Kuhn-Tucker

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... 

Sufficiency conditions to assure that a Kuhn-Tucker point is a local minimum point require one to prove that the objec tive function will increase for any feasible move away from such a point. To carry out such a test, one has to generate the matrix of second derivatives of the Lagrange func tion with respect to all the variables z evaluated at z. The test is seldom done, as it requires too much work. 

Now consider the imposition of inequality [g(x) < 0] and equality constraints 7i(x) = 0] in Fig. 3-55. Continuing the kinematic interpretation, the inequality constraints g(x) < 0 act as fences in the valley, and equality constraints h(x) = 0 act as "rails. Consider now a ball, constrained on a rail and within fences, to roll to its lowest point. This stationary point occurs when the normal forces exerted by the fences [- Vg(x )] and rails [- V/i(x )] on the ball are balanced by the force of gravity [— Vfix )]. This condition can be stated by the following Karush-Kuhn-Tucker (KKT) necessary conditions for constrained optimality ... 

The first-order necessary conditions for problems with inequality constraints are called the Kuhn-Tucker conditions (also called Karush-Kuhn-Tucker conditions). The idea of a cone aids the understanding of the Kuhn-Tucker conditions (KTC). A cone is a set of points R such that, if x is in R, Tx is also in R for X 0. A convex cone is a cone that is a convex set. An example of a convex cone in two dimensions is shown in Figure 8.2. In two and three dimensions, the definition of a convex cone coincides with the usual meaning of the word. 

The Kuhn-Tucker conditions are predicated on this fact At any local constrained optimum, no (small) allowable change in the problem variables can improve the value bf the objective function. To illustrate this statement, consider the nonlinear programming problem ... 

Relations (8.23) and (8.24) are the form in which the Kuhn-Tucker conditions are usually stated. 

The Kuhn-Tucker necessary conditions are satisfied at any local minimum or maximum and at saddle points. If (x, A, u ) is a Kuhn-Tucker point for the problem (8.25)-(8.26), and the second-order sufficiency conditions are satisfied at that point, optimality is guaranteed. The second order optimality conditions involve the matrix of second partial derivatives with respect to x (the Hessian matrix of the... 

The second order sufficiency conditions show that the first two of these three Kuhn-Tucker points are local minima, and the third is not. The Hessian matrix of the Lagrangian function is... 

Finally, we should mention that in addition to solving an optimization problem with the aid of a process simulator, you frequently need to find the sensitivity of the variables and functions at the optimal solution to changes in fixed parameters, such as thermodynamic, transport and kinetic coefficients, and changes in variables such as feed rates, and in costs and prices used in the objective function. Fiacco in 1976 showed how to develop the sensitivity relations based on the Kuhn-Tucker conditions (refer to Chapter 8). For optimization using equation-based simulators, the sensitivity coefficients such as (dhi/dxi) and (dxi/dxj) can be obtained directly from the equations in the process model. For optimization based on modular process simulators, refer to Section 15.3. In general, sensitivity analysis relies on linearization of functions, and the sensitivity coefficients may not be valid for large changes in parameters or variables from the optimal solution. 

Quadratic programming (QP) is a special problem including a product of two decision variables in the objective function e.g. maximization of turnover max p x with p and x both variable requiring a concave objective function and that can be solved if the so-called Kuhn-Tucker-Conditions are fulfilled, e g. by use of the Wolf algorithm (Dom-schke/DrexI 2004, p. 192)... 

These conditions are the well known Kuhn-Tucker (KT) conditions ... 

Necessary conditions for a local solution to (2) are given by the following Kuhn-Tucker conditions ... 

For these time periods, the ODEs and active algebraic constraints influence the state and control variables. For these active sets, we therefore need to be able to analyze and implicitly solve the DAE system. To represent the control profiles at the same level of approximation as for the state profiles, approximation and stability properties for DAE (rather than ODE) solvers must be considered. Moreover, the variational conditions for problem (16), with different active constraint sets over time, lead to a multizone set of DAE systems. Consequently, the analogous Kuhn-Tucker conditions from (27) must have stability and approximation properties capable of handling ail of these DAE systems. 

Under this assumption, (X,y) e S x R " must be an optimal solution that is, the solution that maximizes and minimizes the functions, respectively, for Eqs. (l)-(2) if and only if it satisfies the Karush-Kuhn-Tucker condition ... 

The basic idea of the active constraint strategy is to use the Kuhn-Tucker conditions to identify the potential sets of active constraints at the solution of NLP (4) for feasibility measure ip. Then resilience test problem (6) [or flexibility index problem (11)] is decomposed into a series of NLPs with a different set of constraints (a different potential set of active constraints) used in each NLP. 

The potential sets of active constraints are identified by applying the Kuhn-Tucker conditions to NLP (4) for feasibility measure tft (Grossmann and Floudas, 1985, 1987) ... 

Assuming that constraint functions fm, m G M, are all monotonic in z, the potential sets of active constraints can be determined from Kuhn-Tucker conditions (35b) and (35e) as follows (Grossmann and Floudas, 1987). If the constraint functions fm are monotonic in z, then every component of dfm/dz is one signed for all z for each possible value of 8. Since Am 0 must hold for each constraint m M [Eq. (35e)], Eq. (35b) identifies the different sets of n2 + 1 constraints which can satisfy the Kuhn-Tucker conditions (different potential sets MA of nz + 1 active constraints). 

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... 

Thus, from Kuhn-Tucker condition (35b), the potential sets of active constraints are identified as shown in Table VIII. For each potential set of active constraints MA(k), NLP (34 ) is formulated to determine trial resilience measure For example, for potential set MA(1) = (/i,/2), the following NLP is solved ... 

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)... 

Remark 1 Note that in the Kuhn-Tucker constraint qualification w(t) is a once-differentiable arc which starts at x. Then, the Kuhn-Tucker constraint qualification holds if z is tangent to w(r) in the constrained region. 

Remark 2 The linear independence constraint qualification as well as the Slater s imply the Kuhn-Tucker constraint qualification. 

Geometric Interpretation of Karush-Kuhn-Tucker Necessary Conditions From the gradient KKT conditions we have that... 

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