Slack variables inequality constraints

The foregoing inequality constraints must be converted to equality constraints before the operation begins, and this is done by introducing a slack variable q, for each. The several equations are then combined into a Lagrange function F, and this necessitates the introduction of a Lagrange multiplier, X, for each constraint. 

Note that all the constraints in Equation (7.4) are equalities. It is necessary to place the problem in this form to solve it most easily (equations are easier to work with here than inequalities). If the original system is not of this form, it may easily be transformed by use of so-called slack variables. If a given constraint is an inequality, for example,... 

The following figure shows the constraints. If slack variables jc3, x4 and x5 are added respectively to the inequality constraints, you can see from the diagram that the origin is not a feasible point, that is, you cannot start the simplex method by letting x x2 = 0 because then x3 = 20, x4 = -5, and x5 = -33, a violation of the assumption in linear programming that x > 0. What should you do to apply the simplex method to the problem other than start a phase I procedure of introducing artificial 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. 

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. 

A method of formatting the equations and inequality constraints. Slack variables are used to transform the inequality constraints into equality constraints. 

Compensating slack variables accounting for shortfall and/or surplus in production are introduced in the stochastic constraints with the following results (i) inequality constraints are replaced with equality constraints (ii) numerical feasibility of the stochastic constraints can be ensured for all events and (iii) penalties for feasibility violations can be added to the objective function. Since a probability can be assigned to each realization of the stochastic parameter vector (i.e., to each scenario), the probability of feasible operation can be measured. In this... 

As discussed, a constraint with is transformed into an equality by adding a (nonnegative) slack variable to its left-hand side. The same can be done in an inequality with >, this time by substracting a (nonnegative) slack variable from its left-hand side. 

With inequality constraints of the form < only, the columns corresponding to the slack variables can be used as a starting basis. This does not work for the generalized problem, and we must proceed in two phases. 

This minimax problem can be converted to a simpler nonlinear program (NLP) by introducing a slack variable /3 to measure violations of the inequality constraints ... 

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... 

The SIMPLEX method introduces slack and surplus variables to transform the inequality constraints into equalities. For example, if... 

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. 

The computational method to determine the optimum, the simplex method, requires equality constraints. The inequalities are converted to equalities by introducing slack and surplus variables. This is illustrated by converting the inequality equation given here to an equality by adding a slack variable V3. 

A basic solution of the constraint equations is a solution obtained by adding slack and surplus variables to the inequality constraints, then by setting (n — m) the variables equal to zero, and solving the constraint set for the remaining m variables. From this set of basic solutions, the groups of solutions where the... 

Select an initial basic feasible solution. If all the constraint equations were inequalities of the dess than or equal to form, the slack variables can be used as the initially feasible basis. 

General inequality constraints can be converted to equations by adding nonnegative slack (or surplus) variables. So the only essential inequalities are the nonnegativity conditions x > 0. The idea of the barrier function approach is to start from a point in the strict interior of the inequalities (x° > 0 for aU f) and construct a barrier that prevents any variable from reaching the boundary (x = 0). 

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. 

To obtain the stationarity conditions, the Lagrangian is formed and differentiated, with the details of this procedure described by McMillan (1970) and by Beveridge and Schecter (1970). The development begins by converting the inequality constraints to equality constraints through the addition of slack variables, zj,i= 1,...,, ..., ..1, such that constraints (18.4) become... 

Note that V L = 0 gives Eqs. (18.21), which are the definitions of the slack variables and need not be expressed in the KKT conditions. Note also that L = 2A.jZ, = 0, and, using Eqs. (18.21), Eqs. (18.26) result. These are the so-called complementary slackness equations. For constraint i, either the residual of the constraint is zero, g, = 0, or the Kuhn-Tucker multiplier is zero, X., = 0, or both are zero that is, when the constraint is inactive (gj > 0), the Kuhn-Tucker multiplier is zero, and when the Kuhn-Tucker multiplier is greater than zero, the constraint must be active (g, = 0). Stated differently, there is slackness in either the constraint or the Kuhn-Tucker multiplier. Finally, it is noted that V c x is the Jacobian matrix of the equality constraints, J x, and V g i is the Jacobian matrix of the inequality constraints, K[x). 

The variables added to transform an inequality constraint into an equality constraint are generally called Slack Variables. 

As demonstrated in the previous section, if any inequality constraints were present, it might be sufficient to introduce a new slack variable for each of them and a lower bound for normegativity. The upper bound will have an adequately large value. Thus, the presence of inequality constraints enters into the formulation case of the problem (12.35) and (12.36). 

The generalized reduced gradient (GRG) transforms inequality constraints into equality constraints by introducing slack variables [24]. Hence all the constraints in (P) are of equality form and can be represented as follows ... 

To find Xmin, we again use the augmented Lagrangian method, writing each inequality constraint hj x) > 0 in an equivalent form similar to an equality constraint by introducing a slack variable Sj,... 

The augmented Lagrangian method is not the only approach to solving constrained optimization problems, yet a complete discussion of this subject is beyond the scope of this text. We briefly consider a popular, and efficient, class of methods, as it is used by fmincon, sequential quadratic programming (SQP). We wUl find it useful to introduce a common notation for the equality and inequality constraints using slack variables. 

Introduce slack and surplus variables to convert inequalities to equalities and adjust the constraint equations to have positive right-hand sides. 

In the generalized reduced gradient method, the independent variables are separated into basic and nonbasic ones. There are m basic variables Xb, and (n - m) nonbasic variables x b from Eqs. (2) and (3) with the inequalities converted to equalities using slack and surplus variables. In theory, the m constraint equations could be solved for the m basic variables in terms of the n - m) nonbasic variables, i.e.. 

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