Constraints violation

Step 5. Since the constraint was violated, the non-monotonic planner attempts to identify the proper operator among the set of available operators, which can alleviate the constraint violation, if this operator were forced to be applied before the constraint violation. In this example such an operator is (PURGE X). Thus, (PURGE X) must be applied before (ESTABLISH-FLOW METHANE). The variable X remains temporarily unbound to any particular value. 

Step 2. Identification of potential constraint violations, resulting from selected operations during the construction of partial plans. 

The detection of potential mixing constraints violations at an intermediate operating state is based on a worst-case scenario with the maximum potential for the species to coexist, and which corresponds to the following situation ... 

Theorem 8. The algorithm WHITE-KNIGHT will detect every manipulation, that will possibly avoid a quantitative constraint violation, as a White Knight. 

We use a starting point of (0.75, 0). The feasible region is shown in Figure 8.15 as the dashed line segment. At the initial point constraint 1 is strictly satisfied, but constraints 2 and 3 are violated. GRG constructs the phase I objective function as the sum of the absolute values of all constraint violations. For this case the sum of the infeasibilities (sinf) is... 

This weighted sum of absolute values in e(x) was also discussed in Section 8.4 as a way of measuring constraint violations in an exact penalty function. We proceed as we did in that section, eliminating the nonsmooth absolute value function by introducing positive and negative deviation variables dpt and dnt and converting this nonsmooth unconstrained problem into an equivalent smooth constrained problem, which is... 

Note that independent variable, time, disappears from this problem. While final time constraints in (16) appear naturally in (17), other constraints that need to be enforced over the time domain are difficult to handle. For example, Sargent and Sullivan (1977) converted these to final time constraints by integrating the square of the constraint violations and forcing these to be less than a tolerance at final time this however, leads to degeneracies in solving the nonlinear program. 

A HEN is feasible for assumed, fixed values of uncertain variables 8 if control variables z can be found to satisfy the reduced set of constraints. The HEN feasibility test can be formulated as follows to minimize the maximum constraint violations (Halemane and Grossmann, 1983) ... 

The solution 8C of max-min problem (8) defines a critical point for feasible operation—it is the point where uncertainty range 0 is closest to feasible region R if (d) 0 (Fig. 3a), or it is the point where maximum constraint violations occur if (d) > 0 (Fig. 3b). In qualitative terms, the critical points in the resilience test are the worst case conditions for feasible operation. 

The two contaminant plumes are represented in the first stage of the optimization formulation with a set of 110 control points along the plume boundaries. These same control point locations are used as starting points for particles when forward tracking is used in the second stage of the solution process. For the unconfined simulation, additional constraints are included to require a minimum saturated thickness of 1.5 m at each well cell. Both confined and unconfined assumptions are simulated under two sets of penalty parameters. Recall that the solution algorithm uses the penalty method for the plume capture constraints, in which each constraint violation is multiplied by a penalty parameter and added to the objective function. 

When the penalty multiplier is large, constraint violations are discouraged and the search algorithm is biased toward satisfying constraints at the expense of minimizing pumping. 

Find the maximum constraint violation over the set V, c, and the corresponding value of v, v ... 

For each vertex, v , in v choose o to minimize the maximum constraint violation for the new design parameters. 

The calculation methods for the flexibility index discussed below can be applied to step 3 of the worst-case design optimization procedure to maximize constraint violation instead of minimizing 8. 

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