What Is a Linear Optimizer?
A linear optimizer is a mathematical optimization tool that can identify the best possible solutions to questions where the outcome has a linear relationship with the variables of interest. It is also known as a linear programming solver.
Linear optimizers are often the first solvers people use in their optimization journeys, as the mathematical concepts are relatively basic (i.e., linear algebra and college-level calculus). This allows users to master optimization concepts before moving to more complicated mathematical concepts.
While linear optimizers solve (relatively) straightforward problems, they are also powerful. Small startups to Fortune 100 companies use linear optimizers to solve pressing business challenges, such as workforce scheduling, resource allocation, and supply chain optimization.
How Does a Linear Optimizer Work?
All problems have constraints, such as budgets, timelines, or production limits. A linear optimizer works by utilizing the linear constraints to define the boundaries of any possible solution. It then explores the feasible region (i.e., the region of solutions that fit within the constraints) until it identifies the optimal solution. How it explores the feasible region for the optimal solution depends on the specific algorithm teams employ. For example, the Simplex Method starts at an initial feasible solution and then follows the exterior of the feasible region until it reaches the most optimal point. The Interior Point Method, however, explores the interior of the feasible region.
For an in-depth explanation of how linear programming models are developed, be sure to read the Gurobi LP Models Primer.
What Are the Main Use Cases For and Benefits of a Linear Optimizer?
Linear optimizers are appropriate for pure linear programs (i.e., programs in which variables are continuous and have linear relationships with the outcome and constraints). Most business challenges have at least some integer variables, which means that most models will require at least mixed-integer linear solvers. However, linear optimizers can still solve many problems.
When deployed appropriately, a linear optimizer can help your team identify the best possible solution in a timely manner. For example, with Gurobi, Lyft was able to reduce their time spent per linear problem by 80% and improve their overall end-to-end time by 92%. This played a key role in helping them be more responsive to changing conditions and scale up driver incentive programs.
Are There Limitations of Linear Optimizers?
The main weakness of linear optimizers is the relative rarity of purely linear optimization problems. To determine if a linear optimizer is appropriate for your challenge, you must determine:
Is the relationship between the objective function, variables, and constraints truly linear?
Are the variables continuous?
Is the data reliable and scalable?
If the answer to any of these questions is “no,” then a linear optimizer may not be the appropriate tool for your challenge and you will need to explore other optimizers.
Is Gurobi a Linear Optimizer?
Yes, the Gurobi Optimizer is a linear optimizer. As noted above, that may not always be the right tool for the job. Luckily, Gurobi is not just a linear optimizer. In addition to linear programming, Gurobi can solve:
Mixed-integer programming problems
Mixed-integer nonlinear programming problems
Quadratic programming problems
Mixed-integer quadratic programming problems
Mixed-integer quadratically-constrained programming problems
General nonlinear problems
How Is a Linear Optimizer Different from Other Optimizers?
A linear optimizer is specifically designed to solve linear problems (i.e., problems in which variables are continuous and have linear relationships with the objective function) by taking advantage of the geometry of the solution space. Other optimizers include:
Mixed-Integer Optimizers: Solvers that can accommodate both continuous and discrete variables.
Nonlinear Optimizers: Solvers that can accommodate problems where variables have nonlinear impacts on objective functions.
Heuristic Algorithms: Solvers that prioritize speed and ease-of-use over perfection, resulting in “good enough” answers.
What Skills Do I Need to Use a Linear Optimizer?
Operations researchers and data scientists need three key skills to use a linear optimizer:
Mathematical Knowledge: The basic math skills optimization teams need are linear algebra, college-level calculus, and optimization theory.
Mathematical Modeling: Optimization teams should understand how mathematical models are designed. Gurobi’s AI Modeling and OptiMods can help build these skills and jumpstart the process.
Programming Experience: Gurobi supports all major programming languages, though Python is the most popular option.
Where Can I Learn More About Linear Optimization?
Gurobi has many resources to learn more about linear optimization, including:
A Linear Programming Primer, which covers basic terms and approaches;
A Linear Optimization Explained blog post, which dives in deeper on concepts and application; and
A 14-chapter Linear Programming Tutorial, complete with in-depth videos.
