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

What is InfiniteOpt?

InfiniteOpt.jl provides a general mathematical abstraction to express and solve infinite-dimensional optimization problems (i.e., problems with decision functions). Such problems stem from areas such as space-time programming and stochastic programming. InfiniteOpt is meant to facilitate intuitive model definition, automatic transcription into solvable models, permit a wide range of user-defined extensions/behavior, and more.

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

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

InfiniteOpt is intended to be used for infinite-dimensional optimization problems. Finite problems (e.g., directly modeling a discrete time model) should instead be modeled using JuMP.

Moreover, InfiniteOpt decouples the infinite-dimensional formulations from the finite transformations typically used to solve them. This readily enables diverse techniques be used to solve these types of problems. By default, we employ direct transcription (i.e., discretization) transformations whose features include:

  • Efficient implementations that scale linearly!
  • Diverse integral approximations (e.g., quadratures, sampling)
  • Diverse derivative approximations (e.g., finite difference, orthogonal collocation)
  • Sophisticated support point management system
  • Compatible with all JuMP-supported solvers

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

Infinite-Dimensional Optimization with InfiniteOpt.jl

See our YouTube overview of infinite-dimensional programming and InfiniteOpt.jl's capabilities from JuliaCon 2021 (note that nonlinear support has since been added): youtube


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

(v1.9) pkg> add InfiniteOpt

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

First Steps

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.

How to Use the Documentation

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 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.

      title = {A unifying modeling abstraction for infinite-dimensional optimization},
      journal = {Computers & Chemical Engineering},
      volume = {156},
      year = {2022},
      issn = {0098-1354},
      doi = {},
      url = {},
      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 our support from the Department of Energy under grant DE-SC0014114.