Large Scale Optimization Algorithm (BIGOPT)

A gradient-based method that consumes less memory and is relatively more computationally efficient, compared to MFD and SQP.

Implementation

Consider an optimization problem that involves minimizing $f (x)$ based on a set of constraints. It is described as:

\begin{array}{l} \min [f (x)] \\ g_{i} (x) = 0, i = 1, ..., m_{e} \\ g_{i} (x) \leq 0, i = (m_{e} + 1), ..., m_{e} \\ x^{L} \leq x \leq x^{U} \end{array}

Where, $f (x)$ is the objective function, $g_{i} (x)$ is the i’th constraint function, $m_{e}$ is the number of equality constraints, $m$ is the total number of constraints, $x$ is the design variable vector, and $x^{L}$ and $x^{U}$ are the lower and upper bound vectors of design variables, respectively.

If BIGOPT is selected, OptiStruct converts this problem to an equivalent problem using the penalty method, as:

\begin{array}{l} \min Φ (x) = f (x) + r (\sum_{i = 1}^{m_{e}} q_{i} {[g_{i} (x)]}^{2} + \sum_{i = m_{e} + 1}^{m} q_{i} {m a x [0, g_{i} (x)]}^{2}) \\ x^{L} \leq x \leq x^{U} \end{array}

Where, $r$ and $q_{i}$ are penalty multipliers.

BIGOPT considers the bound constraints separately, so the original problem is converted to an unconstrained problem. Polak-Ribiere conjugate gradient method is used to generate search direction. After the search direction is calculated, a one-dimensional search can be accomplished by parabolic interpolation (Brent’s method).

Terminating Conditions

Optimization runs, based on a BIGOPT algorithm, will be terminated if one of the following conditions is met:

$‖ \nabla Φ ‖ \leq ε$ and the design is feasible.
$\frac{2 | Φ (x^{k + 1}) - Φ (x^{k}) |}{| Φ (x^{k + 1}) | + | Φ (x^{k}) |} \leq ε$ and the design is feasible.
The number of iteration steps exceeds $N_{\max}$ .

Where, $\nabla Φ$ is the gradient of $Φ (x)$ , $k$ is the $k$ ’th iteration step, $εε$ is the convergence parameter, and $N_{\max}$ is the allowed maximum iterations.