# Design Optimization

## What is Design Optimization?

Design optimization is a process of finding the values of input parameters, which lead to the best performance of analytical or simulation model of a product or a manufacturing process under investigation. Ultimately, it answers the following questions:

- How to improve product or process characteristics?
- Which combination of input parameters is the best?
- How to decrease the effect of input parameters variability on the overall product or process behavior?

*Create model*

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*Define variables and goals*

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*Run optimization*

## Key Advantages of Optimization in pSeven

- Efficient solution of complex engineering problems with up to ten objective functions, hundreds of design variables and constraints with a small computation budget.
- A wide range of easy-to-use proprietary and in-house developed optimization algorithms with a minimum setup required.
- Automatic algorithm selection that allows users with no specialized competence to solve optimization problems easier.
- Robustness of optimization process to random noise in model responses, as well as to undefined model behavior.
- Parallel execution of optimization procedures allowing to reduce computational time of resource-consuming problems solution drastically.

## Optimization Algorithms

pSeven offers a complete set of in-house developed methods and algorithms to conduct optimization of one or multiple model objective functions subject to various constraints. It allows to efficiently solve both design optimization problems with fast to evaluate analytical models and the problems where the key challenge is expensive in terms of computing resources simulations.

In addition to in-house developed methods and algorithms, pSeven also uses popular and well-known implementations of optimization algorithms and supplements them with in-house developed features, which makes them even more efficient and unique. The efficiency of optimization problem solution highly depends on optimization method and proper algorithm selection.

**Supported Problem Types:**

- Single- or Multi-Objective
- Single- or Multi-Objective Robust

**Method Classes:**

- Derivative-Free Optimization
- Gradient-Based Optimization
- Surrogate-Based Optimization
- Robust Design Optimization

**Local Algorithms:**

- Feasible Direction (FD)
- Mixed-Integer Linear Programming (MILP)
- Method of Multipliers (MoM)
- Non-Linear Conjugated Gradients (NCG)
- Non-Linear Simplex (NLS)
- Powell’s Conjugate Direction Method
- Quadratic Programming (QP)
- Sequential QP (SQP)
- Sequential Quadratic Constrained QP (SQ2P)

**Globalization Algorithms:**

- Random Linkage (RL)
- Plain MultiStart (PM)
- MultiStart PI (MS)

With pSeven, the user has to simply set the basic properties of the model (if known), such as model evaluation expensiveness, smoothness of model responses, etc., instead of tedious tuning of optimization algorithm internal parameters. After that automatic and adaptive choice of specific optimization algorithm(s) based on this information is provided by SmartSelection technique. |

Along with the automatic algorithms selection convenient for the users, full control over the whole optimization process is available for expert-level users, making optimization capabilities of pSeven highly customizable.

## Surrogate-Based Optimization (SBO)

Single- and multi-objective surrogate-based optimization in pSeven is based upon a widely-known approach of fitting a response surface to output values of few points evaluated from the model. After first fitting, it balances between finding the minimum/maximum values and improving the response surface by sampling areas where the chance of improvement may be high.

*Create model*

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*Define variables and goals*

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*Run optimization*

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*Approximation*

Implementation of this approach in pSeven is significantly different from other available realizations and is free from the most common drawbacks. The sampling utilizes an in-house developed Design of Experiments strategy, which respects as much feasible domain of the problem as possible. Techniques for fitting the response surface origin to Gaussian Processes (GP) for constructing approximation models, but are adapted for problems with high dimensionality (up to 100 input parameters) and other real-world applications issues.

## Robust Design Optimization (RDO)

Single- and multi-objective robust optimization in pSeven supports virtually all possible formulations, including probabilistic and quantile type constraints. It is based on the state-of-the-art stochastic approach that supports both direct and surrogate-based optimization, so it is suitable for expensive to evaluate problems.

The basis of this approach is a careful adjustment of the distribution realizations number of uncertain parameters. Only a small number of random realizations away from optimal solution needs to be considered to change the uncertain parameter itself. When the optimal solution is approached, the number of distribution realizations will be increased.

*Create model*

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*Define variables and goals*

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*Provide distributions for variables*

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*Run optimization*

The unique feature of this approach is that it provides both the solution and the corresponding uncertainty estimations of objective and constraints values. Another advantage of robust optimization in pSeven for engineering applications is that explicit distribution law of the uncertain parameters is not required, it is enough to provide their distribution empirically.

## Advanced Problem Statements

There is also a set of specific aspects of analytical and simulation models that can and must be taken into account during the design optimization process in pSeven:

**Problem Dimensionality**

This challenge is characterized by a large number of design variables (hundreds), constraints (dozens) and objectives (multiple).

**Problem Statement**

Various types of input parameters (continuous, integer, categorical), constraints and objectives (smooth, nonlinear, multiextremal, non-differentiable, mixed).

**Conflicting objectives**

Optimization of two or more conflicting objectives simultaneously creates a Pareto frontier.

**Noisy Objective Functions**

Finite precision of numbers used in computers and special features of numerical methods, in particular, meshing and convergence rules of finite element methods, produces numerical noise.

**Computational Time**

Calculation of a single design point can take a significant time. For example, a typical CFD simulation takes up to several hours. That is why it is extremely important for an optimization process to use the simulation model as rarely as possible.

**Uncomputable Regions**

Their presence means that the simulation model is unable to obtain results for certain values of model parameters, for example, when the solution of a non-linear structural problem is not able to achieve convergence or if an intense turbulent flow motion makes the solution of Navier-Stokes equations numerically unstable.