Inequality bound constraints

Repeating the previous development for this problem, Newton s method applied to the KTC yields a mixed system of equations and inequalities for the Newton step (Ax, AX). This system is the KTC for the QP in (8.69)-(8.70) with the additional bound constraints... 

A second important difference in the Attic method is the presence of artificial inequality constraints these are constraints on a single variable and, when they are present in the matrix J, they must be treated differently to the active bound constraints since they are not real constraints. 

A third difference is the exploitation of variables that appear only in one equality or inequality constraint, aside from their bound constraints. If such variables lie on an artificial constraint (that is not an active bound), they allow the equality or... 

In the matrix J, only the equality constraints that do not contain variables that appear only in that equation and that are not on a bound constraint, and some (or none of the) active inequality constraints selected from the satisfied inequality constraints can be included. 

The simplest device that requires no factorizations, involves checking whether some variables that are not included in the equality constraints have the same sign in the objective function and in all the satisfied inequality constraints (including the lower and upper bound constraints). This alternative is always adopted in the initial... 

The third problem is knowing which constraints must be kept active at a certain iteration. In line with the philosophy of the Attic method, only the equality constraints are always considered active. The inequality and bound constraints are gradually added one by one until saturation is achieved (no other constraints can be added). If the constraints inserted fulfill the KKT conditions, the solution is achieved otherwise, the inequality constraints or bounds with 1 < 0 are removed. 

It is possible to vwite a similar function even when inequality constraints are present. Below, we vdll consider only bound constraints on the variables, in addition to equality constraints. Whenever inequality constraints are included in the problem, it is sufficient to introduce an auxiliary variable for each inequality constraint to transform them into equality constraints, plus a nonnegativity constraint on the new additional variables. 

Small-medium problems with bound, equality, and inequality linear constraints. The matrices E and D are dense (BzzMatrix) ... 

Small-medium problems with bound, equality, and inequality linear constraints and nonlinear equality constraints. nH is the number of nonlinear equality constraints and HName is the name of the function where the nonlinear equality constraints are calculated. The matrices E and D are dense... 

Large sparse problems with bound, equality and inequality linear constraints, and nonlinear equality constraints. G is a BzzMatrixSparseSymmetricLocked class object where the structure of the Hessian of the function (13.1) is provided. nH is the number of nonlinear equality constraints, HName is the name of the function where the nonlinear equality constraints are calculated, H is the name of the BzzMatrixSparseLocked matrix with the structure of the nonlinear equality constraints and nlH is a BzzVectorInt whose elements indicate which variables are really nonlinear in the system of nonlinear equality constraints. The matrices E and D are sparse (BzzMatrixSparseLocked) ... 

Quantitative inequality constraints, requiring that the values of temperatures, pressures, flowrates, concentrations, and material or energy accumulations remain within a range defined by lower and upper bounds, in order to maintain the safety of personnel and... 

Now consider the influence of the inequality constraints on the optimization problem. The effect of inequality constraints is to reduce the size of the solution space that must be searched. However, the way in which the constraints bound the feasible region is important. Figure 3.10 illustrates the concept of convex and nonconvex regions. 

As the size of a polymerization batch is normalized (the size of a batch is 1 ) and the initial product amounts are defined to be zero (see below), the number of polymerizations N is a nonbinding upper bound for the product amounts. The first inequality constraint forces p to zero if the difference Znm — or if... 

Constraints in optimization arise because a process must describe the physical bounds on the variables, empirical relations, and physical laws that apply to a specific problem, as mentioned in Section 1.4. How to develop models that take into account these constraints is the main focus of this chapter. Mathematical models are employed in all areas of science, engineering, and business to solve problems, design equipment, interpret data, and communicate information. Eykhoff (1974) defined a mathematical model as a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in a usable form. For the purpose of optimization, we shall be concerned with developing quantitative expressions that will enable us to use mathematics and computer calculations to extract useful information. To optimize a process models may need to be developed for the objective function/, equality constraints g, and inequality constraints h. 

We can state these ideas precisely as follows. Consider any optimization problem with n variables, let x be any feasible point, and let act(x) be the number of active constraints at x. Recall that a constraint is active at x if it holds as an equality there. Hence equality constraints are active at any feasible point, but an inequality constraint may be active or inactive. Remember to include simple upper or lower bounds on the variables when counting active constraints. We define the number of degrees of freedom (dof) at x as... 

