What is Optimization? ( Optimization Tree/ Wiki at NEOS)

• Quotes:

  • Two quotes from Tjalling C. Koopmans, (1975 Nobel Memorial Lecture):
    • ``best use of scarce resources''
    • ``Mathematical Methods of Organizing and Planning of Production"
  • Quote from Roger Fletcher (text, 1987):
    "The subject of optimization is a fascinating blend of heuristics and rigour, of theory and experiment."

• Basic Models     (NEOS Solvers)

  • Unconstrained Optimization

    minimizex f(x)

    where x is an n dimensional vector and f is a continuous real valued function.

  • Linear Programming

    minimizex cTx     subject to    Ax = b,   x &ge 0

    where x is an n dimensional vector and A is an m by n matrix.
    (NEOS Sample Submissions)

  • Quadratic Programming

    minimizex xTQ x +cTx     subject to    Ax = b,   Bx &ge d

    where x is an n dimensional vector and A is an m by n matrix and B is a p by n matrix.
    
    Portfolio selection; Efficient Frontier; Formulations; demo
    NEOS Sample Submission (risk management)

  • Semidefinite (and Second Order) Cone Programming

    minimizeX < C,X>     subject to    Aop(X) = b,   X psd

    where X is a symmetric positive semidefinite matrix and Aop is a linear transformation.
    Application to combinatorial optimization - the Max-Cut Problem
    ( NEOS Sample Submissions,    duality (pg 5-30))

  • General Nonlinear Programming

    minimizex f(x) subject to    gk(x) &le bk,   k = 1,...,m

    where x is an n dimensional vector and f and g_k are real valued (sufficiently smooth) functions.

  • Infinite Dimensional (Control) Programming

    (from Luenberger text, 1969)
        A nonconvex problem with strong duality (TRS pg 5-14)

Supplementary Information

• NEOS Wiki

  the ubiquitous online source for optimization.

• NEOS Guide: Case Studies; NEOS Server Stories

• Suggested Texts/References:

  • Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications

    by Aharon Ben-Tal and Arkadi Nemirovski
    MPS/SIAM Series on Optimization 2
    URL: http://www.ec-securehost.com/SIAM/MP02.html

  • Numerical Optimization

    Jorge Nocedal Stephen J. Wright
    URL: http://www.ece.northwestern.edu/~nocedal/book/num-opt.html
  • Lecture Notes for/from Boyd and Vandenberghe book, Convex Optimization.

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