Lagrange inequality constraints

Each of the inequality constraints gj(z) multiphed by what is called a Kuhn-Tucker multiplier is added to form the Lagrange function. The necessaiy conditions for optimality, called the Karush-Kuhn-Tucker conditions for inequality-constrained optimization problems, are... 

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. 

EXAMPLE 8.3 APPLICATION OF THE LAGRANGE MULTIPLIER METHOD WITH NONLINEAR INEQUALITY CONSTRAINTS... 

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,... 

For the geometrical interpretation of the dual problem, we will consider particular values for the Lagrange multipliers i, fi2 associated with the two inequality constraints (fa > 0, fa > 0), denoted as fa, fa. 

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... 

The solution of FP yk), which is convex, provides the Lagrange multipliers fix, fe1 for the inequality constraints. Then, the Lagrange function takes the form ... 

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 classical building blocks that we need are Newton s method (Newton 1687) for unconstrained optimization, Lagrange s method (Lagrange 1788) for optimization with equality constraints, and Fiacco and McCormick s barrier method (Fiacco and McCormick 1968) for optimization with inequality constraints. Let us review these. A good general reference is Bazarra and Shetty 1979. 

In this chapter, we introduce the concept of Lagrange multipliers. We show how the Lagrange Multiplier Rule and the John Multiplier Theorem help us handle the equality and inequality constraints in optimal control problems. 

In most problems, a Lagrange multiplier can be shown to be related to the rate of change of the optimal objective functional with respect to the constraint value. This is an important result, which will be utilized in developing the necessary conditions for optimal control problems having inequality constraints. 

Case 1 Here K u) = ko and the inequality constraint is said to be active. The augmented objective functional is M = J + pK where /it is a Lagrange multiplier. The Lagrange Multiplier Theorem yields... 

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... 

In the case of inequality constraints, beyond all conditions (9.29) and (9.30), it is necessary to impose the additional condition that all Lagrange multipliers of active constraints are nonnegative. 

The seminal idea on which active set methods are based is rather simple and is the same, albeit with several variants, as the one adopted in the Attic method as well as in the Simplex method for linear programming starting from a point where certain constraints are active (all the equality constraints and some inequality constraints), we search for the solution to this problem as if all the constraints are equalities. During the search, it, however, may be necessary to insert other inequality constraints that were previously passive and/or remove certain active inequality constraints as they are considered useless based on their Lagrange parameters. The procedure continues until KKT conditions are fulfilled. 

Linearisation of the nonlinear dynamic equations is performed at each trajectory point so the nonlinear interactions are taken into consideration. The computational technique can explicitly handle active set changes and hard bounds on all variables efficiently. Active set changes require the modification of the equation through the addition (if a bound or inequality constraint become active) or the removal (if a bound or inequality ceases to be binding) of the respective constraints. Optimality is ensured by inspection of the sign of the Lagrange multipliers associated with the active inequalities at every continuation point. The solution technique is quite efficient because an approximation of the optimal solution path of Eq. (9) is sufficient for the purposes of the problem. 

The two Lagrange multipliers must fulfill all of the constraints of the full problem. The inequality constraints cause the Lagrange multipliers to lie in the box. The linear equality constraint causes them to lie on a... 

Se is tile set of equality constraints, Cm sX ) = 0 and Si is the set of inequality constraints, Cm<=Si x) > 0. If /LX is the vector of Lagrange multipliers that enforce the constraints, tiie constrained minimum satisfies... 

The basic idea in OA/ER is to relax the nonlinear equality constraints into inequalities and subsequently apply the OA algorithm. The relaxation of the nonlinear equalities is based upon the sign of the Lagrange multipliers associated with them when the primal (problem (6.21) with fixed y) is solved. If a multiplier A is positive then the corresponding nonlinear equality hi(x) = 0 is relaxed as hi x) <0. If a multiplier A, is negative, then the nonlinear equality is relaxed as -h (jc) < 0. If, however, A = 0, then the associated nonlinear equality constraint is written as 0 ht(x) = 0, which implies that we can eliminate from consideration this constraint. Having transformed the nonlinear equalities into inequalities, in the sequel we formulate the master problem based on the principles of the OA approach discussed in section 6.4. 

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. 

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