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A JuMP extension for modeling and efficiently solving infinite-dimensional optimization problems.

!!! note InfiniteOpt v0.6 introduces quite a few new features and makes some breaking changes to the underlying data structures. The core API remains largely the same.

InfiniteOpt.jl provides a general modeling environment to express and efficiently solve infinite-dimensional optimization problems (e.g., dynamic, stochastic, and PDE-constrained optimization). InfiniteOpt is meant to facilitate intuitive model definition that is compatible with a large suite of solution techniques (including automatic transcription) to tackle large-scale real world problems.

It builds upon JuMP to add support for many complex modeling objects which include:

• Infinite parameters (e.g., time, space, uncertainty, etc)
• Infinite variables (e.g., y(t, x))
• Derivatives (e.g., \frac{\partial y(t, x)}{\partial t})
• Measures (e.g., \int_{t \in \mathcal{D}_t}y(t,x) dt, \mathbb{E}[y(\xi)])
• Chance constraints
• More

Moreover, InfiniteOpt modeling interface decouples infinite-dimensional formulations from the solution/transformations used to solve them. This enables rapid diverse switching between techniques with state-of-the-art implementations. By default, we employ direct transcription (i.e., discretization) transformations whose features include:

• Highly efficient automatic transformation that scales linearly
• Large suite of approximation schemes (e.g., quadratures, orthogonal collocation)
• Compatible with all JuMP-supported solvers
• Compatible with GPU solvers via InfiniteExaModels that make InfiniteOpt orders-of-magnitude faster than state-of-the-art alternatives.

Accepted infinite/finite problem forms currently include:

• Variables
  • Continuous and semi-continuous
  • Binary
  • Integer and semi-integer
  • Semi-definite
• Derivatives
  • Ordinary derivative operators (of any order)
  • Partial derivative operators (of any order)
• Measures
  • Univariate and multivariate integrals
  • Univariate and multivariate expectations
  • Arbitrary measure operators (via general measure API)
• Objectives
  • Linear
  • Quadratic (convex and non-convex)
  • General nonlinear
• Constraints
  • Linear
  • Quadratic (convex and non-convex)
  • General nonlinear
  • Conic
  • Semi-definite
  • Indicator
  • Anything else supported by JuMP
• Machine learning surrogate models (via [MathOptAI](@ref mathoptai_guide))
• Generalized disjunctive programming (via DisjunctiveProgramming.jl)

See our YouTube overview of infinite-dimensional programming and InfiniteOpt.jl's capabilities from our JuMP-dev 2025 talk:

InfiniteOpt.jl is a registered Julia package and can be added simply by inputting the following in the package manager:

(v1.12) pkg> add InfiniteOpt

Please visit our Installation Guide for more details and information on how to get started.

InfiniteOpt is extension of JuMP, thus new users should start by familiarizing themselves with how to use JuMP first. See JuMP's documentation to learn more.

Once you're familiar with JuMP check out our Quick Start Guide to get started. From there we provide extensive documentation to help you along, please see How to Use the Documentation for more info.

InfiniteOpt is intended to serve both as a high-level interface for infinite-dimensional optimization and as a highly customizable/extendable platform for implementing advanced techniques. With this in mind, we provide the User Guide sections to walk through the ins and outs of InfiniteOpt. Each page in the User Guide typically contains the following:

• An Overview section describing the purpose of the page.
• A Basic Usage section to guide using InfiniteOpt at a high level.
• Other sections offering more in-depth information/guidance beyond basic usage.

We also provide a technical manual in the API Manual sections which are comprised of the docstrings of all the public methods and types that comprise InfiniteOpt. These detail the technical aspects of each function, method, macro, and composite type.

Details, instructions, templates, and tutorials on how to write user-defined extensions in InfiniteOpt are provided on the Extensions page.

Finally, case study examples are provided in the Examples sections.

For additional help please visit and post in our Discussion Forum.

InfiniteOpt is a powerful tool with a broad scope lending to a large realm of possible feature additions and enhancements. So, we are thrilled to support anyone who would like to contribute to this project in any way big or small.

For small documentation fixes (such as typos or wording clarifications) please do the following:

1. Click on Edit on GitHub at the top of the documentation page
2. Make the desired changes
3. Submit a pull request

For other contributions, please visit our [Developers Guide](@ref contribute_guide) for step-by-step instructions and to review our style guide.

If you use InfiniteOpt.jl in your research, we would greatly appreciate your citing it.

@article{pulsipher2022unifying,
      title = {A unifying modeling abstraction for infinite-dimensional optimization},
      journal = {Computers & Chemical Engineering},
      volume = {156},
      year = {2022},
      issn = {0098-1354},
      doi = {https://doi.org/10.1016/j.compchemeng.2021.107567},
      url = {https://www.sciencedirect.com/science/article/pii/S0098135421003458},
      author = {Joshua L. Pulsipher and Weiqi Zhang and Tyler J. Hongisto and Victor M. Zavala},
}

A pre-print version is freely available though arXiv.

We acknowledge previous support from the Department of Energy under grant DE-SC0014114.