A method combining genetic algorithm with simultaneous perturbation stochastic approximation for linearly constrained stochastic optimization problems

Abstract

This paper considers the optimization of linearly constrained stochastic problem which only noisy measurements of the loss function are available. We propose a method which combines genetic algorithm (GA) with simultaneous perturbation stochastic approximation (SPSA) to solve linearly constrained stochastic problems. The hybrid method uses GA to search for optimum over the whole feasible region, and SPSA to search for optimum at local region. During the GA and SPSA search process, the hybrid method generates new solutions according to gradient projection direction, which is calculated based on active constraints. Because the gradient projection method projects the search direction into the subspace at a tangent to the active constraints, it ensures new solutions satisfy all constraints strictly. This paper applies the hybrid method to nine typical constrained optimization problems and the results coincide with the ideal solutions cited in the references. The numerical results reveal that the hybrid method is suitable for multimodal constrained stochastic optimization problem. Moreover, each solution generated by the hybrid method satisfies all linear constraints strictly.

Appendix

• 1. Test function 1

  $$\begin{aligned} \begin{array}{l@{\quad }l} \min &{} f = - 3{\left( {1 - {x_1}} \right) ^2}{e^{ - x_1^2 - {{\left( {{x_2} + 1} \right) }^2}}} + 10\left( {0.2{x_1} - x_1^3 - x_2^5} \right) {e^{ - x_1^2 - x_2^2}}\\ &{} + \frac{1}{3}{e^{{{\left( {1 - {x_1}} \right) }^2} - x_2^2}}\\ \hbox {Subject to: } &{} {x_1} - {x_2} \ge - 6,\, {x_1} + {x_2} \ge - 6,\, - 4 \le {x_1} \le 4,\, - 4 \le {x_2} \le 4. \end{array} \end{aligned}$$

  The global solution is ${x^ * } = \left( {\begin{array}{cc} {0.000474}&{1.580535} \end{array}} \right)$, and $f\left( x^* \right) = - 8.105417$.

• 2. Test function 2

  $$\begin{aligned} \begin{array}{ll} \min &{} f = \left( {1 + {{\left( {{x_1} + {x_2} + 1} \right) }^2}\left( {19 - 14{x_1} + 3x_1^2 + 6{x_1}{x_2} + 3x_2^2} \right) } \right) \\ &{} \times \left( {30 + {{\left( {2{x_1} - 3{x_2}} \right) }^2}\left( {18 - 32{x_1} + 12x_1^2 + 48{x_2} - 36{x_1}{x_2} + 27x_2^2} \right) } \right) \\ \hbox {Subject to:} &{} - 2 < {x_i} < 2,\, i = 1,2. \end{array} \end{aligned}$$

  The global solution is ${x^ * } = \left( {0, - 1} \right)$, and $f\left( {{x^ * }} \right) = 3$.

• 3. Test function 3

  $$\begin{aligned} \begin{array}{ll} \min &{} f = {\left( {{x_2} - \frac{5.1}{4{\pi ^2}}x_1^2 + \frac{5}{\pi }{x_1} - 6} \right) ^2} + 10\left( {1 - \frac{1}{{8\pi }}} \right) \cos \left( {{x_1}} \right) + 10 \\ \hbox {Subject to:} &{} - 5 \le {x_1} \le 10,\, 0 \le {x_2} \le 15 \end{array} \end{aligned}$$

  The problem has three global solutions:

  $$\begin{aligned} x_1^ * = \left( { - \pi ,12.275} \right) ,\, x_2^ * = \left( {\pi ,2.275} \right) ,\, x_3^ * = \left( {3\pi ,2.475} \right) . \end{aligned}$$

  In all cases $f\left( {x_i^ * } \right) = \frac{5}{{4\pi }},\, i = 1,2,3$.

• 4. Test function 4

  $$\begin{aligned} \begin{array}{ll} \min &{} f = \left\{ {\begin{array}{l} {{x_2} + {{10}^{ - 5}}{{\left( {{x_2} - {x_1}} \right) }^2} - 1,0 \le {x_1} < 2}\\ {\frac{1}{{27\sqrt{3} }}\left( {{{\left( {{x_1} - 3} \right) }^2} - 9} \right) x_2^3,2 \le {x_1} < 4}\\ {\frac{1}{3}{{\left( {{x_1} - 2} \right) }^3} + {x_2} - \frac{11}{3},4 \le {x_1} < 6} \end{array}} \right. \\ \hbox {subject to:} &{} \frac{1}{\sqrt{3} }{x_1} - {x_2} \ge 0,\, - {x_1} - \sqrt{3} {x_2} + 6 \ge 0,\, 0 \le {x_1} \le 6,\, {x_2} \ge 0. \end{array} \end{aligned}$$

  The problem has three global solutions: $x_1^ * = \left( {0,0} \right) ,\, x_2^ * = \left( {3,\sqrt{3} } \right) ,\, x_3^ * = \left( {4,0} \right)$. in all cases $f\left( {x_i^ * } \right) = - 1,\, i = 1,2,3$.

• 5. Test function 5

  $$\begin{aligned} \begin{array}{ll} \min &{} f = \left( {4 - 2.1x_1^2 + \frac{1}{3}x_1^4} \right) x_1^2 + {x_1}{x_2} + \left( { - 4 + 4x_2^2} \right) x_2^2 \\ \hbox {Subject to:} &{} - 3 \le {x_1} \le 3,\, - 2 \le {x_2} \le 2. \end{array} \end{aligned}$$

  The problem has two global solutions: $x_1^ * = \left( { - 0.0898,0.7126} \right) ,\, x_2^ * = \left( {0.0898, - 0.7126} \right)$, and $f\left( {x_1^ * } \right) = f\left( {x_2^ * } \right) = - 1.0316$.