Formulate the preceding problem as a linear programming problem. How many variables are there How many inequality constraints How many equality constraints How many bounds on the variables ... 

The vector x can contain slack variables, so the equality constraints (8.33) may contain some constraints that were originally inequalities but have been converted to equalities by inserting slacks. Codes for quadratic programming allow arbitrary upper and lower bounds on x we assume x>0 only for simplicity. 

Now examine how GRG proceeds when some of the constraints are inequalities and there are bounds on some or all of the variables. Consider the following problem ... 

GRG converts inequality constraints to equalities by introducing slack variables. If s is the slack in this case, the inequality x - y > 0 becomes x — y — s = 0. We must also add the bound for the slack, ssO, giving the new problem ... 

Let the starting point be (1, 0), at which the objective value is 6.5 and the inequality is satisfied strictly, that is, its slack is positive (s = 1). At this point the bounds are also all satisfied, although y is at its lower bound. Because all of the constraints (except for bounds) are inactive at the starting point, there are no equalities that must be solved for values of dependent variables. Hence we proceed to minimize the objective subject only to the bounds on the nonbasic variables x and y. There are no basic variables. The reduced problem is simply the original problem ignoring the inequality constraint. In solving this reduced problem, we do keep track of the inequality. If it becomes active or violated, then the reduced problem changes. 

We can show the various subproblems developed from the stated problem by a tree (Figure E9.1). The objective function and inequality constraints are the same for each subproblem and so are not shown. The upper bound and lower bound for / are represented by ub and lb, respectively. 

The logical condition, called a disjunction, means that exactly one of the three sets of conditions in brackets must be true the logical variable must be true, the constraint must be satisfied, and c must have the specified value. Note that c appears in the objective function. There are additional constraints on x here these are simple bounds, but in general they can be linear or nonlinear inequalities. The single inequality constraint in each bracket may be replaced by several different inequalities. There... 

The inequality constraints. Various kinds of inequality constraints exist, such as requiring that all of the xiJk yiJc, Qk, Fk, Wk, and so on be positive, that upper and lower bounds be imposed on some of the product stream concentrations, and specification of the minimum recovery factors. A recovery factor for stage k is the ratio... 

As explained in Chapter 9, a branch-and-bound enumeration is nothing more than a search organized so that certain portions of the possible solution set are deleted from consideration. A tree is formed of nodes and branches (arcs). Each branch in the tree represents an added or modified inequality constraint to the problem defined for the prior node. Each node of the tree itself represents a nonlinear optimization problem without integer variables. 

Thus, the model has eight inequality constraints in addition to the three equality constraints and the upper and lower bounds on all of the variables. 

The uniformity inequality constraints [Equations (/)-(/)] were again included in the problem. Additionally, the bounds on the variables were... 

Process simulators contain the model of the process and thus contain the bulk of the constraints in an optimization problem. The equality constraints ( hard constraints ) include all the mathematical relations that constitute the material and energy balances, the rate equations, the phase relations, the controls, connecting variables, and methods of computing the physical properties used in any of the relations in the model. The inequality constraints ( soft constraints ) include material flow limits maximum heat exchanger areas pressure, temperature, and concentration upper and lower bounds environmental stipulations vessel hold-ups safety constraints and so on. A module is a model of an individual element in a flowsheet (e.g., a reactor) that can be coded, analyzed, debugged, and interpreted by itself. Examine Figure 15.3a and b. 

In summary, the problem consists of 34 bounded variables (both upper bound and lower bounds) associated with the process, 12 linear equality constraints, 18 nonlinear equality constraints, and 3 linear inequality 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. 

Natural equality constraints exist in many real systems. For example, consider a chemical reaction in which a binary mixed solvent is to be used (see Figure 2.15). We might specify two continuous factors, the amount of one solvent (represented by X,) and the amount of the other solvent ( 2). These are clearly continuous factors and each has only a natural lower bound. However, each of these factors probably should have an externally imposed upper bound, simply to avoid adding more total solvent than the reaction vessel can hold. If the reaction vessel is to contain 10 liters, we might specify the inequality constraints... 

Give definitions for the following maximum, minimum, optimum, unimodal, multimodal, local optimum, global optimum, continuous, discrete, constraint, equality constraint, inequality constraint, lower bound, upper bound, natural constraint, artificial constraint, degree of freedom, feasible region, nonfeasible region, factor tolerance. 

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