Maryam Fazel Research Scientist Control And Dynamical Systems, California Institute of Technology Address: Mail Code 107-81 California Institute of Technology Pasadena, CA 91125 Phone: (626) 395-3368 Fax: (626) 796-8914 E-mail: maryam (at) cds(dot)caltech(dot)edu URL: http://www.cds.caltech.edu/~maryam

Education

• Ph.D. Electrical Engineering, Stanford University, CA, March 2002
• M.S. Electrical Engineering, Stanford University, CA, June 1997
• B.Sc. Electrical Engineering, Sharif University of Technology, Tehran, Iran, 1995

General Research Areas

• Optimization: Convex and robust optimization; convex relaxations; decomposition methods. Algebraic methods and Sum-of-Squares programming. Matrix rank minimization problems.
• Systems and Control: Nonlinear system analysis, identification, and invalidation using Sum-of-Squares programming.
• Applications: Networks (utility maximization); systems biology (cell signaling); control; finance (portfolio optimization); quantum physics (channel design for error correction).

Teaching

• CDS270: Convex optimization and applications, Spring 2003, Caltech.
  course webpage.

Ph.D. Thesis

• Matrix Rank Minimization with Applications. Elec. Eng. Dept, Stanford University, March 2002.
  Thesis: (ps file), (pdf file), (gzipped ps file)
  Talk: slides (ps file)
  

  
  My PhD work concerns the problem of minimizing the rank of a matrix over a convex set. In many engineering applications, notions such as order, complexity, or dimension of a model or design can be expressed as the rank of an appropriate matrix. If the set of feasible models or designs is convex, then choosing the simplest model can be cast as a Rank Minimization Problem (RMP). For example, a low-rank matrix could correspond to a low-order controller for a plant, a low-degree statistical model fit for a random process, a shape that can be embedded in a low-dimensional space, or a design with a small number of components. It is thus not surprising that rank minimization has diverse applications: following Occam's razor, we often seek simple explanations and designs.

  The RMP is known to be NP-hard. If the variable in an RMP is a positive semidefinite matrix, a simple and effective relaxation commonly used is trace minimization. We prove that any general RMP involving non-positive semidefinite or even non-square matrices, can be embedded in a larger, positive semidefinite RMP. This semidefinite embedding lemma is then used to extend the trace relaxation, as well as other methods using semidefiniteness, to general matrices. The generalized trace relaxation minimizes the dual of the matrix 2-norm, referred to as the nuclear norm. Furthermore, we show that this relaxation is equivalent to minimizing the convex envelope of the rank function over the unit ball, thus providing theoretical support for its use. We also propose another heuristic based on (locally) minimizing the logarithm of the determinant (log-det) of the matrix, to refine the result of the trace relaxation and reduce the rank further. This heuristic is in fact equivalent to iteratively minimizing a weighted trace objective, and thus also admits an interpretation in terms of convex envelopes. Finally, applying these methods to application examples shows they work well in practice.

  Related papers:

  • Log-det Heuristic for Matrix Rank Minimization with Applications to Hankel and Euclidean Distance Matrices (M. Fazel, H. Hindi, and S. Boyd).
  • A Rank Minimization Heuristic with Application to Minimum Order System Approximation (M. Fazel, H. Hindi, and S. Boyd).
  • Portfolio Optimization with Linear and Fixed Transaction Costs and Bounds on Risk (M. Lobo, M. Fazel, and S. Boyd).
  • Rank Minimization and Applications in System Theory (M. Fazel, H. Hindi, and S. Boyd). A short tutorial paper on rank minimization.

Recent publications

• Resorce allocation in networks (and economics):

  Network Utility Maximization with Nonconcave Utilities Using Sum-of-Squares Method (M. Fazel, M. Chiang).

• Analysis of biological systems:

  Application of Robust Model Validation Using SOSTOOLS to the Study of G-Protein Signaling in Yeast (T.-M. Yi, M. Fazel, X. Liu, T. Otitoju, A. Papachristodoulou, S. Prajna, and J. Doyle).

  Stochastic Reachability Analysis in Complex Biological Networks (H. El-Samad, M. Fazel, X. Liu, A. Papachristodoulou, S. Prajna).

• Communication systems:

  Amplitude and Sign Adjustment for Peak to Average Power Reduction (M. Sharif, C. Florens, M. Fazel, B. Hassibi).

  Transient Analysis for Wireless Power Control (M. Fazel, D. Gayme, M. Chiang). To appear in Globecom'06.

• Complexity and robustness:

  Complexity and Robustness in Lattice Percolation Models and Linear Programs (M. Fazel, X. Liu, J. Doyle). In preparation.

  Complexity in Automation of SOS Proofs: An Illustrative Example (D. Gayme, M. Fazel, J. Doyle). To appear in CDC'06.

• Others:

  A Convex Relaxation for Maximum-purity Quantum Encoding Channel Design (N. Yamamoto, M. Fazel). To be submitted.

  A Methodology for Optimization of Shell and Tube Heat Exchangers in Series (A. Fakheri, M. Fazel).

*** Page under construction! ***
Last modified: Aug 2006