### Constrained Optimization in Engineering Design

We now begin our discussion of gradient-based constrained optimization. Recall that we looked at gradient-based unconstrained optimization and learned about the necessary and sufficient conditions for an unconstrained optimum, various search directions, conducting a line search, and quasi-Newton methods. We will build on that foundation as we extend the theory to problems with constraints.

At an unconstrained local optimum, there is no direction in which we can move to improve the objective function. We can state the necessary conditions mathematically as Grad(f)=0. At a constrained local optimum, there is no feasible direction in which we can move to improve the objective. That is, there may be directions from the current point that will improve the objective, but these directions point into infeasible space.

The necessary conditions for a constrained local optimum are called the Kuhn-Tucker Conditions, and these conditions play a very important role in constrained optimization theory and algorithm development.

In the previous section we examined the necessary and sufficient conditions for a constrained optimum. We did not, however, discuss any algorithms for constrained optimization. The purpose of this section is to review three popular techniques for constrained optimization:

A refinery must produce 100 gallons of gasoline and 160 gallons of diesel to meet customer demands. The refinery would like to minimize the cost of crude and two crude options exist. The less expensive crude costs $80 USD per barrel while a more expensive crude costs$95 USD per barrel. Each barrel of the less expensive crude produces 10 gallons of gasoline and 20 gallons of diesel. Each barrel of the more expensive crude produces 15 gallons of both gasoline and diesel. Find the number of barrels of each crude that will minimize the refinery cost while satisfying the customer demands.