06. Engineering Modeling — Turning Real Problems into Optimization Problems

// BACKGROUND 6/6 — input-output model, simulation, ODE/PDE, objective definition, constraint definition

The difficulty of engineering optimization begins with modeling, before the algorithm. The process of translating a real design problem into equations determines what becomes a variable, what becomes the objective function, and what becomes a constraint.

Input-output models

An input-output model is a relationship in which an input $x$ produces an output $y$:

$$y=M(x).$$

Here $M$ may be an empirical formula, simulator, neural network, or physical model. Optimization usually calculates performance measures from $M(x)$.

Simulation

Simulation executes a real system as a computational model. CFD, structural analysis, circuit simulation, and process simulation all belong here.

The problem is that simulation is expensive. If one evaluation takes minutes, hours, or days, methods that require many evaluations, such as gradient descent, become difficult to use. Expensive optimization and surrogate modeling are needed in that case.

ODE/PDE

Dynamics or physical laws can be expressed as ODEs or PDEs.

ODE example:

$$\frac{dz}{dt}=f(z(t),x,t).$$

PDE example:

$$\frac{\partial u}{\partial t}=\mathcal{L}(u,x).$$

If the optimization variable $x$ represents system coefficients, boundary conditions, or geometry parameters, the objective is calculated from the solution of the ODE/PDE.

Objective definition

Defining the objective means expressing "what counts as good" in equations.

Examples:

• Minimize weight
• Minimize energy consumption
• Maximize output efficiency
• Minimize a weighted sum of error and cost

When several objectives exist, combine them into one scalar objective or handle them with multi-objective optimization.

Constraint definition

Defining constraints means specifying "conditions that must be respected."

Examples:

• Material stress must not exceed the allowable stress.
• Temperature must stay within a safe range.
• Design variables must stay within manufacturable ranges.
• Budget and time must not exceed their upper bounds.

:::tip Modeling order First choose the decision variables, then identify what the simulation outputs, and finally separate the objective from the constraints. Mixing "something that is better when improved" with "something that must not be violated" obscures the interpretation of the problem. :::

Next: Problem Formulation