Main Content

Student Learning Objectives

• SLO#1: Discuss/Analyze and Implement in software — backtracking linesearch using the (1) steepest descent and (2) Newton search directions; carefully quantify convergence rates, and total mount of work required to optimize the two-dimensional Rosenbrock function from carefully selected initial approximations.
  • Assessment#1: Homework, with Practical Software Implementation, and Theoretical analysis/discussion
  • Activity: Lecture

• SLO#2: Discuss/Analyze and Implement in software — a polynomial interpolation based improvement of the backtracking linesearch strategy; carefully quantify and discuss the improvement (or lack thereof) of convergence properties when applied to the (1) steepest descent and (2) Newton search directions for the two-dimensional Rosenbrock function.
  • Assessment#2: Homework, with Practical Software Implementation, and Theoretical analysis/discussion
  • Activity: Lecture

• SLO#3: Discuss/Analyze and Implement in software — the solution of the trust-region sub-problem, using a strategy more sophisticated than the Dogleg method; visualize the approximate the optimal path, by applying the algorithm to a range of trust region radii.
  • Assessment#3: Homework, Software & Theoretical
  • Activity: Lecture

• SLO#4: Discuss/Analyze and Implement in software — the standard Conjugate Gradient method, and use it to to solve (1) discretizations (using as many points as the current computational environment allows for) of the Poisson Equation in one, two, and three dimensions using the Helical Coordinate Preconditioner for the Laplacian; and (2) linear systems involving the Hilbert matrix of sizes 5-by-5 to 20-by-20. Discuss strengths and limitations of the algorithm.
  • Assessment#4: Homework, with Practical Software Implementation, and Theoretical analysis/discussion.
  • Activity: Lecture

• SLO#5: Discuss/Analyze and Implement in software, — the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm; use it to find the optimum of a version of the 18 (or higher) dimensional version of the Rosenbrock function; compare convergence against the full Newton optimization; and compute the sequence corresponding to the Quasi-Newton convergence criteria.
  • Assessment#5: Homework, with Practical Software Implementation, and Theoretical analysis/discussion.
  • Activity: Lecture

• SLO#6: Demonstrate, through an interpretive dance, how the modern computational optimization methods discussed in the class can be applied to a reseach problem of your interest.
  • Assessment#6: Project presentation covering theoretical and practical aspects of the research problem
  • Activity: Lecture, Independent Research