• 6. Test function 6

  $$\begin{aligned} \begin{array}{ll} \min &{} f = x_1^{0.6} + x_2^{0.6} - 6{x_1} - 4{x_3} + 3{x_4} \\ \hbox {Subject to:} &{} - 3{x_1} + {x_2} - 3{x_3} = 0,\, {x_1} + 2{x_3} \le 4,\, {x_2} + 2{x_4} \le 4,\, {x_1} \le 3,\\ &{} {x_4} \le 1,\, {x_i} \ge 0,\, i = 1,2,3,4. \end{array} \end{aligned}$$

  The best known global solution is ${x^ * } = \left( {\frac{4}{3},4,0,0} \right)$, and $f\left( {{x^ * }} \right) = - 4.5142$.

• 7. Test function 7

  $$\begin{aligned} \begin{array}{ll} \min &{} f = 6.5{x_1} - 0.5x_1^2 - {x_2} - 2{x_3} - 3{x_4} - 2{x_5} - {x_6} \\ \hbox {Subject to:} &{} {x_1} + 2{x_2} + 8{x_3} + {x_4} + 3{x_5} + 5{x_6} \le 16,\, - 8{x_1} - 4{x_2}\\ &{}\quad - 2{x_3} + 2{x_4} + 4{x_5} - {x_6} \le - 1\\ &{} 2{x_1} + 0.5{x_2} + 0.2{x_3} - 3{x_4} - {x_5} - 4{x_6} \le 24,\, 0.2{x_1}\\ &{}\quad + 2{x_2} + 0.1{x_3} - 4{x_4} + 2{x_5} + 2{x_6} \le 12\\ &{} - 0.1{x_1} - 0.5{x_2} + 2{x_3} + 5{x_4} - 5{x_5} + 3{x_6} \le 3,\, {x_4} \le 1,\\ &{}\quad {x_5} \le 1,\, {x_6} \le 2,\, {x_i} \ge 0,\, i = 1,2,3,4,5,6. \end{array} \end{aligned}$$

  The global solution is ${x^ * } = \left( {0,6,0,1,1,0} \right)$, and $f\left( {{x^ * }} \right) = - 11$.

• 8. Test function 8

  $$\begin{aligned} \begin{array}{ll} \min &{} f = \sum \nolimits _{j = 1}^{10} {{x_j}\left( {c_j} + \ln \frac{x_j}{\sum \nolimits _{i = 1}^{10} {{x_i}}} \right) } \\ \hbox {subject to:} &{} {x_1} + 2{x_2} + 2{x_3} + {x_6} + {x_{10}} = 2,\, {x_4} + 2{x_5} + {x_6} + {x_7} = 1, \\ &{} {x_3} + {x_7} + {x_8} + 2{x_9} + {x_{10}} = 1,\, {x_i} \ge 0.000001,\, i = 1, \cdots ,10. \end{array} \end{aligned}$$

  Where $c = ( - 6.089, - 17.164, - 34.054, - 5.914, - 24.721, - 14.986, - 24.1, - 10.708, - 26.663, - 22.179)$. The global solution is

  $$\begin{aligned} {x^ * }&= (0.04034785,0.15386976,0.77497089,0.00167479,0.48468539,\\&\quad 0.00068965,0.02826479,0.01849179,0.03849563,0.10128126),\\&\quad \hbox { and } f\left( {{x^ * }} \right) = - 47.760765. \end{aligned}$$

• 9. Test function 9

  $$\begin{aligned} \begin{array}{ll} \min &{} f\left( x \right) = \sum \nolimits _{k = 0}^4 \left( 2.3{x_{3k + 1}} + 0.0001x_{3k + 1}^2 + 1.7{x_{3k + 2}}\right. \\ &{}\quad \quad \quad \quad \ \left. +\, 0.0001x_{3k + 2}^2 + 2.2{x_{3k + 3}} + 0.0001x_{3k + 3}^2\right) \\ \hbox {subject to:} &{} 0 \le {x_{3k + 1}} - {x_{3k - 2}} + 7 \le 13,\, 0 \le {x_{3k + 2}} - {x_{3k - 1}} + 7 \le 14 \\ &{} 0 \le {x_{3k + 3}} - {x_{3k}} + 7 \le 13,\, {x_1} + {x_2} + {x_3} - 60 \ge 0 \\ &{} {x_4} + {x_5} + {x_6} - 50 \ge 0, {x_7} + {x_8} + {x_9} - 70 \ge 0 \\ &{} {x_{10}} + {x_{11}} + {x_{12}} - 85 \ge 0, {x_{13}} + {x_{14}} + {x_{15}} - 100 \ge 0 \\ &{} 8 \le {x_1} \le 21, 43 \le {x_2} \le 57, 3 \le {x_3} \le 16 \\ &{} 0 \le {x_{3k + 1}} \le 90, 0 \le {x_{3k + 2}} \le 120, 0 \le {x_{3k + 3}} \le 60, k = 1, \cdots ,4 \end{array} \end{aligned}$$

  The global solution is

  $$\begin{aligned} {x^ * } = \left( {8,49,3,1,56,0,1,63,6,3,70,12,5,77,18} \right) , \hbox { and } f\left( {{x^ * }} \right) = 664.82045. \end{aligned}$$