Constrained Minimization: Inequality Constraints
The minimization of a function whose variables must satisfy inequality constraints is considered here. Because of their nature, any number of inequality constraints can be imposed. To introduce the subject, conditions for locating a boundary minimal point of a function of one independent variable (see Section 2.2) are determined using the direct approach. Next, a device for converting an inequality constraint into an equality constraint, called a slack variable, is used to solve the same problem. Then, inequality constraints are treated more formally by minimizing a function of two variables subject to one inequality constraint. For a minimal point on the boundary, two second variation conditions must be satisfied. The optimal point must be a minimum with respect to admissible comparison points on the boundary as if the boundary were an equality constraint, and it must be a minimum with respect to admissible comparison points off the boundary. The latter requires that the Lagrange multiplier associated with the inequality constraint (≤ 0) must be nonnegative at a minimal point on the boundary. Next, eliminating bounded variables in favor of unbounded variables is discussed. This approach is demonstrated by solving some example problems from linear programming. Finally, some comments are made about the general parameter optimization problem.
KeywordsLagrange Multiplier Performance Index Equality Constraint Inequality Constraint Minimal Point
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