Lagrange multiplier linear constraints

By combining the Lagrange multiplier method with the highly efficient delocalized internal coordinates, a very powerfiil algoritlun for constrained optimization has been developed [ ]. Given that delocalized internal coordinates are potentially linear combinations of all possible primitive stretches, bends and torsions in the system, cf Z-matrix coordinates which are individual primitives, it would seem very difficult to impose any constraints at all however, as... 

Further Comments on General Programming.—This section will utilize ideas developed in linear programming. The use of Lagrange multipliers provides one method for solving constrained optimization problems in which the constraints are given as equalities. 

In addition to providing optimal x values, both simplex and barrier solvers provide values of dual variables or Lagrange multipliers for each constraint. We discuss Lagrange multipliers at some length in Chapter 8, and the conclusions reached there, valid for nonlinear problems, must hold for linear programs as well. In Chapter 8 we show that the dual variable for a constraint is equal to the derivative of the optimal objective value with respect to the constraint limit or right-hand side. We illustrate this with examples in Section 7.8. 

Let x be a local minimum or maximum for the problem (8.15), and assume that the constraint gradients Vhj(x ),j — 1,m, are linearly independent. Then there exists a vector of Lagrange multipliers A = (Af,..., A ) such that (x A ) satisfies the first-order necessary conditions (8.17)-(8.18). 

The preceding results may be stated in algebraic terms. For V/ to lie within the cone described earlier, it must be a nonnegative linear combination of the negative gradients of the binding constraints that is, there must exist Lagrange multipliers u such that... 

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,... 

The optimization problem in Eq. (5.146) is a standard situation in optimization, that is, minimization of a quadratic function with linear constraints and can be solved by applying Lagrangian theory. From this theory, it follows that the weight vector of the decision function is given by a linear combination of the training data and the Lagrange multiplier a by... 

The truncation of the series expansion (4) at second order allows the solution to be found by solving a linear system of equations. The only complication is the need to enforce the constraint (2), which can be taken care of with the method of Lagrange multipliers. In this context, the Lagrange multiplier can be interpreted as the chemical potential, and the solution to the constrained problem is the charge distribution and chemical potential which minimizes the free energy... 

A similar calculation and linear solve is needed to compute A +3/4 in order to maintain the hidden constraint for p +3/4, whereas the Lagrange multiplier A +i/2 is responsible for ensuring thatg( +i) = 0, and is typically found using an implicit solve. 

Each constraint equation, is assigned a Lagrange multiplier L,. For a holonomic linear sum... 

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... 

The next step of SMO is to compute the location of the constrained maximum of the objective function in the following equation while allowing only two Lagrange multipliers to change. Under normal circumstances (it 9 0), there will be a maximum along the direction of the linear equality constrain, and k will be less than zero. In this case, SMO computes the maximum along the direction of the constraint. 

These mixed differential-algebraic equations of motion for the robot were solved by a simple numerical algorithm. The secoi time derivatives of the constraint equations were appended to the differential equations of motion to form a linear set, which at any instant can be solved for the system acceleration and Lagrange multipliers. A direct integration method using the Cartesian coordinates and velocities as the state variable of integration, together with a forth-order Rui e-Kuta with an adjustable step size were employed. 

---