Lagrangian linear programming

The time consumer in calculating chemical equilibria by one of the linear or nonlinear programming methods is usually considerably longer than the time needed for methods based on the theory of Lagrangian multipliers For this reason, linear and non-linear programming are rather seldom employed in calculating chemical equilibria. 

Successive quadratic programming (SQP) methods solve a sequence of quadratic programming approximations to a nonlinear programming problem. Quadratic programs (QPs) have a quadratic objective function and linear constraints, and there exist efficient procedures for solving them see Section 8.3. As in SLP, the linear constraints are linearizations of the actual constraints about the selected point. The objective is a quadratic approximation to the Lagrangian function, and the algorithm is simply Newton s method applied to the KTC of the problem. 

Murtagh, B. A. and Saunders, M. A. (1982) Mathematical Programming Study 16, 84. A projected Lagrangian algorithm and its implementation for sparse non-linear constraints. 

Successive quadratic programming solves a sequence of quadratic programming problems. A quadratic programming problem has a quadratic economic model and linear constraints. To solve this problem, the Lagrangian function is formed from the quadratic economic model and linear constraints. Then, the Kuhn-Tucker conditions are applied to the Lagrangian function to obtain a set of linear equations. This set of linear equations can then be solved by the simplex method for the optimum of the quadratic programming problem. 

Problem Type Large-scale linear and nonlinear programs Method Projected Lagrangian... 

Algorithms for the solution of quadratic programs, such as the Wolfe (1959) algorithm, are very reliable and readily available. Hence, these have been used in preference to the implementation of the Newton-Raphson method. For each iteration, the quadratic objective function is minimized subject to linearized equality and inequality constraints. Clearly, the most computationally expensive step in carrying out an iteration is in the evaluation of the Lapla-cian of the Lagrangian, V xL x , X which is also the Hessian matrix of the La-grangian that is, the matrix of second derivatives with respect to X . 

A review of the theoretical basis, finite-element model, and sample applications of the program NOVA are presented. The updated incremental Lagrangian formulation is used to account for geometric nonlinearity (i.e., small strains and moderately large rotations), the nonlinear viscoelastic model of Schapery is used to account for the nonlinear constitutive behavior of the adhesive, and the nonlinear Fickean diffusion model in which the diffusion coefficient is assumed to depend on the temperature, penetrant concentration, and dilational strain is used. Several geometrically nonlinear, linear and nonlinear viscoelastic and moisture... 

The methods may be classified in two groups. The first group of methods is based on the classical mathematical theory of Lagrangian multipliers. The second includes methods in which use is made of the theory of linear or convex programming. 

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