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The most widely used gap penalty function is the affine gap penalty. The affine gap penalty combines the components in both the constant and linear gap penalty, taking the form + (). This introduces new terms, A is known as the gap opening penalty, B the gap extension penalty and L the length of the gap.
From a dynamic programming point of view, Dijkstra's algorithm for the shortest path problem is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method.
Dynamic problem For an initial set of N numbers, dynamically maintain the maximal one when insertion and deletions are allowed. A well-known solution for this problem is using a self-balancing binary search tree. It takes space O(N), may be initially constructed in time O(N log N) and provides insertion, deletion and query times in O(log N).
Stochastic dynamic programming deals with problems in which the current period reward and/or the next period state are random, i.e. with multi-stage stochastic systems. The decision maker's goal is to maximise expected (discounted) reward over a given planning horizon.
The corresponding dynamic programming algorithm takes cubic time. The paper also points out that the recursion can accommodate arbitrary gap penalization formulas: A penalty factor, a number subtracted for every gap made, may be assessed as a barrier to allowing the gap. The penalty factor could be a function of the size and/or direction of the ...
The following is the skeleton of a generic branch and bound algorithm for minimizing an arbitrary objective function f. [3] To obtain an actual algorithm from this, one requires a bounding function bound, that computes lower bounds of f on nodes of the search tree, as well as a problem-specific branching rule.
In dynamic programming, a method of mathematical optimization, backward induction is used for solving the Bellman equation. [3] [4] In the related fields of automated planning and scheduling and automated theorem proving, the method is called backward search or backward chaining. In chess, it is called retrograde analysis.
Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute optimization) is an area of multiple-criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously.