August 9, 2017

New Direct SBO technique for design optimization in pSeven

In optimization, our goal is to find the most suitable variant of design among others. To compare designs, we evaluate responses — objectives and constraints that quantify designs quality and validity.

Time-consuming evaluation and non-smoothness of responses are common features for many engineering problems. These features underpin most of the strategies that either rely on the local smoothness or require a lot of unpractical evaluations.

In pSeven, we provide a unique family of optimization methods — Surrogate-Based Optimization (SBO) — that pretends to be a relief for such engineering problems. SBO is based on the idea to replace time-consuming computational models with lightweight approximation ones (shown in Figure 1).

Figure 1. The general framework of surrogate-based optimization

To build approximation models in pSeven, the Optimizer block uses a hierarchical variant of GP (Gaussian processes) specifically tuned for large-scale optimization problems which makes it possible to apply SBO to solve the problems with hundreds of variables.

Approximation models may have different accuracy depending on the design area and different utmost strategies to build criteria are highlighted:

  1. Exploration: search new designs that improve the accuracy of models.
  2. Exploitation: search new designs that should improve the best known values.

In the Optimizer block, we distinguish between Generic SBO that relies on a trade-off between these two strategies and Direct SBO that does exploitation only. Direct SBO was first introduced in pSeven 6.11.

In theory, good exploration is important for the good global search of optimum. But it cannot be done for free, a lot of evaluations should be made just to investigate the responses. However, in practice, the number of evaluations may be very limited, but we still need to improve design globally.  In such situations, the Direct SBO approach may be the best choice to find a better design. To illustrate it let’s consider a constrained optimization example.

Subject to constraints:

Value ℇ(x,y) represents deterministic noise.  The function to minimize is shown in Figure 2.

Figure 2.  Response surface of the function used for optimization

In the table below, the achieved objectives for different evaluation budgets and approaches are provided.

Evaluations 15 40
Direct SBO -0.26 -0.27
Generic SBO -0.03 -0.29

 

The results show that Direct SBO was able to find a good solution for a small budget, but was unable to improve it in the long run. On the contrary, the exploration-exploitation approach started slow, but in the long run, it was possible to further improve objectives’ values.

Along with single-objective optimization, Direct SBO in pSeven supports multi-objective and robust optimization and integer variables.

 

By Alexis Pospelov, Senior Researcher, DATADVANCE