A multi-stage descent algorithm for discrete and continuous optimization applied to truss structures optimal design

Abstract

In this paper the step by step descent algorithm is extended to continuous optimization problems and applied to the optimal sizing of truss structures. The step by step descent method, recently developed, was applied to the discrete optimization of rigid structures. To solve continuous problems, a multi-stage optimization algorithm is proposed, which involves implementing the step by step descent method in multiple stages. In each stage, the continuous areas of the variables are progressively divided into a finite number of points. The algorithm is based on the search for the steepest gradient descent direction that minimizes the cost function and employs a wise heuristic approach to bypassing local optima. To evaluate the algorithm's performance, a study was carried out on ten mathematical and four truss optimization problems. For unconstrained mathematical problems, the algorithm demonstrates quick converges to optimum solution from any starting point. For truss structures, the starting design point is selected based on structural knowledge to enhance convergence speed. The obtained results surpass those achieved by other state-of-the-art optimization algorithms, affirming the efficiency, robustness and the promptness of the proposed method.

Appendix

• 1. Two-dimensional Rastrigin function

$\begin{gathered} {\text{min}}\;f\left( {x_{1} , \, x_{2} } \right)\, = \,x_{1}^{2} \, + \,x_{2}^{2} - cos(18x_{1} ) - cos(18x_{2} ) \hfill \\ s.t. - 2\, \le \,x_{i} \, \le \,2,\;i\, = \,1, \, 2. \hfill \\ f^{*} \, = \, - 2\;{\text{at }}\left( {0, \, 0} \right) \hfill \\ \end{gathered}$

• 2. Two-dimensional Six-Hump Camel-back

$\begin{gathered} {\text{min}}f\left( {x_{1} , \, x_{2} } \right)\, = \,{4}x_{1}^{{2}} - {2}.{1}x_{1}^{{4}} \, + \,x_{1}^{{6}} /{3}\, + \,x_{1} x_{2} - {4}x_{2}^{{2}} \, + \,{4}x_{2}^{{4}} \hfill \\ {\text{s}}.{\text{t}}. \, - {3}\, \le \,x_{i} \, \le \,{3},\;i\, = \,{1},{2}. \hfill \\ f^{*} \, = \, - {1}.0{\text{316284534 at }}\left( {0.0{898422}, \, - 0.{7126566}} \right){\text{ and }}( - 0.0{898422}, \, 0.{7126566}) \hfill \\ \end{gathered}$

• 3. Treccani function

• $\begin{gathered} {\text{min}}\;f(x_{1} ,x_{2} )\, = \,x_{1}^{{4}} \, + \,{4}x_{1}^{{2}} \, + \,{4}x_{1}^{{2}} \, + \,x_{2}^{{2}} \hfill \\ {\text{s}}.{\text{t}}. \, - {3}\, \le \,x_{1} \, \le \,{3}\;{\text{and}}\, - \,{3}\, \le \,x_{2} \, \le \,{3}. \hfill \\ f^{*} \, = \,0{\text{ at }}\left( { - {2}, \, 0} \right){\text{ and }}(0,0) \hfill \\ \end{gathered}$

• 4. Two-dimensional Shubert II function

  min f(x₁, x₂) = $\left(\sum_{i=1}^{5}i\mathrm{ cos}\left[\left(i + 1\right){\mathrm{x}}_{ 1}+ i \right]\right)\left(\sum_{i=1}^{5}i\mathrm{ cos}\left[\left(i + 1\right){\mathrm{x}}_{ 2} + i\right]\right)$

  s.t. 0 ≤ x_i ≤ 10, i = 1, 2.

  f^* = − 186.730907998

• 5. n-dimensional Sine-square II function

  min f(X) = $\frac{\uppi }{\mathrm{n}}\Bigg\{10{\mathrm{sin}}^{2}\left(\pi {y}_{1}\right)+ {\left({y}_{n}- 1\right)}^{2}+{\sum }_{\mathrm{i}=1}^{\mathrm{n}-1}\Bigg[{({y}_{i} - 1)}^{2}(1 + 10{\mathrm{sin}}^{2}(\pi {y}_{i}+1)\Bigg]\Bigg\}$

  s.t. y_i = 1 + (x_i − 1)/4 and − 10 ≤ x_i ≤ 10, i = 1, 2,..., n.

  f^* = 0 at x_i.^* = 1

• 6. n-dimensional Rastrigin function

  min f(X) = 10n + $\sum_{i=1}^{n}\left[{x}_{ i}^{2}-10 cos(2\pi {x}_{i})\right]$

  s.t. − 5.12 ≤ x_i ≤ 5.12, i = 1,…,n..

  f^* = 0 at x_i.^* = 0

• 7. Branin function

  min f (x₁, x₂) = (x₂ − 1.275 x₁²/π² + 5x₁/π − 6)² + 10(1 − 0.125/π) cos x₁ + 10,

  s.t. − 5 ≤ x₁ ≤ 15, − 5 ≤ x₂ ≤ 15.

  f^* = 0.397887 at (-π, 12.275), (π, 2.275) and (9.42478, 2.475),

• 8. Eason and Fenton’s gear train inertia function

  min f (x₁, x₂) = $\left\{12+{x}_{1}^{2}+ \frac{1+{x}_{2}^{2}}{{x}_{1}^{2}} + \frac{{x}_{1}^{2}{x}_{2}^{2}+100 }{{({x}_{1}{x}_{2})}^{4}}\right\}\left(\frac{1}{10}\right)$

  s.t. 0 < x_i ≤ 10, i = 1, 2..

  f^* = 1.74 at (1.7435, 2.0297)

• 9. Wood function

  min f(X) = 100(10x₂ − x₁²)² + (1 − x₄)² + 90(x₄ − x₃²)² + (1 − x₃)² + 10.1[(x₂ − 1)² + (x₄ − 1)²] + 19.8(x₂ − 1)(x₄ − 1)

  s.t. − 5 ≤ x_i ≤ 5, i = 1,…, 4. 

  f* = 0 at x_i^* = 1

• 10. Ackley’s function

  min f(X) = − 20 exp $\left(\sqrt[-0.2]{\frac{1}{n}\sum_{i=1}^{n}{x}_{ i}^{2}}\right)$-exp $\left(\frac{1}{n}\sum_{i=1}^{n}cos(2\pi {x}_{i})\right)$+20 + exp(1)

  s.t. − 10 ≤ x_i ≤ 10, i = 1, 2,..., n.f^* = 0 at ${x}_{i}^{*}$=0