Semidefinite and Second Order Cone Programming

16:711.530

Fall 2012

This course is a seminar covering semidefinite, Second order and related optimization problems. For more detailed description read the course announcement.

Each student is required to be responsible for a lecture, take careful notes and transcribe them by into LaTeX. Then the Latex document should be e-mailed to me before the Sunday following the lecture. I will edit the scribe notes and post them in in this page. For preparing your lecture notes use this template. Also make sure that all vectors are represented by boldface letters e.g b for vector b. Use the file myDefs.sty for useful predefined commands.

Also each student is required to either read a current research paper, or finish a (possibly computational project) and present it in class in a 30-40 minute talk.

Lecture Notes: last updated on 01/11/13

Here are lecture notes as scribed by you. I have done minimal editing. Be careful, there may be errors in these notes.

Presentations

Possible topics for in-class presentation:

Suggested topics:

  1. Robust optimization, and the possibility of expressing robust problems as SDP or SOCP

  2. Various extensions of the Goemans-Williamson approximation of the MAX-CUT and MAX-2SAT problems including Nesterov's generalization to non-convex or binary quadratic programming

  3. Survey of positive and sum-of-squares polynomials. This topic can be pursued in different sub-topics, for example: Expression of combinatorial optimization problems as positive polynomial problems and their relaxations to sum-of-square problems (Lasserre's and Parillo's papers), nonnegative polynomials and moment problems over intervals(Karlin-Studden, Nesterov, Parrilo), gap between positive and sum-of-square multivariate polynomial. Another direction is review of real algebraic geometry, Schmudgen's theorem, Putinar's lemma, and applications,

  4. Applications in Engineering and finance: for example signal processing, portfolio optimization, statistical design of experiments, matrix completion problems and applications

  5. Applications in control theory: The S procedure, Lypunov operator, Youla's Theorem and applications, KYP Lemma and connections to SDP.

  6. Students suggestion: If you have ideas about project or presentation, let me know.

Reading list:

Following is a list of useful references to read. More may be added as needed:

@book {MR1778223,
     TITLE = {Handbook of semidefinite programming},
    SERIES = {International Series in Operations Research \& Management
              Science, 27},
    EDITOR = {Wolkowicz, Henry and Saigal, Romesh and Vandenberghe, Lieven},
      NOTE = {Theory, algorithms, and applications},
 PUBLISHER = {Kluwer Academic Publishers},
   ADDRESS = {Boston, MA},
      YEAR = {2000},
     PAGES = {xxviii+654},
      ISBN = {0-7923-7771-0},
   MRCLASS = {90-00 (90C22)},
  MRNUMBER = {MR1778223 (2001k:90001)},
MRREVIEWER = {K. G. Murty},
}
                
@article {MR1315703,
    AUTHOR = {Alizadeh, Farid},
     TITLE = {Interior point methods in semidefinite programming with
              applications to combinatorial optimization},
   JOURNAL = {SIAM J. Optim.},
  FJOURNAL = {SIAM Journal on Optimization},
    VOLUME = {5},
      YEAR = {1995},
    NUMBER = {1},
     PAGES = {13--51},
      ISSN = {1052-6234},
     CODEN = {SJOPE8},
   MRCLASS = {90C27},
  MRNUMBER = {MR1315703 (95k:90065)},
MRREVIEWER = {Edward J. Anderson},
}
@article {MR1379041,
    AUTHOR = {Vandenberghe, Lieven and Boyd, Stephen},
     TITLE = {Semidefinite programming},
   JOURNAL = {SIAM Rev.},
  FJOURNAL = {SIAM Review. A Publication of the Society for Industrial and
              Applied Mathematics},
    VOLUME = {38},
      YEAR = {1996},
    NUMBER = {1},
     PAGES = {49--95},
      ISSN = {0036-1445},
     CODEN = {SIREAD},
   MRCLASS = {90-02 (90C25)},
  MRNUMBER = {MR1379041 (96m:90005)},
MRREVIEWER = {Henry Wolkowicz},
}
@article {MR1971381,
    AUTHOR = {Alizadeh, F. and Goldfarb, D.},
     TITLE = {Second-order cone programming},
      NOTE = {ISMP 2000, Part 3 (Atlanta, GA)},
   JOURNAL = {Math. Program.},
  FJOURNAL = {Mathematical Programming. A Publication of the Mathematical
              Programming Society},
    VOLUME = {95},
      YEAR = {2003},
    NUMBER = {1, Ser. B},
     PAGES = {3--51},
      ISSN = {0025-5610},
   MRCLASS = {90C08 (90C25)},
  MRNUMBER = {MR1971381 (2004j:90060)},
}
@article {MR1655138,
    AUTHOR = {Lobo, Miguel Sousa and Vandenberghe, Lieven and Boyd, Stephen
              and Lebret, Herv{\'e}},
     TITLE = {Applications of second-order cone programming},
      NOTE = {ILAS Symposium on Fast Algorithms for Control, Signals and
              Image Processing (Winnipeg, MB, 1997)},
   JOURNAL = {Linear Algebra Appl.},
  FJOURNAL = {Linear Algebra and its Applications},
    VOLUME = {284},
      YEAR = {1998},
    NUMBER = {1-3},
     PAGES = {193--228},
      ISSN = {0024-3795},
     CODEN = {LAAPAW},
   MRCLASS = {90C08 (90C20 90C90 93B40)},
  MRNUMBER = {MR1655138 (99k:90103)},
MRREVIEWER = {Jos F. Sturm},
}
@article {MR1412228,
    AUTHOR = {Goemans, Michel X. and Williamson, David P.},
     TITLE = {Improved approximation algorithms for maximum cut and
              satisfiability problems using semidefinite programming},
   JOURNAL = {J. Assoc. Comput. Mach.},
  FJOURNAL = {Journal of the Association for Computing Machinery},
    VOLUME = {42},
      YEAR = {1995},
    NUMBER = {6},
     PAGES = {1115--1145},
      ISSN = {0004-5411},
     CODEN = {JACOAH},
   MRCLASS = {90C27 (68Q25 90C35)},
  MRNUMBER = {MR1412228 (97g:90108)},
MRREVIEWER = {Nikolay N. Ivanov},
}
@article {MR1464765,
    AUTHOR = {Goemans, Michel X.},
     TITLE = {Semidefinite programming in combinatorial optimization},
      NOTE = {Lectures on mathematical programming (ismp97) (Lausanne,
              1997)},
   JOURNAL = {Math. Programming},
  FJOURNAL = {Mathematical Programming},
    VOLUME = {79},
      YEAR = {1997},
    NUMBER = {1-3, Ser. B},
     PAGES = {143--161},
      ISSN = {0025-5610},
     CODEN = {MHPGA4},
   MRCLASS = {90C27 (90-02)},
  MRNUMBER = {MR1464765 (98g:90028)},
MRREVIEWER = {D. de Werra},
}
@article {MR1618100,
    AUTHOR = {Nesterov, Yu.},
     TITLE = {Semidefinite relaxation and nonconvex quadratic optimization},
   JOURNAL = {Optim. Methods Softw.},
  FJOURNAL = {Optimization Methods and Software},
    VOLUME = {9},
      YEAR = {1998},
    NUMBER = {1-3},
     PAGES = {141--160},
      ISSN = {1055-6788},
     CODEN = {OMSOE2},
   MRCLASS = {90C20 (90C26)},
  MRNUMBER = {MR1618100 (99a:90144)},
}
@incollection {MR1778235,

