Solution of illconditioned problems in non linear programming using lagranges multiplier method

Abstract

ABSTRACT newlineThe Lagrangian Multipliers are those variables which aid in constructing a Lagrange function to investigate problems on conditional extrema. In mathematical optimization, concept of Lagrangian Multipliers is used to find local maxima (or minima) of a function subject to equality constraints. The origin and genesis of Lagrange multiplier along with its advancements in mathematical optimization for solving various nonlinear problems are assessed in this thesis. The Hypothesis of Penalty and Barrier strategies is straight forward. Within the penalty method we have to pay fine for violating the constraints and get inexact solution of our unique issue by adjusting the objective function and a penalty term including the constraints. By expanding the penalty, the surmised arrangement is constrained to approach the feasible. For the barrier method, we have to move around within the interior of the feasible domain and each time we try to approach the boundary. The forces makes on halt, near to the exact solution, in case this happens to be at the boundary as we know, typically the case for compelled issues. By debilitating the barrier steadily, we get approximate solutions which ideally converge to the precise arrangement space, the arrangement of the initial constrained problem. Here we also discuss Exact and Approximation method which further divided into Primal and Dual augmented Lagrangian method. To elaborate primal and dual augmented method we discuss Proximal minimization method and Rockafeller method. In the last section we explain Forward Backward splitting and alternating direction method of multipliers method. newline newline