A genetic algorithm based augmented Lagrangian method for constrained optimization

Abstract

Among the penalty based approaches for constrained optimization, augmented Lagrangian (AL) methods are better in at least three ways: (i) they have theoretical convergence properties, (ii) they distort the original objective function minimally, thereby providing a better function landscape for search, and (iii) they can result in computing optimal Lagrange multiplier for each constraint as a by-product. Instead of keeping a constant penalty parameter throughout the optimization process, these algorithms update the parameters (called multipliers) adaptively so that the corresponding penalized function dynamically changes its optimum from the unconstrained minimum point to the constrained minimum point with iterations. However, the flip side of these algorithms is that the overall algorithm requires a serial application of a number of unconstrained optimization tasks, a process that is usually time-consuming and tend to be computationally expensive. In this paper, we devise a genetic algorithm based parameter update strategy to a particular AL method. The proposed strategy updates critical parameters in an adaptive manner based on population statistics. Occasionally, a classical optimization method is used to improve the GA-obtained solution, thereby providing the resulting hybrid procedure its theoretical convergence property. The GAAL method is applied to a number of constrained test problems taken from the evolutionary algorithms (EAs) literature. The number of function evaluations required by GAAL in most problems is found to be smaller than that needed by a number of existing evolutionary based constraint handling methods. GAAL method is found to be accurate, computationally fast, and reliable over multiple runs. Besides solving the problems, the proposed GAAL method is also able to find the optimal Lagrange multiplier associated with each constraint for the test problems as an added benefit—a matter that is important for a sensitivity analysis of the obtained optimized solution, but has not yet been paid adequate attention in the past evolutionary constrained optimization studies.

Appendix: Problem formulations

These problems are taken from [24].

1.1 A.1 Problem g01

The problem is given as follows:

$$\everymath{\displaystyle}\begin{array}{l@{\quad}l}\mbox{min.} & f(\mathbf{x})=5\sum_{i=1}^4 x_i-5\sum_{i=1}^4 x_i^2 + \sum_{i=5}^{13} x_i, \\[14pt]\mbox{s.t.} & g_1(\mathbf{x}) \equiv2x_1+2x_2+x_{10} + x_{11}-10\le 0,\\[3pt]&g_2(\mathbf{x}) \equiv2x_1+2x_3+x_{10}+x_{12}-10\le0,\\[3pt]&g_3(\mathbf{x}) \equiv2x_2+2x_3+x_{11}+x_{12}-10\le0,\\ [3pt]&g_4(\mathbf{x}) \equiv -8x_1+x_{10}\le0,\\ [3pt]&g_5(\mathbf{x}) \equiv -8x_2+x_{11}\le0,\\ [3pt]&g_6(\mathbf{x}) \equiv -8x_3 + x_{12}\le0,\\ [3pt]&g_7(\mathbf{x}) \equiv -2x_4-x_5+x_{10}\le0,\\ [3pt]&g_8(\mathbf{x}) \equiv -2x_6-x_7 + x_{11}\le0,\\[3pt]&g_9(\mathbf{x}) \equiv -2x_8-x_9+x_{12}\le0,\end{array}$$

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where 0≤x _i ≤1 for i=1,2,…,9, 0≤x _i ≤100 for i=10,11,12, and 0≤x ₁₃≤1.

1.2 A.2 Problem g02

The problem is given as follows:

$$\everymath{\displaystyle}\begin{array}{l@{\quad}l}\mbox{min.} & f(\mathbf{x})={-}\!\left|\frac{\sum_{i=1}^{20}\cos^4(x_i)-2\prod _{i=1}^{20}\cos^2(x_i)}{\sqrt{\sum_{i=1}^{20}ix_i^2}}\right|, \\[20pt]\mbox{s.t.} & g_1(\mathbf{x}) \equiv0.75-\prod_{i=1}^{20}x_i\le0, \\[14pt]& g_2(\mathbf{x}) \equiv\sum_{i=1}^{20}x_i-150\le0, \\[14pt]& \quad 0 \leq x_i \leq10, \quad i=1,2,\ldots,20.\end{array}$$

