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Recognizing and solving convex optimization problems. Convex sets, functions, and optimization problems. Least-squares, linear, and quadratic optimization. Geometric and semidefinite programming. Vector optimization. Duality theory. Convex relaxations. Approximation, fitting, and statistical estimation. Geometric problems. Control and trajectory planning.

From the Preface of Convex Optimization by Stephen Boyd and Lieven Vandenberhghe, page xi: "[C]onvex optimization [is] a special class of mathematical optimization problems, which includes least-squares and linear programming problems...[Convex problems] are more prevalent in practice than was previously thought. Since 1990 many applications have been discovered in areas such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling,statistics, and finance. Convex optimization has also found wide application in combinatorial optimization and global optimization, where it is used to find bounds on the optimal value as well as approximate solutions.......

Graduate Standing

The course requires a mathematical maturity that is uncommon in undergraduates, but is expected of graduate students who use computational mathematics.

- 1. Determine whether a set is convex and whether a function is convex, quasiconvex, or log-convex.
- 2. Determine if an optimization problem is convex or quasiconvex, and specify whether a convex optimization problem is a linear, quadratic, and second-order cone, geometric, or semidefinite programming problem.
- 3. Solve convex vector optimization problems by scalarization and generate optimal trade-off curves which are formed by Pareto optimal points.
- 4. Formulate the Lagrange dual problem and, in the case of convex problems, check Slater's constraint qualification to determine if strong duality holds.
- 5. List Karush-Kuhn-Tucker (KKT) conditions and perform perturbation and sensitivity analysis.
- 6. Perform regularized and robust approximation as well as maximum likelihood estimation.
- 7. Design optimal detectors and experiments, obtain extremal volume ellipsoids, find conters of convex sets, and solve some classification problems.

## Topic |
## Percentage of Course |

1. Convex Sets and Functions. Basic properties and examples; operations that preserve convexity; convexity with respect to generalized inequalities; quasiconvex and log-concave functions. | 20% |

2. Convex Optimization Problems. Linear, quadratic, and geometric programs; generalized inequalities; semidefinite programming; vector optimization and Pareto optimal points. | 20% |

3. Optimality Conditions and Duality theory. Lagrange dual problem; weak and strong duality; geometric interpretation; Karush-Kuhn-Tucker conditions; perturbation and sensitivity analysis. | 15% |

4. Approximation and Fitting. Norm approximation; least-norm problems; regularized approximation; robust approximation. | 15% |

5. Statistical Estimation. Maximum likelihood estimation; optimal detector design; experiment design. | 15% |

6. Geometric problems. Extremal volume ellipsoids; centering; classification; facility location. | 15% |

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