Mathematical optimization

Mathematical optimization (also knwon as optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives (domain).

In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, optimization includes finding “best available” values of some objective function given a defined domain (or a set of constraints), including a variety of different types of objective functions and different types of domains.

The function which gives us the criteria of which element is the best is usually called as objective function, loss function or cost function (minimization), a utility function or fitness function (maximization), or, in certain fields, an energy function or energy functional. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal solution.

The local optimal is the optimal solution of the optimization problem for a local neighborhood arbitrarily defined.

See also

Computational intelligence, Mathematical optimization, Machine learning, Artificial Intelligence

Material

• http://www.pyomo.org/

Books

• Bradley, Hax, and Magnanti (1977). Applied Mathematical Programming. Addison-Wesley
• Diwekar, Urmila (2008). Introduction to Applied Optimization. Springer
• Hromkovic, Juraj (2010). Algorithmics for Hard Problems: Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics. Springer