This is the set of all possible values of the independent variables for
which it even makes sense to talk about the optimization problem. For example, if you
are optimizing profits from selling different quantities of different goods, you do not
expect to find the optimum profit when you sell a negative number of an item. That
negative value is outside the feasibility region.
Feasible solution
After solving the optimization problem, you need to check whether the
solution is inside the feasibility region. If it is, you have a feasible solution.
Gradient function
The gradient of a function of several variables represents a list, each
item of which is one of the partial derivatives of the function. When solving an
unconstrained optimization problem, you are trying to find the value(s) of each
independent variable that make the partial derivatives in this list equal to zero
simultaneously.
Lagrange multipliers
This is a method of solving constrained optimization problems in n
variables with m constraints. This method is similar to that of solving unconstrained
optimization problems, but instead of trying to make n derivatives simultaneously zero
we have to solve n + m equations, a considerably harder task.
Linear programming
This is a technique for solving optimization problems when you
have a single linear objective function and your constraints are all linear functions of
the different variables. It is guaranteed to find a solution, but may not find all the
solutions or the best possible solution if there are multiple solutions possible. There
are many software packages designed to solve linear programming problems. In fact,
Excel’s solver table may be manipulated to solve LP problems.
Dynamic programming
This is a generalized version of linear programming that can be
used to solve optimization problems with more complicated objective and constraint
functions. By default, this is the way solver table in Excel works.
Sensitivity and sensitivity analysis
When solving an optimization problem with
constraints, we often need to find not only the solution to a particular problem but
also what happens if we change some of the information slightly. For example, we can
optimize profits for various anticipated budget and materials constraints, but need
to know how this solution will change if the budget is a little less (or more) or the
materials are a little harder to get. This additional analysis helps us determine how
sensitive the optimum solution is to changes in the inputs and constraints.
Solver Table
This is a program in Excel that enables you to solve various types of
optimization problems
lpSolve
Linear programming package for R similar to Solver Table.