Linear equations allow us to model a wide range of everyday problems, so we can calculate everything from mileage rates to income earned over time.

In many business cases, it is safe to assume linear relationships and work with linear functions to solve for costs, resource consumption or other general KPIs.

However, most real-world relationships are nonlinear, especially in industries like financial services, oil and gas, and power and utilities.

Unlike their linear counterparts, nonlinear relationships introduce complexities that demand a more sophisticated and adaptable algorithmic optimization approach. Thatâ€™s where nonlinear solving comes in.

### What Is Nonlinear Solving?

Nonlinear solving can help us find solutions to equations or systems of equations that involve nonlinear relationships between variables.

In mathematics, common examples of nonlinear expressions include equations with variables that are raised to powers greater than one, or that involve trigonometric functions, exponentials, logarithms, or other nonlinear operations.

### Examples of Nonlinear Relationships

The world is full of nonlinear relationships across all industries. Here are a few examples of how nonlinear solving can be applied to a wide range of problems:

• Portfolio Optimization: In finance, nonlinear optimization can be used to identify the optimal allocation of assets in a portfolio to maximize returns while managing risks. This involves nonlinear relationships between asset returns and risks.
• Engineering Design: Nonlinear systems are often used in engineering design to optimize parameters for better performance, cost-effectiveness, or efficiency. Engineering designs must ensure certain physical properties, and those cannot be stated with linear relationships.
• Electrical Grid Optimization: Nonlinear solving supports electrical grid optimization by addressing the nonlinear properties and constraints of AC systems, ensuring optimal generation, transmission, and distribution for reliable and cost-effective operations.
• Oil and Gas: Nonlinear solving is essential in the oil and gas industry to accurately model and optimize complex processes, since physical propertiesâ€”such as storage capacity under pressureâ€”are nonlinear.
• Machine Learning and Data Analysis: To train machine learning models, nonlinear systems are often used to identify optimal parameters. Nonlinear regression analysis is also used in statistics to model relationships between variables.
• Agricultural Planning: Nonlinear solving can be used in agriculture for crop planning and management to optimize planting schedules, irrigation, and fertilizer application to maximize yield while working with resource constraints.

### Solving Nonlinear Problems with Linear Methods

Despite the many nonlinear relationships that we encounter every day in the real world, the options for solving nonlinear systems of equations have traditionally been few.

A pure linear programming solver will only accept linear expressions as input. So how do you go about solving a complex, nonlinear equation?

By breaking down a nonlinear relationship into multiple linear ones, you can solve your problem using a piecewise linear (PWL) approximation.

The problem is that approximations introduce errorsâ€”which are found in the difference between your true and approximated values. Still, for many nonlinear relationships, it is perfectly sufficient to treat them as linear, as long as youâ€™re willing to work with some degree of error.

But because most business use cases demand a certain degree of accuracy, youâ€™ll need to approximate a large number of â€śpieces,â€ť which will negatively impact your solving performance if youâ€™re using a pure linear solver.

The question then becomes: How much are you willing to sacrifice in accuracy for better solving performance?

### Beyond Best Guesses

Most optimization software systems today rely on approximations, requiring some level of trade-off between performance and accuracy.

But what if you had a solver that could natively support models with nonlinear relationships, without the need for linear approximations?

With the newly released Gurobi 11.0, you can use new algorithms to unlock a whole new level of accuracy. The solver natively supports a selected set of non-linear functions, so you can find a globally optimal solution in as little as a few seconds. Switching between exact and approximated solving is as easy as changing a single solver parameter. Based on use case and performance requirements, you can freely choose which approach is right for you.

To learn more about how you can obtain better, faster solutions to your most challenging business problems, check out the full list of new features included in Gurobi 11.0.

AUTHOR

### Dr. Kostja Siefen

Director Technical Account Management

AUTHOR

### Dr. Kostja Siefen

Director Technical Account Management

Dr. Kostja Siefen leads the global Technical Account Management team at Gurobi Optimization. Kostja holds a Ph.D. in Operations Research from the University of Paderborn (Germany). He joined Gurobi in 2015 after many years of experience in the development and design of decision support systems using mathematical optimization. Before joining Gurobi he worked at Daimler Research & Development and as a lecturer at the University of Paderborn. Since 1998, before focusing on optimization and during his studies he continuously worked as system administrator, software developer and support engineer for an IT service company. Kostja has been active in academic teaching and customer training since 2009. Beyond Gurobi, Kostja enjoys spending time with his family, working as a Les Mills group fitness instructor, traveling and good food.

Dr. Kostja Siefen leads the global Technical Account Management team at Gurobi Optimization. Kostja holds a Ph.D. in Operations Research from the University of Paderborn (Germany). He joined Gurobi in 2015 after many years of experience in the development and design of decision support systems using mathematical optimization. Before joining Gurobi he worked at Daimler Research & Development and as a lecturer at the University of Paderborn. Since 1998, before focusing on optimization and during his studies he continuously worked as system administrator, software developer and support engineer for an IT service company. Kostja has been active in academic teaching and customer training since 2009. Beyond Gurobi, Kostja enjoys spending time with his family, working as a Les Mills group fitness instructor, traveling and good food.

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