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  2. Penalty method - Wikipedia

    en.wikipedia.org/wiki/Penalty_method

    In the above equations, (()) is the exterior penalty function while is the penalty coefficient. When the penalty coefficient is 0, f p = f . In each iteration of the method, we increase the penalty coefficient p {\displaystyle p} (e.g. by a factor of 10), solve the unconstrained problem and use the solution as the initial guess for the next ...

  3. Meshfree methods - Wikipedia

    en.wikipedia.org/wiki/Meshfree_methods

    Several methods have been developed to impose the essential boundary conditions weakly, including Lagrange multipliers, Nitche's method, and the penalty method. As for quadrature , nodal integration is generally preferred which offers simplicity, efficiency, and keeps the meshfree method free of any mesh (as opposed to using Gauss quadrature ...

  4. Replica trick - Wikipedia

    en.wikipedia.org/wiki/Replica_trick

    The cavity method is an alternative method, often of simpler use than the replica method, for studying disordered mean-field problems. It has been devised to deal with models on locally tree-like graphs. Another alternative method is the supersymmetric method. The use of the supersymmetry method provides a mathematical rigorous alternative to ...

  5. Constrained optimization - Wikipedia

    en.wikipedia.org/wiki/Constrained_optimization

    Many constrained optimization algorithms can be adapted to the unconstrained case, often via the use of a penalty method. However, search steps taken by the unconstrained method may be unacceptable for the constrained problem, leading to a lack of convergence. This is referred to as the Maratos effect. [3]

  6. Lagrangian relaxation - Wikipedia

    en.wikipedia.org/wiki/Lagrangian_relaxation

    The penalty method does not use dual variables but rather removes the constraints and instead penalizes deviations from the constraint. The method is conceptually simple but usually augmented Lagrangian methods are preferred in practice since the penalty method suffers from ill-conditioning issues.

  7. Three-body problem - Wikipedia

    en.wikipedia.org/wiki/Three-body_problem

    In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three-body problem. [22] The free-fall formulation starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forward and backward along an open "track".

  8. Physics engine - Wikipedia

    en.wikipedia.org/wiki/Physics_engine

    Penalty methods, where interactions are commonly modelled as mass-spring systems. This type of engine is popular for deformable, or soft-body physics. Constraint based methods, where constraint equations are solved that estimate physical laws. Impulse based methods, where impulses are applied to object interactions. However, this is actually ...

  9. Drift plus penalty - Wikipedia

    en.wikipedia.org/wiki/Drift_plus_penalty

    The drift-plus-penalty method applies to queueing systems that operate in discrete time with time slots t in {0, 1, 2, ...}. First, a non-negative function L(t) is defined as a scalar measure of the state of all queues at time t.