Equality and Inequality Constraints

Inequality constraints involve additional difficulties it is not possible to know a priori which constraints are active at the solution. 

Let us indicate with q(x) the ng inequality constraints that are considered active and with w(x) the n passive inequality constraints. Since q(x) can be considered as equality constraints, the Lagrange function becomes 

1) Some passive constraints are violated. In this case, we have to make them active. 

2) The solution is feasible, but some active constraints must be made passive to allow the search region to be extended to further decrease the function. 

3) The solution is feasible and none of the active constraints can be made passive. In this case, the solution is achieved. 

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Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 

Chapter 9 deals with estimation of parameters subject to equality and inequality constraints whereas Chapter 10 examines systems described by partial differential equations (PDE). Examples are provided in Chapters 14 and 18. 

The objective is to maximize the profit which is calculated by a cost model of sales revenues, production costs, storage costs, and penalties for lateness and for finishing line start-ups and shut-downs. The cost model adds some equality and inequality constraints with associated real valued variables for the sales, deficits, and the storage, but it does not further restrict the feasibility of the production decisions. 

In Section 1.5 we briefly discussed the relationships of equality and inequality constraints in the context of independent and dependent variables. Normally in design and control calculations, it is important to eliminate redundant information and equations before any calculations are performed. Modem multivariable optimization software, however, does not require that the user clearly identify independent, dependent, or superfluous variables, or active or redundant constraints. If the number of independent equations is larger than the number of decision variables, the software informs you that no solution exists because the problem is overspecified. Current codes have incorporated diagnostic tools that permit the user to include all possible variables and constraints in the original problem formulation so that you do not necessarily have to eliminate constraints and variables prior to using the software. Keep in mind, however, that the smaller the dimensionality of the problem introduced into the software, the less time it takes to solve the problem. 

Problems Containing both Equality and Inequality Constraints... 

When both equality and inequality constraints are present, the KTC are stated as follows Let the problem be... 

The general form of the quadratic penalty function for a problem of the form (8.25)-(8.26) with both equality and inequality constraints is... 

Hence the QP subproblem now has both equality and inequality constraints and must be solved by some iterative QP algorithm. 

With feasible path strategies, as the name implies, on each iteration you satisfy the equality and inequality constraints. The results of each iteration, therefore, provide a candidate design or feasible set of operating conditions for the plant, that is, sub-optimal. Infeasible path strategies, on the other hand, do not require exact solution of the constraints on each iteration. Thus, if an infeasible path method fails, the solution at termination may be of little value. Only at the optimal solution will you satisfy the constraints. 

The disadvantage of these NLP algorithms is the large amount of computation time required relative to the successive linearisation algorithm. Nevertheless, their range of application is wider, and they are able to manage nonlinear objective functions, equality and inequality constraints, and bounds on variables. 

P x] = 1 if and only if x satisfies the model equations (equality and inequality constraints)... 

Definition 3.2.1 (Feasible Point(s)) A point jc g X satisfying the equality and inequality constraints is called a feasible point. Thus, the set of all feasible points of /(jc) is defined as... 

In the definition of the Lagrange function L(x, A, m) (see section 3.2.2) we associated Lagrange multipliers with the equality and inequality constraints only. If, however, a Lagrange multiplier Mo is associated with the objective function as well, the definition of the weak Lagrange function L (x, A, fi) results that is,... 

If the primal problem at iteration k is feasible, then its solution provides information on xk, f(xk, yk ), which is the upper bound, and the optimal multiplier vectors k, for the equality and inequality constraints. Subsequently, using this information we can formulate the Lagrange function as... 

The solution of the feasibility problem (FP) provides information on the Lagrange multipliers for the equality and inequality constraints which are denoted as Ak,fik respectively. Then, the Lagrange function resulting from on infeasible primal problem at iteration k can be defined as... 

Remark 2 Note that using the above basic equivalence relations we can systematically convert any arbitrary propositional logic expression into a set of linear equality and inequality constraints. The basic idea in an approach that obtains such an equivalence is to reduce the logical expression into its equivalent conjunctive normal form which has the form ... 

Activation and deactivation of inequality and equality constraints can be obtained in a similar way. For instance, let us consider the model of a process unit i that consists of one inequality g(x) < 0 and one equality h(x) = 0. If the process unit i does not exist (i.e., yi = 0), then both the equality and inequality should be relaxed. If, however, the process unit i exists (i.e., yi = 1), then the inequality and equality constraints should be activated. This can be expressed by introducing positive slack variables for the equality and inequality constraint and writing the model as... 

Nonlinear and Mixed-Integer Optimization addresses the problem of optimizing an objective function subject to equality and inequality constraints in the presence of continuous and integer variables. These optimization models have many applications in engineering and applied science problems and this is the primary motivation for the plethora of theoretical and algorithmic developments that we have been experiencing during the last two decades. 

Equality and inequality constraints on any proportion or any group of proportions (e.g., 80% < proportion of support within the catalyst <90%, the proportion of Mg within the active components + the proportion of Mn within the active components =50%). 

Equality and inequality constraints on the number of simultaneously present components or on the number of those among them that belong to a given class of the component types hierarchy (e.g., the number of all component lies between 3 and 6, the number of active components is at least twice that of dopants). 

Constraints, Definition, Classification Equality and Inequality Constraints Based on Chemical or... 

Algebraic optimization with equality and inequality constraints... 

Successive Quadratic Programming (SQP) The above approach to finding the optimum is called a feasible path method, as it attempts at all times to remain feasible with respect to the equality and inequality constraints as it moves to the optimum. A quite different method exists called the Successive Quadratic Programming (SQP) method, which only requires one be feasible at the final solution. Tests that compare the GRG and SQP methods generally favor the SQP method so it has the reputation of being one of the best methods known for nonlinear optimization for the type of problems considered here. 

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