Optimization

Optimization is a process of finding the values of model parameters (design variables), which lead to the best performance of model under investigation. Model performance is quantified in terms of several fitness functions constructed from measured or simulated model responses. 

Optimization

pSeven offers a variety of methods to conduct optimization of one or multiple model performance criteria subject to various constraints. It allows to efficiently solve both engineering optimization problems with cheap to evaluate semi-analytical models and the problems where the key challenge is expensive (in terms of computing resources) evaluations of objective functions and constraints. pSeven provides a smart optimization manager, which automatically and adaptively selects the most suitable optimization algorithm for a given optimization problem from a pool of powerful optimization methods and algorithms. This allows addressing a wide range of applications.

Key features of pSeven’s optimization functionality

Algorithms efficiency

For any type of optimization problem, methods implemented in pSeven are the most efficient in terms of required number of model evaluations. Other criteria, such as CPU time, are considered to be of lower priority. 

Availability for non-experts

The efficiency of optimization problem solution highly depends on proper algorithm selection. With pSeven, user has to simply set the basic properties of the model (such as model evaluation expensiveness, smoothness of model responses, expected degree of multi-modality, etc.) instead of tedious tuning of optimization algorithm internal parameters. The automatic and adaptive choice of specific optimization algorithm(s) based on this information is provided by SmartSelection™ technology.

Highest customization degree

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.

Best practice and unique features combination

In addition to the complete in-house development of mathematical methods, pSeven development team reuses popular and well-known implementations of optimization algorithms and supplements them with in-house developed features, which makes them the most efficient and unique. 

Focus on real-life engineering applications

pSeven optimization algorithms are specifically developed for real-life industrial applications. They are robust to numerical noise in model response and stable with respect to occasional undefined model behavior. Several optimal solution candidates are always provided instead of one. Thus, the user avoids solving the same problem repeatedly if the proper formulation is found later, and saves computational resources.


Formal definition of optimization problem solved by pSeven optimizer (this formulation does not concerns robust optimization cases):

\begin{array}{cl}\min\limits_x \, f^i(x) & i=1,…,K & \\ c^j_L \,\le\, c^j(x)\,\le\, c^j_U & j=1,…,M \\x^k_L \,\le\, x^k\,\le\, x^k_U & k=1,…,N\end{array}

N - number of design variables that can be subject to upper and lower bounds (box bounds)

K - number of criteria (objective functions) to consider

- number of (functional) constraints defining feasible domain, excluding box bounds on design variables.


 

Main classes of optimization problems supported in pSeven

Main classes of problems, which are supported in pSeven (but not limited by this list) are summarized in the table below (numbers are approximate):

Type of problem   N  K  M
Linear problems (including mixed-integer linear problems O(106) 1 [linear] O(106) [linear]
Quadratic indefinite problems  O(104) 1 [quadratic] O(104) [linear]
Unconstrained single-objective problems O(104) 1 0
Constrained single-objective problems O(104) 1 O(104)
Constraints satisfaction problems O(104)   0 O(104)
Multi-objective constrained optimization problems O(104) O(10) O(104)
Computationally expensive problems O(10) O(1) O(10)

Within this table, relevant problem data (objectives/constraints) are assumed to be generic non-linear (albeit sufficiently regular) functions unless other information is provided (primarily via hints mechanism described here). Note that characteristic dimensionalities cited above are estimated conservatively, for instance, the number of design variables in computationally intensive problems might be as large as 100.

Efficient solution of these problems requires implementing of the variety of numerical algorithms. Brief summary of selected numerical methods implemented in pSeven to solve optimization problems is given below.

Selected optimization methods implemented in pSeven

Surrogate Based Optimization

Family of optimization strategies based on surrogate modeling are proven to be most efficient in the majority of engineering applications. Surrogate based optimization in pSeven supports all types of problems, including multi-objective robust formulations.

Related publications

See also: featured webinar "Surrogate Based Optimization - a unique solution for optimization with a minimum of iterations"

Robust and Reliability Based Design Optimization

Objective functions and constraints often depend upon external parameters with uncertain values, e.g. wind force or external temperature. In this case, optimal designs with minimal sensitivity to variations of these uncertain parameters are needed. This requires problem reformulation to get robust optimization task. pSeven provides efficient robust and reliability based optimization functionality which naturally complements uncertainty management capabilities.

Mixed-Integer Optimization

Mixed-integer black-box optimization is acknowledged as much more complex than continuous one. pSeven provides this capability based on surrogate modeling approach, which covers all types of problems mentioned above, including robust formulations. Special optimization algorithms are implemented for mixed-integer linear problems.

More detailed list and description of methods available in pSeven can be found here.

Multidisciplinary Design Optimization

Thank to powerful workflow engine pSeven supports various types of multidisciplinary optimization strategies, including bi-level optimization, collaborative optimization, analytical target cascading and many others.

Summary

Key advantages of performing optimization in pSeven:

  • Efficient solution of complex engineering problems with up to ten criteria, hundreds of design variables and constraints within a short timeframe.
  • Parallel execution of optimization procedures, allowing to greatly reduce computational time of resource-consuming problems solution.
  • Robustness of optimization process to random noise present in model responses, as well as to sporadic undefined model behavior.
  • Wide range of easy to use proprietary optimization algorithms with a minimum setup required.
  • Built-in mechanism of automatic method selection, allowing users with no specialized competence to successfully solve optimization tasks.