Friday, October 25, 2013

Some types of mathematical problems

Analytic optimization
Analytic optimization problems are defined by the optimization of a function over a set of real vectors to the set of real numbers $\mathbb{R}^n \to \mathbb{R}$. The most generally applicable type of analytic optimization is linear programming so it is worth starting out with that and then continuing on to describe the nonlinear cases afterwards.

Linear programming:
Real linear programming problems can be solved using the simplex method, however, most integer programming problems don't have exact solutions so they require the use of heuristics. A variety of combinatorial problems including TSP, covering problems, and boolean satisfiability can be solved using integer linear programming.

Nonlinear programming:
Given some general multivariable real function we can use the combination of partial derivatives and equation solving to solve unconstrained optimization problems. Otherwise, there are special kinds of nonlinear programming such as fractional programming and quadratic programming that have their own sorts of solutions.

Goal oriented optimization:
In general goal based problems include a discrete temporal model and a predicate that determines rather or not the current state is the goal state. In planning problems the problem is generally to find the shortest sequence of moves to reach the goal state and in turn selection problems the problem is to find a move that maximizes the frequency of wins and minimizes the frequency of loses in the game tree.

Planning problems:
The purpose of planning problems is to find a sequence of moves to achieve the goal state that is optimal with respect to some ordering condition such as the size of the sequences. Planning problems include shortest path problems such as the problem of finding the shortest path in a maze and combination puzzles such as the 15-puzzle and the Rubik's cube puzzle.

Turn selection problems:
In any combinatorial game such as chess, checkers, go, connect four, or tictactoe players select moves each turn out of the set of valid moves based upon rather or not their move is going to achieve the end goal taking into account all possible future moves.

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