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1.3 A.3 Problem g04

The problem is given as follows:

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1.4 A.4 Problem g06

The problem is given as follows:

$$\everymath{\displaystyle}\begin{array}{l@{\quad}l}\mbox{min.} & f(\mathbf{x}) = (x_1 - 10)^3 + (x_2 - 20)^3, \\[3pt]\mbox{s.t.} & g_1(\mathbf{x}) \equiv-(x_1-5)^2-(x_2-5)^2 + 100\le 0, \\[3pt]& g_2(\mathbf{x}) \equiv(x_1 - 6)^2 + (x_2 - 5)^2 - 82.81 \le0, \\[3pt]&\quad 13 \le x_1 \le100, \quad 0 \le x_2 \le100.\end{array}$$

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1.5 A.5 Problem g07

The problem is given as follows:

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1.6 A.6 Problem g08

The problem is given as follows:

$$\everymath{\displaystyle}\begin{array}{l@{\quad}l}\mbox{min.} & f(\mathbf{x})= -\frac{\sin^3(2\pi x_1)\sin(2\pi x_2)}{x_1^3(x_1+x_2)}, \\[3pt]\mbox{s.t.} & g_1(\mathbf{x}) \equiv x_1^2-x_2+1\le0, \\[3pt]& g_2(\mathbf{x}) \equiv1-x_1+(x_2-4)^2\leq0, \\[3pt]& \quad 0\le x_i \le10, \quad i=1,2.\end{array}$$

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1.7 A.7 Problem g09

The problem is given as follows:

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1.8 A.8 Problem g10

The problem is given as follows:

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1.9 A.9 Problem g12

The problem is given as follows:

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The parameters p _i , q _j and r _k take one of nine interger values in [1, 9]. Thus, the minimum value of 9³ or 243 terms within brackets is used as the constraint value. As long as, a point lies inside any of the 243 spheres, the point is feasible.

1.10 A.10 Problem g18

The problem is given as follows:

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1.11 A.11 Problem g24

The problem is given as follows:

$$\everymath{\displaystyle}\begin{array}{l@{\quad}l}\mbox{min.} & f(\mathbf{x}) =-x_1 - x_2, \\[3pt]\mbox{s.t.} & g_1(\mathbf{x}) \equiv-2x^4_1 + 8x^3_1-8x^2_1+x_2-2\leq0, \\[3pt]& g_2(\mathbf{x}) \equiv -4x^4_1 + 32x^3_1 - 88x^2_1 + 96x_1 + x2 - 36 \leq0, \\[3pt]& \quad 0 \leq x_1 \leq3, \quad0 \leq x_2 \leq4.\end{array}$$

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1.12 A.12 Problem weld

The problem is given as follows (x=(h,l,t,b)^T) [10, 31]:

$$\everymath{\displaystyle}\begin{array}{l@{\quad}l}\mbox{min.} & f_1(\mathbf{x})=1.10471h^2l+0.04811tb(14.0+l), \\[3pt]\mbox{s.t.} & g_1(\mathbf{x})\equiv13{,}600-\tau(\mathbf{x}) \geq0,\\[3pt]& g_2(\mathbf{x})\equiv30{,}000-\sigma(\mathbf{x}) \geq0, \\[3pt]& g_3(\mathbf{x})\equiv b-h \geq0, \\[3pt]& g_4(\mathbf{x})\equiv P_c(\mathbf{x})-6{,}000 \geq0, \\[3pt]& g_5(\mathbf{x})\equiv0.25-\delta(\mathbf{x}) \geq0, \\[3pt]&\quad 0.125 \leq (h, b) \leq5, \quad 0.1 \leq (l, t) \leq10,\end{array} $$

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where,