Research Interests

My research program lies in the areas of mathematical optimization, mathematics of operations research and foundations of computational mathematics. The focus of the program is on commonly encountered optimization problems (including linear and nonlinear programming and combinatorial optimization problems). The research program aims at understanding the structure of some problems in the aforementioned class and provably efficient methods for solving them. Currently the focus of my interest lies with linear optimization over convex cones and interior-point algorithms.

Publications

The file list.html includes a list of my publications and pointers to the ps and/or pdf files of some papers. The following is a list of recent research reports.

  • Optimization problems over unit distance representations of graphs (with M. K. De Carli Silva), January 2012 arXiv:1010.6036.pdf

  • Complexity Analyses of Bienstock-Zuckerberg and Lasserre Relaxations on the Matching and Stable Set Polytopes (with Y. H. (Gary) Au), Proceedings of the 15th International Conference on Integer Programming and Combinatoral Optimization (IPCO 2011) New York, NY, USA, June 15-17, 2011, Lecture Notes in Computer Science 6655, Springer 2011, pp. 14-26.

  • Sufficient conditions for low-rank matrix recovery, translated from sparse signal recovery (with L. Kong and N. Xiu), June 2011 arXiv:1106.3276.pdf

  • Near-perfect graphs with polyhedral N_+(G) (with S. M. Bianchi, M. S. Escalante and G. L. Nasini), Electronic Notes on Discrete Mathematics 37 (2011) 393-398.

  • Some advances on Lovász-Schrijver N_+(.) relaxations on the fractional stable set polytope (with S. M. Bianchi, M. S. Escalante and G. L. Nasini), Electronic Notes on Discrete Mathematics 37 (2011) 189-194.

  • Local quadratic convergence of polynomial-time interior-point methods for conic optimization problems (with Yu. Nesterov), CORE Discussion Paper 2009/72, Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (UCL), Belgium, November 2009 CORE-2009-72.pdf

  • Homogeneous cone complementarity problems and P properties (with L. Kong and N. Xiu), April 2009 (revised: November 2010) arXiv:0904.1827.pdf

  • Equivalent conditions for Jacobian nonsingularity in linear symmetric cone programming (with L. Kong and N. Xiu), Journal of Optimization Theory and Applications 148 (2011) 364-389 corr2008-12.pdf

  • Strong duality and minimal representations for cone optimization (with H. Wolkowicz), Research Report CORR 2008-07, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, August 2008 (revised: December 2011) corr2008-07.pdf.

  • Clarke generalized Jacobian of the projection onto symmetric cones (with L. Kong and N. Xiu), Set-Valued and Variational Analysis 17 (2009) 135-151 journal-version.pdf, initial-version.pdf

  • On the polyhedral lift-and-project methods and the fractional stable set polytope ( Y. H. (Gary) Au), Discrete Optimization 6 (2009) 206-213 corr2008-03.pdf.

  • The Fischer-Burmeister complementarity function on Euclidean Jordan algebras (with L. Kong and N. Xiu), Pacific Journal of Optimization 6 (2010) 423-440 corr2007-17.pdf.

  • Maximum utility product pricing models and algorithms based on reservation prices (with R. Shioda and T. G. J. Myklebust), Computational Optimization and Applications 48 (2011) 157--198 corr2007-08.pdf journal-version.pdf.

  • Monotonicity of Löwner operators and its applications to symmetric cone complementarity problems (with L. Kong and N. Xiu), Mathematical Programming A, to appear corr2007-07.pdf.

  • Self-concordant barriers for convex approximations of structured convex sets (with A. Nemirovski), Foundations of Computational Mathematics 10 (2010) 485-525 corr2007-03.pdf, article.

  • Applications of deterministic optimization techniques to some probabilistic choice models for product pricing using reservation prices (with R. Shioda and B. Hui), Research Report CORR 2007-02, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, February 2007 (revised: February 2009) corr2007-02.pdf.

  • Vector-valued implicit Lagrangian for symmetric cone complementarity problems (with L. Kong and N. Xiu), Asia-Pacific Journal of Operational Research 26 (2009) 199-233 corr2006-24r.pdf.

  • Large scale portfolio optimization with piecewise linear transaction costs (with M. Potaptchik and H. Wolkowicz), Optimization Methods and Software 23 (2008) 929-952 corr2006-19.pdf.

  • Lehman matrices (with G. Cornuéjols and B. Guenin), Journal of Combinatorial Theory, Series B 99 (2009) 531-556 corr2006-18.pdf.

  • Invariance and efficiency of convex representations (with C. B. Chua), Mathematical Programming B 111 (2008) 113-140 corr2004-18.pdf. The original publication is available at http://www.springerlink.com.

  • Unification of lower-bound analyses of the lift-and-project rank of combinatorial optimization polyhedra (with S.-P. Hong), Discrete Appl. Math. 156 (2008) 25-41 corr2004-05.ps.

  • Clustering via minimum volume ellipsoids (with R. Shioda), Computational Optimization and Applications 37 (2007) 247-295 corr2005-12.pdf.

  • Some applications of symmetric cone programming in financial mathematics (with A. Li), Transactions on Operational Research 17 (2006) 1-19 survey-TOR1.pdf.

  • Strengthened existence and uniqueness conditions for search directions in semidefinite programming, (with H. Wolkowicz), Linear Algebra and its Applications 400 (2005) 31-60 corr2003-20.ps.

  • Lift-and-project ranks and antiblocker duality (with L. Lipták), Operations Research Letters 33 (2005) 35-41 corr2003-16.ps.

  • ''Cone-free'' primal-dual path-following and potential reduction polynomial time interior-point methods (with A. Nemirovski), Mathematical Programming A 102 (2005) 261-294 corr2002-32.ps.