    AUTHOR = {Nesterov, Yuri and Wolkowicz, Henry and Ye, Yinyu},
     TITLE = {Semidefinite programming relaxations of nonconvex quadratic
              optimization},
 BOOKTITLE = {Handbook of semidefinite programming},
    SERIES = {Internat. Ser. Oper. Res. Management Sci.},
    VOLUME = {27},
     PAGES = {361--419},
 PUBLISHER = {Kluwer Acad. Publ.},
   ADDRESS = {Boston, MA},
      YEAR = {2000},
   MRCLASS = {90C22 (90C26)},
  MRNUMBER = {MR1778235},
}
@book {MR0204922,
    AUTHOR = {Karlin, Samuel and Studden, William J.},
     TITLE = {Tchebycheff systems: {W}ith applications in analysis and
              statistics},
    SERIES = {Pure and Applied Mathematics, Vol. XV},
 PUBLISHER = {Interscience Publishers John Wiley \& Sons, New
              York-London-Sydney},
      YEAR = {1966},
     PAGES = {xviii+586},
   MRCLASS = {41.00},
  MRNUMBER = {MR0204922 (34 \#4757)},
MRREVIEWER = {G. G. Lorentz},
}
@incollection {MR1748764,
    AUTHOR = {Nesterov, Yurii},
     TITLE = {Squared functional systems and optimization problems},
 BOOKTITLE = {High performance optimization},
    SERIES = {Appl. Optim.},
    VOLUME = {33},
     PAGES = {405--440},
 PUBLISHER = {Kluwer Acad. Publ.},
   ADDRESS = {Dordrecht},
      YEAR = {2000},
   MRCLASS = {90C22 (12D15 90C34 90C51)},
  MRNUMBER = {MR1748764 (2001b:90063)},
MRREVIEWER = {Jos F. Sturm},
}
@article {MR1814045,
    AUTHOR = {Lasserre, Jean B.},
     TITLE = {Global optimization with polynomials and the problem of
              moments},
   JOURNAL = {SIAM J. Optim.},
  FJOURNAL = {SIAM Journal on Optimization},
    VOLUME = {11},
      YEAR = {2000/01},
    NUMBER = {3},
     PAGES = {796--817 (electronic)},
      ISSN = {1052-6234},
   MRCLASS = {90C26 (90C22)},
  MRNUMBER = {MR1814045 (2002b:90054)},
MRREVIEWER = {Jos F. Sturm},
}
@article {MR1993050,
    AUTHOR = {Parrilo, Pablo A.},
     TITLE = {Semidefinite programming relaxations for semialgebraic
              problems},
      NOTE = {Algebraic and geometric methods in discrete optimization},
   JOURNAL = {Math. Program.},
  FJOURNAL = {Mathematical Programming. A Publication of the Mathematical
              Programming Society},
    VOLUME = {96},
      YEAR = {2003},
    NUMBER = {2, Ser. B},
     PAGES = {293--320},
      ISSN = {0025-5610},
   MRCLASS = {90C22 (13J30 14P10 90C10)},
  MRNUMBER = {MR1993050 (2004g:90075)},
MRREVIEWER = {Franz Rendl},
}
@article {MR2218121,
    AUTHOR = {Ben-Tal, Aharon and Boyd, Stephen and Nemirovski, Arkadi},
     TITLE = {Extending scope of robust optimization: comprehensive robust
              counterparts of uncertain problems},
   JOURNAL = {Math. Program.},
  FJOURNAL = {Mathematical Programming. A Publication of the Mathematical
              Programming Society},
    VOLUME = {107},
      YEAR = {2006},
    NUMBER = {1-2, Ser. B},
     PAGES = {63--89},
      ISSN = {0025-5610},
   MRCLASS = {90C31 (90C05 90C25 90C34 93C55)},
  MRNUMBER = {MR2218121 (2006m:90169)},
MRREVIEWER = {Oliver Stein},
}