University of Wisconsin–Madison

University of Wisconsin–Madison

Optimization2026-08-20T16:24:49-05:00

Optimization

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Graduate Studies

The optimization group at the University of Wisconsin-Madison is a great place to do graduate studies in optimization. Our faculty have diverse interests and appointments in numerous departments on campus. The group’s position in the Wisconsin Institute of Discovery in the Discovery Building provides a nucleus for research activities with collaborators around campus. Many of our graduate students from various departments work, study and meet with their research themes in the optimization space of the Wisconsin Institute of Discovery.

Over the years, the optimization group at UW-Madison has been distinguished by its contributions to the mathematical theory of optimization, its collaborations with the faculty at UW-Madison – and beyond – on interdisciplinary research projects, and its contributions to the modeling and computational aspects of optimization. The Wisconsin Institute of Discovery in Madison provides a special environment that encourages interdisciplinary research, which in turn motivates new core research in optimization.

Your Graduate Career in Optimization@UW

Graduate studies in optimization at UW-Madison offer:

Support

Contact professors for Research Assistantship availability. Departments offer some Teaching Assistantships and grading support (differs according to department and budget). Availability depends on current research funding, positions typically include tuition remission and stipend. Limited opportunities for Tutorial Assistantships are also available in some departments.

 

  • COMP SCI /​ I SY E / ​MATH 425:   Introduction to Combinatorial Optimization
    • Optimization problems over discrete structures, such as shortest paths, spanning trees, flows, matchings, and the traveling salesman problem.
  • COMP SCI / ​MATH / ​STAT 475:   Introduction to Combinatorics
    • Problems of enumeration, distribution, and arrangement; inclusion-exclusion principle; generating functions and linear recurrence relations. Potential applications in the social, biological, and physical sciences.
  • COMP SCI / ​E C E / ​I SY E 524:   Introduction to Optimization
    • Introduction to mathematical optimization from a modeling and solution perspective.
  • COMP SCI / ​I SY E/ ​MATH / ​STAT 525:   Linear Programming Methods
    • Real linear algebra over polyhedral cones; theorems of the alternative for matrices; formulation of linear programs; duality theory and solvability; the simplex method and related methods for efficient computer solution.
  • COMP SCI / ​I SY E 526:   Advanced Linear Programming
    • Polynomial time methods for linear programming; quadratic programs and linear complementarity problems and related solution techniques; solution sets and their continuity properties.
  • COMP SCI /​ E C E /​ M E 532:   Matrix Methods in Machine Learning
    • An introduction to machine learning that focuses on matrix methods and features real-world applications ranging from classification and clustering to denoising and data analysis.
  • COMP SCI / ​I SY E 719:   Stochastic Programming
    • Stochastic programming is concerned with decision making in the presence of uncertainty, where the eventual outcome depends on a future random event. Topics include modeling uncertainty in optimization problems, risk measures, stochastic programming algorithms, approximation and sampling methods, and applications.
  • COMP SCI /​ I SY E 723:   Dynamic Programming and Associated Topics
    • General and special techniques of dynamic programming are developed by means of examples. Shortest-path algorithms; deterministic equipment replacement models; resource allocation problem; traveling-salesman problem; general stochastic formulations; Markovian decision processes and more
  • COMP SCI /​ I SY E /​ MATH / ​STAT 726:   Nonlinear Optimization I
    • This course emphasizes continuous, nonlinear optimization and could be taken with only a background in mathematical analysis.
  • COMP SCI /​ I SY E 727:   Convex Analysis
    • Convex sets in finite-dimensional spaces: relative interiors, separation, set operations. Convex functions: conjugacy, subdifferentials and directional derivations, functional operations, Fenchel-Rockafellar duality.
  • COMP SCI /​ I SY E /​ MATH 728:   Integer Optimization
    • Introduction to optimization problems over integers and survey of the theory behind the algorithms used in state-of-the-art methods for solving such problems. Special attention is given to the polyhedral formulations of these problems, and to their algebraic and geometric properties.
  • COMP SCI / ​I SY E /​ MATH 730:   Nonlinear Optimization II
    • Theory and algorithms for nonlinearly constrained optimization; relevant geometric concepts, including tangent and normal cones, theorems of the alternative, and separation results.
  • COMP SCI 733:   Computational Methods for Large Sparse Systems
    • Algorithms and theory for large scale systems in engineering and science, with emphasis on sparse matrices and iterative methods.

Here are more details on these courses, including information on credits and requisites.

To embark on either master’s degree or Ph.D programs in optimization, prospective graduate students must enroll in one of the departments with which the group is affiliated. You’re Ph.D. advisor should be a regular or affiliate faculty member of the department that graduate students are enrolled in. Graduate admission requirements and procedures are different among departments.

Refer to the graduate admissions pages for each of the following departments for specific information about the rules of the department:

 

The Optimization Qualifying Exam is a research-oriented milestone in which students showcase their original research to a faculty committee. It consists of two components: a written research report and a public oral presentation.

Coursework Requirements. Before taking the exam, students should have built a solid foundation in optimization by completing two courses from MATH/ISyE 425, 524, and 525, and two optimization courses at the 700 level (719, 726, 727, 728, or 730), with an average GPA of at least 3.5. Students who have taken equivalent courses elsewhere may apply for a waiver from the faculty committee.

Report. The student prepares a written report for the committee presenting their research progress. The report should articulate a compelling research question, motivate its importance, situate it within the relevant literature, and describe the results obtained, reflecting work that is on track for submission to a reputable journal or conference.

Presentation. The student delivers a public oral presentation to the committee, sharing their research question, its motivation, connections to the literature, and their results. This is a great opportunity to communicate research to a broader audience and receive valuable feedback from faculty.

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