# Lecture 0 (Day 1): 1. Introduction ; 2. MATLAB; 3. Examples and Applications of Optimization

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3. ### Projects/Case Studies ( NEOS Case Studies)

1. Financial Optimization: Using MATLAB, solve the problem given in the link: Diversity in Portfolio Optimization (extension of Markowitz portfolio optimization). What can you say about the sparsity of the solution? Would adding a norm constraint help? Can you solve the problem with an additional l-2 norm constraint; ||x||2 &le u? without the nonnegativity constraint, x &ge 0, can you solve the problem with an additional l-1 norm constraint; ||x||1 &le u? Generate random data for the expected return p and the covariance matrix S, e.g. S=cov(randn(m,n)), m>n. Can you solve large scale problems? Can you test the robustness of the solution with respect to perturbations in the data? Can you find an improved model under the assumption that others are making their investment decisions based on the Markowitz model?
2. QQPs: Using MATLAB, solve the problem given in the link: Quadratic Constrained Quadratic Program, i.e. assume that all the matrices are positive semidefinite; find the optimal solution; and show that there is a zero duality gap. Then, for the case that some of the matrices are not positive semidefinite, find the semidefinite relaxation and its solution.
3. SDP and Max-Cut problem: Find the SDP relaxation of the Max-Cut problem; solve this relaxation using e.g. SeDuMi with MATLAB. (Use randomly generated, weighted, undirected graphs, e.g. W=sprandsym(60,.5)) Can you strengthen the relaxation by adding additional constraints? (Hint: consider constraints of the type xi xk2 xj = xi xj.) (Possibly useful: MATLAB code for Max-Cut problem)
4. Best Convex Interpolation: Reference: "A simple constraint qualification in infinite dimensional programming", authors J.M. Borwein and H. Wolkowicz, Mathematical Programming 35 (1986) 83-96. The gzipped pdf file is available at: http://orion.math.uwaterloo.ca/%7Ehwolkowi/henry/reports/PAPER11.pdf.gz Consider the best interpolation problem in equation (1.1). Derive the dual program. Write a MATLAB code to solve this interpolation problem using Newton's method. Can you extend this approach to best convex surfaces?
5. Inverse (Inf. Dim.) Problem: Solve the problems outlined in this assignment.
6. Moment Problems: Given the constants Mi, i=1,2,...,k, solve the moment problem for x, i.e. solve the k nonlinear equations:
x1i + .... + xki = Mi, i=1,...,k. Use the MATLAB (symbolic toolbox) to get explicit solutions up to k=4.

## Supplementary Information

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