Feasible Direction

An optimization calculation usually consists of the minimization (or maximization) of one objective subject to one or more constraints. To that end the sensitivity of the objective is modified to take the constraint sensitivities into account leading to one combined sensitivity. This process takes place in a *FEASIBLE DIRECTION procedure. The ensuing sensitivity determines where the structure has to be thickened or thinned in order to reach a minimum of the objective function while not violating any constraint. Constraints can be taken into account by using the Gradient Projection Method or the Gradient Descent Method.

In the Gradient Projection Method the constraints are taken into account by projecting the unconstrained sensitivities on the complement of the subspace consisting of the normals on the active constraint hyperplanes. Suppose the domain is n-dimensional (n design variables) and the subspace has the dimension m (m constraints). Then the sensitivities of the constraints can be arranged as basis vectors (the normal directions on the hyperplanes) in a $n \times m$ matrix. The projection p of a vector b on the subspace satisfies the orthogonality condition:

$$N^T (b-p)=0.$$ (815)

Since p belongs to the subspace it can be written as a linear combination of the basis vectors $p=Nx$, where $x$ is a $m \times 1$ vector of coefficients. Consequently:

$$N^T N x = N^T b,$$ (816)

from which $x$ can be solved yielding:

$$p=N(N^T N)^{-1} N^T b.$$ (817)

The complement of the projection vector is $I-N(N^T N)^{-1} N^T$. Denoting $A=(N^T N)^{-1}$, the constrained sensitivies $c$ are obtained from the unconstrained sensitivities $b$ by:

$$c=(I-NAN^T) b,$$ (818)

or, in component notation:

$$c_i=b_i-\sum_k w_{ik},$$ (819)

where

$$w_{ik}=\left( \sum_j N_{ij} A_{j \underline{k}} \right) \left( \sum_l (N^T)_{\underline{k}l} b_l \right)$$ (820)

(no summation over k in the last equation).

Active constraints are constraints which

• are fulfilled AND
• for which the Lagrange multiplier points to the non-feasible part of the space

To this end the algorithm starts with all constraints which are fulfilled and removes the constraints one-by-one for which the Lagrange multiplier, satisfying

$$\boldsymbol{\lambda } = -(\boldsymbol{N}^T \boldsymbol{N})^{-1} \boldsymbol{N}^T \nabla g_i.$$ (821)

points to the feasible part of the space.

The Gradient Descent Method is an interior point method which means that the starting point for an optimization has to lie in the feasible domain. The influence of the constraints on the feasible direction is always taken into account independent of the distance to the constraint bound. This methods aims to stay away from the constraint boundaries as long as possible on the way to the local optimum. The feasible direction is calculated with the following formulas (the detailed derivation can be found in [18]):

$$\boldsymbol{s}_{\xi}= - \frac{\nabla f}{\vert\vert \nabla f \vert\vert} - \xi \frac{\nabla g}{\vert\vert \nabla g \vert\vert}.$$ (822)

Here, $\nabla f$ is the gradient of the objective function, $\nabla g$ is the gradient of the constraint and $\xi$ is a value between 0 and 1 defining the weight of the constraint gradient on the descent direction. Choosing a value for $\xi$ between 0.9 and 1.0 leads to meaningful results. By subtracting a portion of the gradient of the constraint one tends to stay away from the constraint boundary (the constraint becomes more negative). In case more than one constraint is defined, the assembly of the gradient vector of all constraint functions $n$ is done using the logarithmic Barrier function:

$$\nabla g = \sum_{i=1}^{n} \frac{1}{-g_i(x)} \nabla g_i.$$ (823)