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## Multi-objective optimization: Finding the best solution

March 10, 2023
by
Luis Fernando Cusicanqui Lopez

### Introduction

The problem of finding an optimal design that satisfies certain constraints is mathematically described by a constrained optimization problem where the solution $$x_{opt}$$ is the one that satisfies the constraints $$C(x)$$ and at the same time minimizes (or maximizes) the objective function $$J(x)$$. In many cases, it suffices to optimize one function $$J(x)$$, a single-objective optimization problem. However, in many real-world applications, we are also interested in simultaneously minimizing multiple objective functions $$J_i(x)$$ . These types of issues are known as multi-objective optimization problems.

### Multi-Objective vs. Single-objective

The main difference between multi-objective and single-objective optimization problems is, as previously mentioned, the number of objective functions that are considered. Furthermore, there exist many other differences that are crucial for the proper understanding of the problem:

• Pareto Optimality: The solution found in multi-objective optimization is Pareto-optimal. This means that for a given set of objective functions $$J_i(x)$$, there is no further improvement possible in one objective without worsening at least one of the other objectives. In single-objective, it is possible to improve the objective until a global minimum is found for the optimization problem.
• Trade-offs: The objective functions may be conflicting in the sense that they cannot be simultaneously optimized. It is important to be aware of the existing trade-offs and to carefully give a correct relative importance to each objective in order to find the desired result. These trade-offs are absent in single-objective optimization, where only one objective function exists.
• Computational complexity: Practically all traditional optimization algorithms require the computation of the derivative of each objective function $$J_i(x)$$ to all the optimization variables. Depending on the problem, the computation for each objective function may become computationally expensive. Therefore, when the number of objective functions is large, this greatly affects the overall computational requirements.

### Examples

In this section, we illustrate the capabilities of multi-objective optimization with two examples to gain more insight.

##### 1. Combination of relative volume minimization and temperature minimization

The problem is the following: There is a need for an optimal design to minimize the mean temperature of the geometry adjacent to the design. However, the material used is highly conductive but also very expensive and not easy to find leading as well to a relative volume minimization (relative compared to the volume available for our design). This problem easily translates to a multi-objective optimization problem with two objective functions. These functions are conflicting because the design that minimizes the temperature fills the space available, while the design that minimizes the relative volume minimization is just an empty design. Therefore, it is necessary to set the weights to find a design that finds a good trade-off for the application between the total cost of material and performance measured by the component's temperature.

##### 2. Combination of power dissipation minimization and a temperature minimization

The problem set is the following: A fluid is driven by a pump, enters a region (design region) through a small inlet, and exits through another outlet. This region is on top of a heated solid layer. The goal is to find an optimal design where the fluid will pass through, to minimize the mean temperature over the heated solid layer. At the same time, we need to ensure that the power provided by the pump is as minimal as possible.

The problem can be translated to a multi-objective optimization problem with two objective functions: A power dissipation minimization in the fluid and a temperature minimization in the solid layer. Likewise, in this situation, the resulting design will vary depending on the relative importance given to each objective function.

### Multi-Objective on ColdStream

ColdStream platform offers the possibility to solve both of the problems defined above. After uploading the geometries to the platform and selecting the desired objective functions, we can choose a value between 0 and 1 which gives the relative importance of the corresponding objective. If defining a multi-objective optimization may be cumbersome to you, the support engineers at Diabatix can also help you set up a feasible design case. More about weighting functions can be found on our documentation page.

If you are also interested in multi-optimization functions for battery cold plates, check out the previous article here. Don’t hesitate to contact us to know how you can benefit from ColdStream.