Optimization

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Optimization Problem

Problem Defination

A problem of the form

\[\min f(x) \\ \text{subject to } x \in C\]
  • $f(x)$ is the objective function
  • C is called the constrained set
  • find $x$ s.t. $f(x)$ is minimized

Problem Formulation

  • Model the problem as optimization (Modelling)
    • objective function
    • constraints
  • Solve using optimization (Solver)
    • Problem class
      • linear/non-linear
      • smooth/non-smooth
      • convex/non-convex
    • use suitable solver based on problem class

Convexity

  • Indication function relates convex functions and convex sets
  • Epi-graph of $f$ is a set of all points above the function.
    • if a fucntion is convex, then its epi-graph is convex and vice-versa

Convex sets

Convex functions

  • Max functions

    \[f(x) = \text{ max } \{x_1, x_2, ..., x_n\} \text{ is convex on } \mathbb{R}^n\]

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