Linear equality constraints

The point z is tested to see if it could be a minimum point. It is necessary that F be stationary for all infinitesimal moves for z that satisfy the equality constraints. Linearize the m equality constraints around z,... 

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 ... 

From the above species constraints (Equations 17.4i to 17.4iii), we also notice that we have four unknown variables, and that the constraints provide us with only three equations we therefore have one degree of freedom in our process. This allows us to evaluate various options for the process. From the above equality constraints (Equations 17.4i to 17.4iii), we also note that the amount of water is fixed simply by the species balance, and that these species (constraints) relationships are linear. 

With 12 variables and 9 independent linear equality constraints, 3 degrees of freedom exist that can be used to maximize profits. Note that we could have added an overall material balance, xn + xl2 + 7 = 8 + x9 + 10, but this would be a redundant equation since it can be derived by adding the material balances. 

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 ... 

Chapter 1 presents some examples of the constraints that occur in optimization problems. Constraints are classified as being inequality constraints or equality constraints, and as linear or nonlinear. Chapter 7 described the simplex method for solving problems with linear objective functions subject to linear constraints. This chapter treats more difficult problems involving minimization (or maximization) of a nonlinear objective function subject to linear or nonlinear constraints ... 

One method of handling just one or two linear or nonlinear equality constraints is to solve explicitly for one variable and eliminate that variable from the problem formulation. This is done by direct substitution in the objective function and constraint equations in the problem. In many problems elimination of a single equality constraint is often superior to an approach in which the constraint is retained and some constrained optimization procedure is executed. For example, suppose you want to minimize the following objective function that is subject to a single equality constraint... 

Problem (8.15) must satisfy certain conditions, called constraint qualifications, in order for Equations (8.17)-(8.18) to be applicable. One constraint qualification (see Luenberger, 1984) is that the gradients of the equality constraints, evaluated at x, should be linearly independent. Now we can state formally the first order necessary conditions. 

The KTC comprise both the necessary and sufficient conditions for optimality for smooth convex problems. In the problem (8.25)-(8.26), if the objective fix) and inequality constraint functions gj are convex, and the equality constraint functions hj are linear, then the feasible region of the problem is convex, and any local minimum is a global minimum. Further, if x is a feasible solution, if all the problem functions have continuous first derivatives at x, and if the gradients of the active constraints at x are independent, then x is optimal if and only if the KTC are satisfied at x. ... 

Many real problems do not satisfy these convexity assumptions. In chemical engineering applications, equality constraints often consist of input-output relations of process units that are often nonlinear. Convexity of the feasible region can only be guaranteed if these constraints are all linear. Also, it is often difficult to tell if an inequality constraint or objective function is convex or not. Hence it is often uncertain if a point satisfying the KTC is a local or global optimum, or even a saddle point. For problems with a few variables we can sometimes find all KTC solutions analytically and pick the one with the best objective function value. Otherwise, most numerical algorithms terminate when the KTC are satisfied to within some tolerance. The user usually specifies two separate tolerances a feasibility tolerance Sjr and an optimality tolerance s0. A point x is feasible to within if... 

For problems with only equality constraints, we could simply solve the linear equations (8.66)-(8.67) for (Ax, AX) and iterate. To accommodate both equalities and inequalities, an alternative viewpoint is useful. Consider the quadratic programming problem... 

Circular objective contours and the linear equality constraint for the GRG example. 

The geometry of this problem is shown in Figure 8.11. The linear equality constraint is a straight line, and the contours of constant objective function values are circles centered at the origin. From a geometric point of view, the problem is to find the point on the line that is closest to the origin at x = 0, y = 0. The solution to the problem is at x = 2, y = 2, where the objective function value is 8. 

Each component of the vector of equality constraint functions h(x) is linear. 

Equality constraints. The equality constraints (30 in all) are the linear and nonlinear material and energy balances and the phase relations. 

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. 

Figure 3.12) and is the substance of linear programming. Let us take a simple example in n = 2 dimensions and assume that we are looking for the minimum of a linear function /(x), where x is the vector [x,x2]T, with the equality constraint brx = q (b and q being known constants) and the inequality constraints x1 0,x2>0.ln geochemistry or thermodynamics, the equality constraints are typically conservation equations, while phase or end-member... 

Minimization subject to linear equality constraints chemical equilibrium composition in oas mixtures... 

The Lagrange multipliers in a constrained nonlinear optimization problem have a similar interpretation to the dual variables or shadow prices in linear programming. To provide such an interpretation, we will consider problem (3.3) with only equality constraints that is,... 

Let x be a global minimum of (3.6) at which the gradients of the equality constraints are linearly independent (i.e., x is a regular point). Perturbing the right-hand sides of the equality constraints, we have... 

Illustration 3.2.4 Consider the following convex quadratic problem subject to a linear equality constraint ... 

The objective function /( ) and the inequality constraint g(x) are convex since f(x) is separable quadratic (sum of quadratic terms, each of which is a linear function of xi, x2,X3, respectively) and g(x) is linear. The equality constraint h(x) is linear. The primal problem is also stable since v(0) is finite and the additional stability condition (Lipschitz continuity-like) is satisfied since f(x) is well behaved and the constraints are linear. Hence, the conditions of the strong duality theorem are satisfied. This is why... 

Note that the objective function is convex since it has linear and positive quadratic terms. The only nonlinearities come from the equality constraint. By introducing three new variables w1,w2)w3, and three equalities ... 

Remark 2 Under the aforementioned assumptions (i) and (ii), problem (6.13) satisfies property (P) of Geoffrion (1972), and hence the OA corresponds to a subclass of vl-GBD (see sections 6.3.5.1) Furthermore, as we have seen in section 6.3.5.2, assumptions (i) and (ii) make the assumption imposed in v2-GBD valid (see remark of section 6.3.5.2) and therefore the OA can be considered as equivalent to v2-GBD with separability in jc andy and linearity iny. Note though that the vl-GBD can handle nonlinear equality constraints. 

Remark 1 The nonlinear equalities h(x) = 0 and the set of linear equalities which are included in h(x) = 0, correspond to mass and energy balances and design equations for chemical process systems, and they can be large. Since the nonlinear equality constraints cannot be treated explicitly by the OA algorithm, some of the possible alternatives would be to perform ... 

Alternative (ii) involves the numerical elimination of the nonlinear equality constraints at each iteration of the OA algorithm through their linearizations. Note though that these linearizations may cause computational difficulties since they may result in singularities depending on the selection of decision variables. In addition to the potential problem of singularities, the numerical elimination of the nonlinear equality constraints may result in an increase of the nonzero elements, and hence loss of sparsity, as shown by Kocis and Grossmann (1987). 

Remark 3 If linear equality constraints in exist in the MINLP formulation, then these are treated as a subset of the h(x) = 0 with the difference that we do not need to compute their corresponding T matrix but simply incorporate them as linear equality constraints in the relaxed master problem directly. 

Remark 5 We can also treat the heat loads of each match as variables since they participate linearly in the energy balances. The penalty that we pay, however, is that the objective function no longer satisfies the property of convexity, and hence we will have two possible sources of nonconvexities the objective function and the bilinear equality constraints. 

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