What is equality constrained optimization?
In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables.
What is the goal of constrained optimization?
Constrained optimization is a set of methods designed to identify efficiently and systematically the best solution (the optimal solution) to a problem characterized by a number of potential solutions in the presence of identified constraints.
Which is an optimization problem subject to inequality constraints?
The optimization problems subject to inequality constraints can be generally formulated as: (185) Again, to visualize the problem we first consider an example with and , as shown in the figure below for the minimization (left) and maximization (right) of subject to .
Is the gradient unrestricted in an inequality constrained problem?
For an equality constrained problem, the direction of the gradient is of no concern, i.e., the sign of is unrestricted; but here for an inequality constrained problem, the sign of needs to be consistent with those shown in Table 188, other wise the constraints may be inactive.
When is an inequality constraint no longer valid?
Second, if the unconstrained extremum is inside the feasible region, i.e., the inequality constraint is inactive, then the problem is actually unconstrained and the results above are no longer valid. The solution of this unconstrained problem is
Which is an example of an inequality constraint?
Consider the following two possible cases. First, if the unconstrained extremum at which is outside the feasible region, i.e., the inequality constraint is active, then the constrained solution must be on the boundary of the feasible region, i.e., , different from the unconstrained solution, i.e., ;