Constrained optimization Lagrange multiplier method

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... 

We solve the nonlinear formulation of the semidefinite program by the augmented Lagrange multiplier method for constrained nonlinear optimization [28, 29]. Consider the augmented Lagrangian function... 

Bertsekas DP (1982) Constrained optimization and Lagrange multiplier methods. Academic Press, New York... 

Koga and Morokuma treated the MEXP search as a constrained optimization. In their Lagrange multiplier method, the energy of the fth state, E (R), is minimized with a constraint that the energy difference between two states, / and /, becomes zero. The Lagrangian is written 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. 

A similar constrained optimization problem has been solved in Section 2.5.4 by the method of Lagrange multipliers. Using the same method we look for the stationary point of the Lagrange function... 

Such constrained optimizations are conveniently carried out by Lagrange s method of undetermined multipliers. Introducing one Lagrange multiplier for each constraint in equation (33), we arrive at the SCF Lagrangian ... 

The important aspect of (13.70b) is that each pa=Pa(U, V, N) has maximal ( most probable ) character with respect to the natural control variables of S. The constrained maximization procedure to find this optimal distribution by the method of Lagrange undetermined multipliers [see Schrodinger (1949), Sidebar 13.4, for further details] is very similar to that described in Section 5.2. In particular, the pa must be maximal with respect to variations in each control variable, leading to the usual second-derivative curvature conditions such as... 

Newton and Leibnitz. The foundations of calculus of variations were laid by Bernoulli, Euler, Lagrange and Weierstrass. The optimization of constrained problems, which involves the addition of unknown multipliers, became known by the name of its inventor Lagrange. Cauchy made the first application of the steepest descent method to solve unconstrained minimization problems. In spite of these early contributions, very little progress was made until the middle of the 20th century, when high-speed digital computers made the implementation of the optimization procedures possible and stimulated further research in new methods. 

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