20231215
1. Introduction
- mathematical optimization
- least-squares and linear programing
- convex optimization
- exapmle
- course goals and topics
- nonlinear optimization
- brief history of convex optimization
mathmetical optimization
optimization problem
minimize \(f_0(x)\)
subject to \(f_i(x){\leq}b_i, i=1,...,m\)
- \(x=(x_1,...,x_n)\):optization variables
- \(f_0:R^n{\rightarrow}R\):objective function
- \(f_i:R^n{\rightarrow}R,i=1,...,m\):constraint functions
optimal solution \(x^*\)has smallest value of \(f_0\) among all vectors that satisfy the constraints
Examples
portfolio optimization
- variables:amounts inveated in different assets
- constraints:budget,max./min. investment per asset, minimum return
- objective:overall risk or return variance
device sizing in eletronic circuits
- variables: device widths and lengths
- constraints: manufacturing limits, timing requirements, maximum area
- objective: power consumption
data fiting
- variables: model parameters
- constraints: prior information, parameter limits
- objective: measure of misfit or prediction error
Solving optimization problems
general optimization problem
- very difficult to solve
- methods involve some compromise,e.g.,very long computation time or not always finding the solution
examples:certain problem classes can be solved efficiently and reliably
- least-squares problems
- linaer programming problems
- convex optimization problems
Least-squares
solving least-square problems
- analytical solution: \(x^*=(A^TA)^{-1}A^Tb\)
-reliabe and efficient algorithems and software - computation