Wed 6/27

Basic Convex Sets and Functions. Wotao Yin, Rice University. Definitions and examples of convex sets and functions. Operations that preserve convexity. Fundamental properties of convex sets and functions. Convexity with respect to generalized inequalities.

Thu 6/28

Convex Optimization Problems. Wotao Yin, Rice University. An introduction to linear, quadratic, second-order cone, semi-definite, geometric, and general convex programming. Convex duality theory.

Tue 7/3

Convex Optimization Software. Wotao Yin, Rice University. An introduction to available solvers in MATLAB toolboxes and at NEOS. The use of convex programming modeling languages: CVX and YALMIP.

Tue 7/17

Optimization theory and algorithms. Mark Abramson, Air Force Institute of Technology. Optimality conditions, descent directions, derivative-based methods for unconstrained problems, globalization via line searches and trust regions, algorithms for constrained problems.

Wed 7/18

Direct Search Algorithms for Derivative-Free Optimization. Mark Abramson, Air Force Institute of Technology. Pattern search and mesh adaptive direct search (MADS) algorithms, Clarke calculus, first- and second-order convergence properties.

Thu 7/19

Extensions of Direct Search Methods. Mark Abramson, Air Force Institute of Technology. Mixed variable optimization problems, filters for constrained problems, exploiting some derivative information, surrogates for computationally expensive functions.

Tue 7/31

Introduction to C^{2} Regular Level Surfaces. David Dreisigmeyer HPC-4, LANL. We'll define what C^{2} regular level surfaces are. From here we'll go on to derive useful formulas for dealing with these surfaces. Specifically we'll see how to differentiate a function on these surfaces (i.e., covariant differentiation) and move over them in straight lines (i.e., geodesics). Definition of a KKT point in terms of covariant derivatives.

Wed 8/1

Extending Direct Search Methods to C^{2} Regular Level Surfaces. David Dreisigmeyer HPC-4, LANL. Building on what we learned about C^{2} regular level surfaces, we'll see how to extend direct search methods to them. We'll define what a pull-back is and why it's important for us. Then a general method for direct searches over C^{2} regular level sets is given. A detailed example will be presented to give a concrete realization of the general method.

Wed 8/1 1-2:30pm

Iterative Methods for Convex Programming. Wotao Yin, Rice University. The concepts of convergence. An introduction to the iterative methods: for "regularization" type of energy minimization. Analysis to an iterative method for l1-minimization.

Thu 8/2

Direct Search Methods Using Triangulations. David Dreisigmeyer HPC-4, LANL. Piecewise-linear approximation algorithms for manifolds are introduced. These are based on an implicit triangulation of the ambient space and pivoting in and out of the triangles. A slight modification of the pivoting algorithm allows us to derive a direct search method. Introduction to some frame-based direct search routines.

Mon 8/6 1-2:30pm

Optimization on Networks. Wotao Yin, Rice University. An introduction to networks, the short path problem, and the max-flow problem. Duality on networks. The max-flow/min-cut algorithms for total variation minimization.

Tue 8/7

Introduction to Interior-Point Methods for Constrained Optimization. Dominique Orban, Ecole Polytechnique de Montreal. Whether we are interested in linear, quadratic or general nonlinear programming, interior-point methods are setting today's standard in numerical optimization. We explore the basic ideas and underlying subproblems of those methods, specify a typical algorithm and identify what its different tasks are. Numerical issues are discussed.

Tue 8/7 2:30 coyote

Measuring Progress in Constrained and Unconstrained Optimization. Dominique Orban, Ecole Polytechnique de Montreal. We introduce the concept and usage of trust-regions in the context of unconstrained optimization. This helps us understand a way to measure progress towards a local solution. In order to transpose this concept to constrained optimization, we review the most popular merit functions and the recent notion of filter.

Thur 8/9

Solving Trust-Region Subproblems and Constrained Optimization Problems. Dominique Orban, Ecole Polytechnique de Montreal. In this lecture, we fill in the blanks left in our typical interior-point algorithm. We show how to solve its subproblems intuitively and efficiently. We next combine trust-regions and either a merit function or a filter to ensure global convergence to a critical point

Tue 8/14

The Implicit Function Theorem. David Dreisigmeyer HPC-4, LANL. The C^{1} and Lipschitz version of the implicit function theorem will be stated. We'll then be able to give a definition of a Lipschitz manifold. We'll examine the pull-back procedure in this context. Some elements of Clarke's non-smooth analysis will be explored along the way.

Wed 8/15

Direct Search Methods on Lipschitz Manifolds. David Dreisigmeyer HPC-4, LANL. Using the previously defined pull-back procedure, we can now extend direct searches to Lipschitz manifolds. Numerical implementation issues will also be discussed.