Constrained equality/inequality

Equality- and Inequality-Constrained Problems—Kuhn-Tucker Multipliers Next a point is tested to see if it is an optimum one when there are inequality constraints. The problem is... 

Inequality Constrained Problems To solve inequality constrained problems, a strategy is needed that can decide which of the inequality constraints should be treated as equalities. Once that question is decided, a GRG type of approach can be used to solve the resulting equality constrained problem. Solving can be split into two phases phase 1, where the go is to find a point that is feasible with respec t to the inequality constraints and phase 2, where one seeks the optimum while maintaining feasibility. Phase 1 is often accomphshed by ignoring the objective function and using instead... 

In parameter estimation we are occasionally faced with an additional complication. Besides the minimization of the objective function (a weighted sum of errors) the mathematical model of the physical process includes a set of constrains that must also be satisfied. In general these are either equality or inequality constraints. In order to avoid unnecessary complications in the presentation of the material, constrained parameter estimation is presented exclusively in Chapter 9. 

An important class of the constrained optimization problems is one in which the objective function, equality constraints and inequality constraints are all linear. A linear function is one in which the dependent variables appear only to the first power. For example, a linear function of two variables x and x2 would be of the general form ... 

Now consider the imposition of inequality [g(x) < 0] and equality constraints 7i(x) = 0] in Fig. 3-55. Continuing the kinematic interpretation, the inequality constraints g(x) < 0 act as fences in the valley, and equality constraints h(x) = 0 act as "rails. Consider now a ball, constrained on a rail and within fences, to roll to its lowest point. This stationary point occurs when the normal forces exerted by the fences [- Vg(x )] and rails [- V/i(x )] on the ball are balanced by the force of gravity [— Vfix )]. This condition can be stated by the following Karush-Kuhn-Tucker (KKT) necessary conditions for constrained optimality ... 

The inequality constraints. The operation of each compressor is constrained so that the discharge pressure is greater than or equal to the suction pressure... 

Finally in this chapter, an alternative approach for nonlinear dynamic data reconciliation, using nonlinear programming techniques, is discussed. This formulation involves the optimization of an objective function through the adjustment of estimate functions constrained by differential and algebraic equalities and inequalities and thus requires efficient and novel solution techniques. 

In some problems the possible region of independent variables is defined by equality or inequality constraints. As you have seen in Section 1.2, such constrained extremum problems are easy to solve if both the constraints and the... 

In this approach, the process variables are partitioned into dependent variables and independent variables (optimisation variables). For each choice of the optimisation variables (sometimes referred to as decision variables in the literature) the simulator (model solver) is used to converge the process model equations (described by a set of ODEs or DAEs). Therefore, the method includes two levels. The first level performs the simulation to converge all the equality constraints and to satisfy the inequality constraints and the second level performs the optimisation. The resulting optimisation problem is thus an unconstrained nonlinear optimisation problem or a constrained optimisation problem with simple bounds for the associated optimisation variables plus any interior or terminal point constraints (e.g. the amount and purity of the product at the end of a cut). Figure 5.2 describes the solution strategy using the feasible path approach. 

Brannock, Verneuil and Wong (5 ) and Boston (55) considered equality and inequality constrained distillation calculations. 

Boston, J. F., "Algorithms for Distillation Calculations with Bounded-Variable Design Constraints and Equality-or Inequality-Constrained Optimization", Paper presented at Houston AIChE Meeting, April 1979. 

After the inequality constraints have been converted to equalities, the complete set of restrictions becomes a set of linear equations with n unknowns. The linear-programming problem then will involve, in general, maximizing or minimizing a linear objective function for which the variables must satisfy the set of simultaneous restrictive equations with the variables constrained to be nonnegative. Because there will be more unknowns in the set of simultaneous equations than there are equations, there will be a large number of possible solutions, and the final solution must be chosen from the set of possible solutions. 

