Kuhn-Tucker multipliers

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... 

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... 

Since there is one control variable, by Theorem 3 feasibility measure is determined by two active constraints. Since for each pair of active constraints the corresponding Kuhn-Tucker multipliers Am must be nonnegative [Eq. (35e)], each pair of active constraint functions must have gradients of opposite sign [Eq. (35b)]. Thus the potential sets of active constraints are... 

Vector of uncertain variables (supply temperatures, flow rates, and/or heat transfer coefficients) Hyperrectangular uncertainty range A Kuhn-Tucker multiplier (35) f Kuhn-Tucker slack variables (35) aL Load surplus, kW (26) crT surplus, K (25)... 

Aia is the vector of Kuhn-Tucker multipliers for the inactive design constraints gjA(x,m) < 0. 

The set points of the released constraints c p and m (if available) are the only free variables to adjust at each optimization step. Whenever new constraints become active at a search step, constraint sets are modified to define the regulatory and the servo loops. If there remains no search direction at a search point due to encountering new constraints, Kuhn-Tucker multipliers are reevaluated at that point. If they are all negative, an apparent minimal is found and the search is terminated otherwise, it is continued with the new search direction selected. 

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). 

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