**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: