Linear and Convex Optimization

Convexity and Optimization – Part II

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166 pages
This book presents the mathematical basis for linear and convex optimization with an emphasis on the important concept of duality. The simplex algorithm is also described in detail.
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Lars-Åke Lindahl obtained his mathematical education at Uppsala University and Institut Mittag-Leffler and got a Ph.D. in Mathematics in 1971 with a thesis on Harmonic Analysis. Shortly thereafter he was employed as senior lecturer in Mathematics at Uppsala University, where he remained until his reti...


This book, the second in a series of three on Convexity and Optimization, presents classical mathematical results for linear and convex optimization with an emphasis on the important concept of duality. Equivalent ways of formulating an optimization problem are presented, the Lagrange function and the dual problem are introduced, and conditions for strong duality are given. The general results are then specialized to the linear case, i.e. to linear programming, and the simplex algorithm is described in detail.

  1. Optimization
    1. Optimization problems
    2. Classification of optimization problems
    3. Equivalent problem formulations
    4. Some model examples
  2. The Lagrange function
    1. The Lagrange function and the dual problem
    2. John’s theorem
  3. Convex optimization
    1. Strong duality
    2. The Karush-Kuhn-Tucker theorem
    3. The Lagrange multipliers
  4. Linear programming
    1. Optimal solutions
    2. Duality
  5. The simplex algorithm
    1. Standard form
    2. Informal description of the simplex algorithm
    3. Basic solutions
    4. The simplex algorithm
    5. Bland’s anti cycling rule
    6. Phase 1 of the simplex algorithm
    7. Sensitivity analysis
    8. The dual simplex algorithm
    9. Complexity