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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]
A restraint algorithm is used to ensure that the distance between mass points is maintained. The general steps involved are: (i) choose novel unconstrained coordinates (internal coordinates), (ii) introduce explicit constraint forces, (iii) minimize constraint forces implicitly by the technique of Lagrange multipliers or projection methods.
Many constrained optimization methods are based on methods for unconstrained optimization that are adapted to deal with constraints. Such is, for example, the class of projected gradient methods wherein the unconstrained minimization inner step "leads" the process and a projection onto the whole constraints set (the feasible region) is ...
A penalty method replaces a constrained optimization problem by a series of unconstrained problems whose solutions ideally converge to the solution of the original constrained problem. The unconstrained problems are formed by adding a term, called a penalty function , to the objective function that consists of a penalty parameter multiplied by ...
The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function or Lagrangian. [2]
When the objective function is differentiable, sub-gradient methods for unconstrained problems use the same search direction as the method of steepest descent. Subgradient methods are slower than Newton's method when applied to minimize twice continuously differentiable convex functions.
Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: An optimization problem with discrete variables is known as a discrete optimization , in which an object such as an integer , permutation or graph must be found from a countable set .
In constrained least squares one solves a linear least squares problem with an additional constraint on the solution. [ 1 ] [ 2 ] This means, the unconstrained equation X β = y {\displaystyle \mathbf {X} {\boldsymbol {\beta }}=\mathbf {y} } must be fit as closely as possible (in the least squares sense) while ensuring that some other property ...