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Classical optimization

Classical optimization

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Before the arrival of the digital computer age, several optimization approaches and theories were utilized to solve simple problems with one or few variables. Powerful tools such as the single-variable Taylor expansion, the multivariable Taylor expansion, and the Karush-Kuhn-Tucker (KKT) conditions were utilized in solving unconstrained and constrained optimization problems. Substitution methods were also used to convert constrained optimization problems to unconstrained optimization ones. Other methods such as the method of constrained variations were also used for solving simple constrained optimization problems. In this chapter, I focus on explaining some of these classical approaches. The theoretical bases of these techniques are relevant for numerical optimization approaches to be addressed in the following chapters.

Chapter Contents:

  • 3.1 Introduction
  • 3.2 Single-variable Taylor expansion
  • 3.3 Multidimensional Taylor expansion
  • 3.4 Meaning of the gradient
  • 3.5 Optimality conditions
  • 3.6 Unconstrained optimization
  • 3.7 Optimization with equality constraints
  • 3.7.1 Method of direct substitution
  • 3.7.2 Method of constrained variation
  • 3.8 Lagrange multipliers
  • 3.9 Optimization with inequality constraints
  • 3.10 Optimization with mixed constraints
  • A3.1 Quadratic programming
  • A3.2 Sequential quadratic programming
  • References
  • Problems

Inspec keywords: optimisation

Other keywords: multivariable Taylor expansion; KKT conditions; classical optimization approach; numerical optimization approach; substitution methods; unconstrained optimization problems; Karush-Kuhn-Tucker conditions; constrained optimization problems; digital computer; single-variable Taylor expansion

Subjects: Optimisation techniques; Optimisation techniques; Optimisation

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