbf{0}, \\ & g_{3}(\mathbf{x})= \frac{1}{x_{1}+\sqrt{2} x_{2}}2-2 \leq \mathbf{0} \end{array} $$

where 0 ≤ x₁ ≤ 1, 0 ≤ x₂ ≤ 1.

Appendix B: Test problems with linear and nonlinear constraints

Table 11 Key characteristics of the constrained global optimization test problems