Linear Programming The combined term linear programming is given to any method for finding where a given linear function of several variables takes on an extreme value, and what that value is, when the variables are nonnegative and are constrained by linear equalities or inequalities. A very general problem consists of maximiz-ing / = 2)/=i CjZj subject to the constraints Zj > 0 (j = 1, 2,. . . , n) and 2)"=i a Zj < b, (i = 1, 2,. . ., m). With S the set of all points whose coordinates Zj satisfy all the constraints, we must ask three questions (1) Are the constraints consistent If not, S is empty and there is no solution. (2) If S is not empty, does the function/become unbounded on S If so, the problem has no solution. If not, then there is a point B of S that is optimal in the sense that if Q is any point of S then/(Q) ifP)- (3) How can we find P ... 

The conditions yielding the unconstrained maximum centerline deposition rate give a deposition uniformity of only about 25%. While this may well be acceptable for some fiber coating processes, there are likely applications for which it is not. We now consider the problem of maximizing the centerline deposition rate, subject to an additional constraint that the deposition uniformity satisfies some minimum requirement. Assuming that the required uniformity is better than that obtained in the unconstrained case, the constrained maximum centerline deposition rate should occur when the uniformity constraint is just marginally satisfied. This permits replacing the inequality constraint of a minimum uniformity by an equality constraint that is satisfied exactly. 

In summary, condition 1 gives a set of n algebraic equations, and conditions 2 and 3 give a set of m constraint equations. The inequality constraints are converted to equalities using h slack variables. A total of M + m constraint equations are solved for n variables and m Lagrange multipliers that must satisfy the constraint qualification. Condition 4 determines the value of the h slack variables. This theorem gives an indirect problem in which a set of algebraic equations is solved for the optimum of a constrained optimization problem. 

This technique, combined with Fenske shortcut calculations for generating initial estimates of temperature profiles and stage liquid or vapor flow rates, is a robust method that can solve a large percentage of different types of separation processes. The algorithm also has provision for handling inequality specifications (Brannock et al., 1977). For each inequality specification, an alternate equality specification is required to ensure a unique solution. In this manner, the so-called over-constrained problems may be solved since inequality specifications are not subject to degrees of freedom restrictions. 

Linear Programming The combined term linear programming is given to any method for finding where a given linear function of several variables takes on an extreme value, and what that value is, when the variables are nonnegative and are constrained by linear equalities or inequalities. A very general problem consists of maximiz-... 

In an attempt to avoid the ill-conditioning that occurs in the regular pentilty tuid bturier function methods, Hestenes (1969) and PoweU (1969) independently developed a multiplier method for solving nonhnearly constrained problems. This multiplier method was originally developed for equality constraints and involves optimizing a sequence of unconstrained augmented Lagrtuigitui functions. It was later extended to handle inequality constraints by Rockafellar (1973). 

When solving an inequality-constrained optimal control problem numerically, it is impossible to determine which constraints are active. The reason is one cannot obtain a p, exactly equal to zero. This difficulty is surmounted by considering a constraint to be active if the corresponding p < a where a is a small positive number such as 10 or less, depending on the problem. Slack variables may be used to convert inequalities into equalities and utilize the Lagrange Multiplier Rule. 

This section deals with optimal control problems constrained by algebraic equalities and inequalities. 

The simplest optimization problems are those without equality constraints, inequality constraints, and lower and upper bounds. They are referred to as unconstrained optimization. Otherwise, if one or more constraints apply, the problem is one in constrained optimization. 

Constrained optimization is broached starting from Chapter 9. The constraints are split into three categories bounds, equality constraints, and inequality constraints. The relationship between primal and dual problems is discussed in further depth. 

As mentioned earlier, the developed algorithm employs dynopt to solve the intermediate problems associated with the local interaction of the agents. Specifically, dynopt is a set of MATLAB functions that use the orthogonal collocation on finite elements method for the determination of optimal control trajectories. As inputs, this toolbox requires the dynamic process model, the objective function to be minimized, and the set of equality and inequality constraints. The dynamic model here is described by the set of ordinary differential equations and differential algebraic equations that represent the fermentation process model. For the purpose of optimization, the MATLAB Optimization Toolbox, particularly the constrained nonlinear rninimization routine fmincon [29], is employed. 

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