Performing constrained optimization

Simulation design problems tend to have conflicting goals. Usually there is a desired ideal response that cannot be obtained exactly due to either modeling or physical constraints. The system response is determined by a few variables that may in themselves be bounded. Mathematically, such a situation is represented by:

Minimize or maximize g(X) subject to

glb_i ≤ g_i(X) ≤ gub_i              for i=1,...,m,
xlb_j ≤ x_j ≤ xub_j                     for j=1,...,n,

X is a vector on n variables, x₁ ,...,x_n, and the functions g₁ ,...,g_m all depend on X.

The function to be minimized g(X) is called the objective (or cost) function. Constraints are given by the function g_i(X). The decision variables x₁ ,...,x_n, may have bounds. The analogous data structures are represented by the following blocks:

Upper and lower bounds are set in the globalConstraint block for the constraint functions, and upper and lower bounds are set in the parameterUnknown block for the decision variables.

Embed uses first partial derivatives of each function g_i with respect to each variable x_j. These are automatically computed by finite difference approximations. After an initial data entry segment, the program operates in two phases. If the initial values of the variables you supply do not satisfy all g_i constraints, a Phase I optimization is started. The Phase I objective function is the sum of the constraint violations. This optimization run terminates either with a message that the problem is infeasible or with a feasible solution. Beware that if an infeasibility message is produced, it may be because the program became stuck at a local minimum of the Phase I objective. In this case, the problem may actually have feasible solutions.

Phase II begins with a feasible solution — either found by Phase I or with you providing a starting point — and attempts to optimize the objective function. After Phase II, a full optimization cycle has been completed and summary output is provided.