Simply put, constrained optimization is the set of numerical methods used to solve problems where one is looking to find minimize total cost based on inputs whose constraints, or limits, are unsatisfied. In business, finance, and economics, it is typically used to find the minimum, or set of minimums, for a cost function where the cost varies depending on the varying availability and cost of inputs, such as raw materials, labor, and other resources. It is also used to find the maximum return or set of returns that depends on varying values of financial resources available and their limits, such as the amount and cost of capital and the absolute minimum or maximum value these variables can reach. Linear, non-linear, multi-objective and distributed constraint optimization models exist. Linear programming, matrix algebra, branch and bound algorithms, and Lagrange multipliers are some of the techniques commonly used to solve such problems.
The choice of constrained optimization method depends on the specific type of problem and function to be solved. More broadly, such methods are related to constraint satisfaction problems, which require the user to satisfy a set of given constraints. Constrained optimization problems, in contrast, require the user to minimize the total cost of the unsatisfied constraints. The constraints can be an arbitrary Boolean combination of equations, such as f(x)=0, weak inequalities such as g(x)>=0, or strict inequalities, such as g(x)>0. What are known as global and local minimums and maximums may exist; this depends on whether or not the set of solutions is closed, i.e., a finite number of maximums or minimums, and/or bounded, meaning that there is an absolute minimum or maximum value.
Constrained optimization is used widely in finance and economics. For example, portfolio managers and other investment professionals use it to model the optimal allocation of capital among a defined range of investment choices to come up with a theoretical maximum return on investment and minimum risk. In microeconomics, constrained optimization may be used to minimize cost functions while maximizing output by defining functions that describe how inputs, such as land, labor and capital, vary in value and determine total output, as well as total cost. In macroeconomics, constrained optimization may be used to formulate tax policies; this may include finding a maximum value for a proposed gasoline tax that minimizes consumer dissatisfaction or yields a maximum level of consumer satisfaction given the higher cost.