Blog
AMPL Team
December 9, 2023
Nonlinear optimization involves problems where the objective function or constraints are nonlinear. While CPLEX is not a dedicated nonlinear solver, it can handle certain types of nonlinearities, particularly those that can be transformed into linear or quadratic forms. This feature opens avenues for solving a wider range of problems.
One common approach is to linearize the nonlinear components of a problem. This involves approximating nonlinear functions with linear ones, a technique that is particularly effective for mild nonlinearities.
For problems with more pronounced nonlinearities, piecewise linear approximation is a useful strategy. This involves dividing the domain of a nonlinear function into segments and approximating each segment with a linear function.
When dealing with quadratic nonlinearities, CPLEX can directly handle them as it supports quadratic programming. This is useful in problems like portfolio optimization where the variance is a quadratic function.
Ensure that the nonlinear problem is accurately represented in its transformed linear or quadratic form. This accuracy is crucial for the relevance of the solution.
While transforming nonlinear problems, it’s important to maintain a balance between accuracy and solver efficiency. Overly complex transformations can lead to inefficient solving processes.
Due to the approximations involved in handling nonlinear problems, validating the solutions against the original nonlinear model is a best practice to ensure solution quality.
QuickDeliver, a fictional logistics company, faces a complex routing problem. Their challenge is to minimize the total travel time for their delivery vehicles, considering that travel time between two points is not strictly proportional to the distance (due to varying traffic conditions).
By using CPLEX, QuickDeliver approached this nonlinear problem through piecewise linear approximation. They divided the city map into zones, approximating travel times within each zone linearly based on historical traffic data.
The optimized routes, computed by CPLEX, led to a 15% reduction in average delivery times and a 10% decrease in fuel costs, enhancing operational efficiency.
EcoManufacture, a fictional sustainable manufacturing firm, aims to optimize its production process. The challenge is the nonlinear relationship between the production rate and energy consumption due to varying efficiencies at different production levels.
EcoManufacture used CPLEX to linearize this nonlinear relationship. They created a model where energy consumption at different production levels was approximated using linear segments, allowing CPLEX to find the optimal production rate that minimized energy costs while meeting demand.
This optimization resulted in a 20% reduction in energy costs and a more efficient production process, aligning with the company’s sustainability goals.
WealthGrow, a fictional investment firm, sought to optimize a portfolio for their clients. The challenge was the quadratic nature of the risk (variance) in portfolio optimization.
WealthGrow utilized CPLEX’s capability to handle quadratic programming. They modeled the portfolio’s risk-return profile as a quadratic problem, allowing CPLEX to directly find the optimal asset allocation.
The solution provided by CPLEX enabled WealthGrow to construct a portfolio that maximized returns for a given level of risk, significantly enhancing client satisfaction.
While CPLEX is not inherently designed for general nonlinear optimization problems, its capabilities in linearization and quadratic programming allow it to effectively tackle a range of nonlinear challenges. The key lies in accurately transforming the problem into a form CPLEX can handle efficiently. By following best practices in problem transformation and validation, users can leverage CPLEX to extend beyond linear and quadratic domains, addressing a wider spectrum of optimization challenges as illustrated in the hypothetical examples of QuickDeliver, EcoManufacture, and WealthGrow. This adaptability makes CPLEX a versatile tool in the optimization toolkit, capable of providing solutions in varied and complex real-world scenarios.
Each post is a collaborative effort by the AMPL development team – a group of dedicated developers, mathematicians, and optimization experts. We combine our diverse expertise to bring you insights into the world of mathematical optimization, sharing our experiences, challenges, and innovations in the field.
Yes, CPLEX can tackle certain types of nonlinear optimization problems, especially those that can be transformed into linear or quadratic forms. While it is not a dedicated nonlinear solver, it offers capabilities for solving a wider range of problems by employing strategies such as linearization, piecewise linear approximation, and direct handling of quadratic forms.
The main strategies include:
Best practices include:
Technical Development Team