optimization

Examples of optimization in the following topics:

• Optimization

  • This application is called design optimization.
  • Economics is closely linked to optimization of agents.
  • Modern optimization theory includes traditional optimization theory but also overlaps with game theory and the study of economic equilibria.
  • Asset prices are also modeled using optimization theory, though the underlying mathematics relies on optimizing stochastic processes rather than on static optimization.
  • The optimization of market portfolios is an example of multi-objective optimization in economics.

• Optimization in Several Variables

  • To solve an optimization problem, formulate the function $f(x,y, \cdots )$ to be optimized and find all critical points first.
  • An optimization process that involves only a single variable is rather straightforward.
  • The same strategy applies for optimization with several variables.
  • In this atom, we will solve a simple example to see how optimization involving several variables can be achieved.
  • Therefore, the goal of the optimization is to minimize a function $S(x,y,z) = 2(xy + yz+zx)$.

• Lagrange Multiplers

  • In mathematical optimization, the method of Lagrange multipliers (named after Joseph Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints.
  • For instance, consider the following optimization problem: Maximize $f(x,y)$ subject to $g(x,y)=c$.
  • Thus, the method of Lagrange multipliers yields a necessary condition for optimality in constrained problems.

• Maximum and Minimum Values

  • Finding global maxima and minima is the goal of mathematical optimization.